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\newtheorem{hhh}{Definition (2.8)} \renewcommand{\thehhh}{}
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\newtheorem{ded}{Dedication} \renewcommand{\theded}{}

\begin{document}
\pagenumbering{Roman}
\title[Topological Entropy for AA C$^{*}$-Algebras]{Topological Entropy for 
Appropriately Approximated C$^{*}$-Algebras}
\author{Thomas Hudetz}
\address{Institute for Theoretical Physics\\Vienna University 
\\Boltzmanngasse 5\\A-1090 Wien, Austria}
\curraddr{Department of Mathematics\\University of California at Berkeley\\
CA 94720, USA (until September 1994)}
\email{hudetz@@math.berkeley.edu (until September 1994)}
\thanks{Supported by {\em Fonds zur F\"orderung der wissenschaftlichen
Forschung in \"Osterreich} as Erwin Schr\"odinger Fellow (J0852-Phy)}
\dedicatory{To the memory of Alfred Wehrl: teacher, colleague, friend}
\maketitle
\begin{abstract}
The ``classical'' topological entropy is one of the main numerical invariants
in topological dynamics on compact spaces. Here, the author's recent
development of a non--commutative generalization of topological entropy, in 
the natural setting of general C$^{*}$--algebras as the non--commutative
counterpart of continuous function algebras on compact spaces, is presented
in a slightly modified and improved form. This includes both a survey of
earlier results with some important corrections, and also new general results
in response to (and inspired by) a more recent counter--proposal for a
non--commutative topological entropy by K. Thomsen. Finally, some partially
new examples for the calculation of the defined topological entropy are
shown. The rather self--evident physical interpretation in the framework of
(operator--algebraic) quantum statistical mechanics and of ``chaotic'' quantum
dynamical systems is briefly touched upon.

\bigskip
PACS numbers: 02.30.Sa, 02.40.--k, 05.30.--d, 05.45.+b                         
\end{abstract}
\vfill
\newpage
\setcounter{page}{1}
\pagenumbering{arabic}
\section{Introduction}
The notion of topological entropy had been introduced in topological dynamics
by Adler, Konheim and McAndrew \cite{akm} in 1965, first by purely formal
analogy with the Kolmogorov--Sinai (KS) entropy of measure--theoretic ergodic
theory that had been created by the two named mathematicians about ten years
earlier. Since then, however, on the one hand the topological entropy has
become one of the main numerical invariants in topological dynamics, and until
quite recently still, it has been more and more successfully applied to
deterministically chaotic {\em classical} physical systems, see in particular
\cite{kk}.

On the other hand, the KS entropy (among other measure--theoretic 
entropy--like quantities) has been more and more successfully generalized to
``quantum ergodic theory'' in the framework of the operator--algebraic 
approach to quantum statistical mechanics (cf.\ for example \cite{br,thirr}).
We can already now refer the reader to at least two recent books on these
quantum generalizations of measure--theoretic (dynamical) entropy: From the
more mathematical point of view, the book by Petz and Ohya \cite{op} is a
complete introduction to the subject of quantum entropy theory, whereas the
book by Benatti \cite{fabio} concentrates on the quantum ergodic theory 
aspects, more from the point of view of mathematical physics. In particular,
the work by Connes \cite{connes} and Connes, Narnhofer and Thirring \cite{cnt}
(CNT for short, and also subsequent work) reviewed in both books has been a
breakthrough in the non--commutative generalization of the classical
measure--theoretic KS entropy; see also \cite{stoe} for a short overview of
these recent mathematical developments.

This latter CNT theory of dynamical (state--dependent, as 
``measure--theoretic'') entropy for non--commutative C$^{*}$--algebras (or
von Neumann algebras) was the star\-ting point for the work on ``quantum
topological entropy'' presented here. As is well known (cf.\ for example
\cite{br,tak}), {\em commutative} C$^{*}$--algebras with unit correspond
exactly to the compact Hausdorff spaces, and in this sense parts of the theory
of non--commutative C$^{*}$--algebras (of bounded operators on Hilbert space)
can be considered as ``quantized topology'' (cf.\ \cite{eff}).

As to the physical interpretation of the theory in the following sections II
and III, within the framework of operator--algebraic quantum statistical
mechanics \cite{br,thirr}, we have to refer the reader to the still very
condensed section (1.2) of the (unpublished, and German) Thesis \cite{h2},
which section had to be omitted in the published form of the preliminary
short version \cite{h1} because of lack of space there (as it is actually also
the case here, for the lengthy discussion of the physical interpretation in
all its details). A very short account of this interpretation for this theory
first published in \cite{h1} can be found in \cite[p.\ 207]{fabio} (after a
whole section on the main parts of this theory in Benatti's book); and the
forthcoming publication \cite{h?} by the present author will contain this
interpretation in full detail. But for the experienced reader, the physical
interpretation will presumably be almost self--evident; even more so if we
now cite already here the two closely related notions of ``operations''
(\cite{kr}, used in a {\em certain} sense) respectively of ``instruments''
(see \cite{lin} and the references there). 

The mathematical prerequisites required from the reader are on the one hand
rather elementary set--theoretic topology as used in the classical theory of
topological entropy (see (1.1) and (1.2) below); and on the other hand a good
knowledge of the (mainly) linear algebra aspects of basic C$^{*}$--algebra
theory, for which all notions not explained or cited in the text here can be
easily found in \cite{br,thirr,tak} for example, or also in 
\cite{op,cnt,fabio} for special aspects of the C$^{*}$--entropy theory.

As basic references for the mathematical entropy theory in the classical case
we recommend for example \cite{me} particularly for the measure--theoretic
(KS) entropy theory (which is not necessary for the understanding of what
follows, but as secondary reading), and \cite{dgs} for the {\em topological}
entropy theory, our starting point for the non--commutative generalization in
the following two sections. For the convenience of the reader and to fix our 
notation for further use, we briefly recall the definitions and general
properties of the ``classical'' topological entropy as in the original paper
by Adler, Konheim and McAndrew \cite{akm}, which contains already most of the 
essential parts of the classical theory (in this approach).

\begin{aaaa}
Let $(X,T)$ be a topological dynamical system, given by a compact Hausdorff
space $X$ and a continuous selfmap $T:X\to X$. By definition, any open cover
of  $X$ possesses a {\em finite} subcover, and we can restrict ourselves to
the latter from the very beginning, denoting by $\cal{O}(X)$ the set (or 
class) of finite open covers of $X$.

For $\cal{U},\cal{V}\in\cal{O}(X)$ we define their join $\cal{U}\vee\cal{V}=
\{U\cap V|U\in\cal{U},\enspace V\in\cal{V}\}\in\cal{O}(X)$, and the action of
$T$ on $\cal{U}\in\cal{O}(X)$ by $T^{-1}(\cal{U})=\{T^{-1}(U)|U\in\cal{U}\}\in
\cal{O}(X)$ (as $T$ is onto).
\begin{enumerate}
\item[(i)] $N(\cal{U})=\min\{\card\cal{U}'|\cal{U}'\subset\cal{U}:\cal{U}'\in
\cal{O}(X)\}$ denotes the cardinality of a minimal subcover $\cal{U}'$ of
$\cal{U}\in\cal{O}(X)$. The ``topological'' {\em entropy} of $\cal{U}\in
\cal{O}(X)$ is defined by $H(\cal{U})=\log N(\cal{U})$. 
\item[(ii)] The entropy of $T$ with respect to $\cal{U}\in\cal{O}(X)$ is
defined as \\
$h(T,\cal{U})=\lim_{n\to\infty}\frac{1}{n}H(\cal{U}\vee T^{-1}\cal{U}\vee\dots
\vee T^{-n+1}\cal{U})$.
\item[(iii)] The {\em topological entropy} of $T$ is $h(T)=\sup_{\cal{U}\in
\cal{O}(X)}h(T,\cal{U})$.
\end{enumerate}
A cover $\cal{V}\in\cal{O}(X)$ is said to be a {\em refinement} of a cover
$\cal{U}\in\cal{O}(X)$ if $\forall V\in\cal{V}\enspace\exists U\in\cal{U}$
such that $V\subset U$, which gives a partial pre--order on $\cal{O}(X)$. 
\end{aaaa}
\begin{aaab} The entropy functionals (1.1,i--iii) have the following general
properties: $\forall \cal{U},\cal{V}\in\cal{O}(X)$,
\begin{enumerate}
\item[(i)] $N(\cal{U}\vee\cal{V})\leqslant N(\cal{U})\cdot N(\cal{V})$, or
$H(\cal{U}\vee\cal{V})\leqslant H(\cal{U})+H(\cal{V})$.
\item[(ii)] $H(T^{-1}\cal{U})\leqslant H(\cal{U})$, and so equality holds if
$T$ is a homeomorphism.
\item[(iii)] $h(T,\cal{U})\leqslant H(\cal{U})$.
\item[(iv)] If $\cal{V}$ is a refinement of $\cal{U}$, then $H(\cal{U})
\leqslant H(\cal{V})$ (the same for $N$), and $h(T,\cal{U})\leqslant
h(T,\cal{V})$. From the latter monotonicity it follows that if $(\cal{U}_{n}
)_{n\in\Bbb{N}}$ is a {\em cofinal} and {\em refining} sequence of open covers 
$\cal{U}_{n}\in\cal{O}(X)$ with respect to the natural partial order from
(1.1) above, then $h(T)=\lim_{n\to\infty}h(T,\cal{U}_{n})$. This is always
achievable in a {\em metric} space $(X,d)$, where each sequence $(\cal{V}_{k}
)_{k\in\Bbb{N}}$ with diameters $d(\cal{V}_{k})\equiv\max_{V\in\cal{V}_{k}}
d(V)$ shrinking to zero, $d(\cal{V}_{k})\to 0$, gives a {\em cofinal} and
then even {\em refining} sequence $(\cal{U}_{n})_{n\in\Bbb{N}}$ with $
\cal{U}_{n}=\bigvee_{k=1}^{n}\cal{V}_{k}$ by Lebesgue's covering lemma.
\item[(v)] $h(T^{k})=k\cdot h(T)\enspace\forall k\in\Bbb{N}$, and if $T$ is
a homeomorphism, then $h(T^{-1})=h(T)$.
\item[(vi)] For a {\em factor} system $(Y,S)$ of $(X,T)$, i.e.\ a continuous 
surjection $\Phi:X\to Y$ such that $\Phi\circ T=S\circ\Phi$, we have
$h_{Y}(S)\leqslant h_{X}(T)$ (where we add the subscripts for the spaces).
\item[(vii)] For a closed, $T$--invariant subset $Y\subseteq X$, i.e.\ such 
that $T(Y)\subseteq Y$, we have $h_{Y}(T\restriction_{Y})\leqslant h_{X}(T)$.
\item[(viii)] Let $\Phi:X\to X'$ be a homeomorphism onto a compact (Hausdorff)
space $X'$, then $h(T)=h(\Phi\circ T\circ\Phi^{-1})$, as follows from (vi).
\item[(ix)] Denote by $M(X,T)$ the (non--empty) convex compact set of $T
$--invariant Borel probability measures $\mu$ on $X$, i.e.\ such that $\mu
\circ T^{-1}=\mu$. Then $h(T)=\sup_{\mu\in M(X,T)}h_{\mu}(T)$, where the right
hand side is the measure--theoretic KS--entropy of $T$ with respect to $\mu$.
\end{enumerate}
\end{aaab}
See now \cite{h1,h2} for a very detailed motivation of the approach chosen
for the non--commutative generalization of this classical theory. But here
we plunge directly into the realm of non--Abelian C$^{*}$--algebras and will
return to the classical case not earlier than in (2.10) below. The paper is
organized in three main sections II-IV: In section II, the definitions and 
basic properties are given, together with the corrected proof of the main
Theorem (2.15). All other proofs are essentially left as simple exercises for
the reader, who can find some more details in the Thesis \cite{h2}. We also
review the earlier versions \cite{h1,h2} as compared to the modified and
improved theory presented here. In section III, we prove additional properties
of the defined topological entropy for C$^{*}$--algebras, all of which are the
direct analogues of the additional properties proved by K. Thomsen \cite{t} 
for his counter--proposal to the theory presented here. In order to prove the
analogous modest continuity properties as in \cite{t}, we have to impose a 
restriction on the original theory of section II, which is of independent
interest, though (particularly concerning the physical interpretation). After
that, we compare our (two) definitions with Thomsen's \cite{t} and add some
partly critical remarks concerning his theory. In section IV, we first
reproduce a slightly different example from \cite{t} with our definitions,
and then we proceed by reviewing the examples for shift automorphisms on AF
algebras from \cite{h1,h2,t}, now also for endomorphisms. Finally, we conclude
by some remarks concerning the non--commutative analogue of the classical 
so--called ``variational principle'' (1.2,ix) above (and some side--remarks).
\section{Definitions and basic properties}
Throughout this paper, $\cal{A}$ will denote a unital C$^{*}$--algebra with
unit $\1\in\cal{A}$.
\begin{aaa}We define the following elementary structures and notations:
\begin{enumerate}
\item[(i)] An {\em operator cover} for $\cal{A}$ is a finite subset $\alpha
\subset\cal{A}$ such that for $\alpha=\{A_{i}\in\cal{A}|i=1,\dots,n\}$, both
$\sum_{i=1}^{n}A_{i}^{*}A_{i}>0$ ({\em strictly} positive, i.e.\ invertible)
{\em and} $\sum_{i=1}^{n}A_{i}^{*}A_{i}\leqslant \1$, the latter condition 
being a suitable normalization. The set (or class) of all operator covers for 
$\cal{A}$ is denoted by $\cal{O}(\cal{A})$.
\item[(ii)] A {\em positive} operator cover for $\cal{A}$ is a finite set
$\alpha\subset\cal{A}^{+}$ such that with above notations $\sum_{i=1}^{n}A_{i}
>0$ and $\sum_{i=1}^{n}A_{i}\leqslant \1$; and $\cal{O}^{+}(\cal{A})$ denotes 
the collection of all these.
\item[(iii)] For two finite subsets $\alpha,\beta\subset\cal{A}$, we denote by
$\alpha\Vec\vee\beta$ their {\em ordered} operator product, where we choose
the order $\alpha\Vec\vee\beta=\{B\cdot A|\forall B\in\beta, A\in\alpha\}$.
\item[(iv)] For $\alpha\in\cal{O}(\cal{A})$, set ${\Bar N}(\alpha)=\min\{\card
\alpha'| \alpha'\subseteq\alpha,\enspace\alpha'\in\cal{O}(\cal{A})\}$.
\end{enumerate}
\end{aaa}
\begin{bbb}For the corresponding parts of (2.1), the following is important
to note:
\begin{enumerate}
\item[ad (i)] For $\alpha=\{A_{i}\in\cal{A}|i=1,\dots,n\}$ with $A\equiv
\sum_{i=1}^{n}A_{i}^{*}A_{i}>0$ but $A\nleqslant \1$, define $\Hat\alpha=
\{\Hat A_{i}\equiv A_{i}\cdot A^{-1/2}|i=1,\dots,n\}$, then obviously
$\Hat\alpha\in\cal{O}(\cal{A})$ and even $\sum_{i=1}^{n}\Hat A_{i}^{*}\Hat 
A_{i}=\1$, illustrating the mentioned normalization for $\cal{O}(\cal{A})$.
Let us denote by $\cal{O}_{1}(\cal{A})$ the set of all such $\Hat\alpha$ (with
the corresponding sum equal to $\1$).

On the other hand, for a general $\alpha\in\cal{O}(\cal{A})$ with elements
as above, the normalization condition in (i) implies that $\1-A\equiv B
\geqslant 0$, meaning that for $\alpha\in\cal{O}(\cal{A})$ there exists
$\Check\alpha\in\cal{O}_{1}(\cal{A})$ such that $\alpha\subseteq\Check\alpha$,
and we define $\Check\alpha$ by the obvious choice $\Check\alpha=\alpha\cup
\{B^{1/2}\}$.
\item[ad (ii)] Rather obviously, for $\alpha\in\cal{O}^{+}(\cal{A})$ also its
element--wise square $\alpha^{2}\equiv\{A_{i}^{2}|i=1,\dots,n\}$ stays 
$\alpha^{2}\in\cal{O}^{+}(\cal{A})$ as well, i.e.\ we have $\cal{O}^{+}(
\cal{A})\subset\cal{O}(\cal{A})$.

On the other hand, there exists a positive number $c(\alpha)<1$ such that also
$c(\alpha)\cdot\alpha^{1/2}\equiv\{c(\alpha)\cdot A_{i}^{1/2}|i=1,\dots,n\}$
is still in $\cal{O}^{+}(\cal{A})$, where the $c(\alpha)$ is necessary only
for the normalization condition in $\cal{O}^{+}({\cal A})$.
\item[ad (iii)] Again rather obviously, for $\alpha,\beta\in\cal{O}(\cal{A})$
(respectively in $\cal{O}_{1}(\cal{A})$), also $\alpha\Vec\vee\beta\in\cal{O}
(\cal{A})$ (resp.\ in $\cal{O}_{1}(\cal{A})$); but note that $\Vec\vee:
\cal{O}^{+}(\cal{A})\times\cal{O}^{+}(\cal{A})\to\cal{O}(\cal{A})$ with image
{\em more} than $\cal{O}^{+}(\cal{A})$ for non--Abelian $\cal{A}$.
\item[ad (iv)] Note that in the first remark above, it is easily seen that
$\Bar N(\Hat\alpha)=\Bar N(\alpha)\enspace\forall\alpha\in\cal{O}(\cal{A})$;
but on the other hand, only $\Bar N(\Check\alpha)\leqslant\Bar N(\alpha)$
holds, where strict inequality is well possible. And the second remark above
amounts to the observation that $\Bar N(\alpha^{2})=\Bar N(\alpha)\enspace
\forall\alpha\in\cal{O}^{+}(\cal{A})$, where we should add the further
observation that for $\alpha\in\cal{O}^{+}(\cal{A})$, equivalently $\Bar N
(\alpha)=\min\{\card\alpha'| \alpha'\subseteq\alpha,\enspace\alpha'\in
\cal{O}^{+}
(\cal{A})\}$ (instead of $\alpha'\in\cal{O}(\cal{A})$ as second condition).
\end{enumerate}
\end{bbb}
\begin{ccc}The following properties are easy to deduce:
\begin{enumerate}
\item[(i)] $\Bar N(\alpha)\leqslant\Bar N(\alpha\Vec\vee\beta)\leqslant
\Bar N(\alpha)\cdot\Bar N(\beta)\enspace\forall\alpha,\beta\in\cal{O}(\cal{A})  
$, but {\em not} necessarily $\Bar N(\beta)\leqslant\Bar N(\alpha\Vec\vee
\beta)$ in general.
\item[(ii)] $\Bar N(\alpha\Vec\vee\alpha\Vec\vee\dots\Vec\vee\alpha)=\Bar N
(\alpha)\enspace\forall\alpha\in\cal{O}^{+}(\cal{A})$, but {\em not}
necessarily for $\alpha\in\cal{O}(\cal{A})\smallsetminus\cal{O}^{+}(\cal{A})$.
Note that this follows already from the observation $\Bar N(\alpha^{2})=\Bar
N(\alpha)$ in (2.2,iv) above, together with (i).
\item[(iii)] For a positive unital map $\gamma:\cal{B}\to\cal{A}$ from a
unital $C^{*}$--algebra $\cal{B}$ into $\cal{A}$, and $\beta\in\cal{O}^{+}(
\cal{B})$, obviously $\gamma(\beta)\in\cal{O}^{+}(\cal{A})$ and $\Bar N(\gamma
(\beta))\leqslant\Bar N(\beta)$.

Note, furthermore, that from Kadison's inequality $\gamma(B^{2})\geqslant
\gamma(B)^{2}\enspace\forall B=B^{*}\in{\cal B}$ for positive maps $\gamma$
\cite{kad}, it follows that $\Bar N(\gamma(\beta^{2}))\leqslant\Bar N(\gamma
(\beta))$ (again using (2.2,iv) above); and the reverse inequality is again
rather obvious because of the positivity of $\gamma$, i.e.\ even $\Bar N
(\gamma(\beta^{2}))=\Bar N(\gamma(\beta))$ holds for $\beta\in\cal{O}^{+}(
\cal{A})$, also if $\gamma$ is not a homomorphism.
\item[(iv)] For a unital $*$--endomorphism $\theta:\cal{A}\to\cal{A}$, we have
that $\Bar N(\theta(\alpha))\leqslant\Bar N(\alpha),\enspace\forall\alpha\in
\cal{O}(\cal{A})$ (not only for $\alpha\in\cal{O}^{+}(\cal{A})$!).
\end{enumerate}
\end{ccc}
\begin{ddd}For a finite--dimensional $C^{*}$--algebra $\cal{B}$, define $D(
\cal{B})$ to be the dimension of a maximal Abelian ($C^{*}$--)subalgebra of
$\cal{B}$.  Then $\Bar N(\alpha)\leqslant D(\cal{B})\enspace\forall\alpha\in
\cal{O}(\cal{B})$; in other words, if ${\cal B}\cong\bigoplus_{k=1}^{n}
M_{d_{k}}(\Bbb{C})$, then $\Bar N(\alpha)\leqslant\sum_{k=1}^{n}d_{k}$.
We leave the proof to the reader as a simple exercise (cf.\ also \cite{h2}).

Obviously, there exists $\beta\in\cal{O}^{+}(\cal{B})$ such that $\Bar N(
\beta)=D(\cal{B})$; and together we have that $\max_{\alpha\in\cal{O}(\cal{B})
}\Bar N(\alpha)=D(\cal{B})=\max_{\beta\in\cal{O}^{+}(\cal{B})}\Bar N(\beta)$
(note that one can obviously choose a $\beta\in\cal{O}^{+}(\cal{B})$ such that
even $\card\beta=\Bar N(\beta)=D(\cal{B})$ holds).
\end{ddd}
\begin{eee}We now proceed to the central definition of the entropy of several
positive unital maps, generalizing the entropy of (the join of) finite open
covers as in (1.1,i).
\begin{enumerate}
\item[(i)] To simplify the notation in the following, we call two positive
unital maps $\gamma_{k}:\cal{B}_{k}\to\cal{A}\enspace(k=1,2)$ from (not yet
necessarily finite-dimensional) C$^{*}$--algebras $\cal{B}_{k}$ into $\cal{A}$
{\em equivalently covering}, denoted by $\gamma_{1}\approxeq\gamma_{2}$, if
$\gamma_{1}=\gamma_{2}\circ\theta_{12}$ with a $*$--isomorphism $\theta_{12}:
\cal{B}_{1}\to\cal{B}_{2}$.
\item[(ii)] Let $(\gamma_{1},\dots,\gamma_{n})$ be a tuple of positive unital
maps $\gamma_{k}:\cal{B}_{k}\to\cal{A}$ from finite--dimensional C$^{*}
$--algebras $\cal{B}_{k}$ into $\cal{A}$, then their {\em entropy} is defined
by
$$\bar H(\gamma_{1},\dots,\gamma_{n})=\max_{\{(\alpha_{1},\dots,\alpha_{n})\}}
\log\Bar N(\gamma_{1}(\alpha_{1})\Vec\vee\dots\Vec\vee\gamma_{n}(\alpha_{n})),
$$
where the maximum is taken over the set of $n$--tuples with entries $\alpha_{
k}\in\cal{O}^{+}(\cal{B}_{k})$, but with the additional restriction:
$\gamma_{k}\approxeq\gamma_{\ell},\enspace k<\ell\Longrightarrow\alpha_{\ell}
\equiv\{\1_{\ell}\in\cal{B}_{\ell}\}$.
\end{enumerate}
\end{eee}
\begin{fff}The following remarks and simple results will eludicate this main
definition:
\begin{enumerate}
\item The additional restriction in the variational expression (2.5,ii) may
be even simpler expressed verbally: Repeated arguments of $\Bar H$ (even
repeated up to covering equivalence only) do not contribute any more, i.e.\
$\Bar H$ is a functional on the {\em ordered set} of its arguments (even of
their covering--equivalence classes only), with the order inherited from the
original tuple of arguments. This additional restriction is necessary for
reasons to become clear in (2.9,iv) and (2.13,iii) below, whereas at first
sight, admittedly, it seems to be unnatural.

Also without this restriction, however, the functional $\Bar H$ would 
generally not be continuous in norm (for the linear maps $\gamma_{k}:\cal{B
}_{k}\to\cal{A}$, $\cal{B}_{k}$ fixed), as we will see in (2.12,2) below.
This seems to be a good excuse for the additional ``discontinuity'' of $\Bar
H$ due to this necessary restriction; and clearly, $\Bar H$ remains unchanged
on the ``generic'' set of $n$--tuples $(\gamma_{1},\dots,\gamma_{n})$ with
$\gamma_{i}\not\approxeq\gamma_{j}$ for all pairs $i\ne j\in\{1,\dots,n\}$.

\item For a {\em single} positive unital map $\gamma:\cal{B}\to\cal{A}$ from
a finite--dimensional C$^{*}$--algebra $\cal{B}$, we get as an immediate
corollary of (2.3,iii) in (2.5,ii), resp.\ of Lemma (2.4): $\Bar H(\gamma)
\leqslant\log D(\cal{B})=\Bar H(\cal{B})$, where we use on the right hand side
the general notation $\Bar H(\cal{B}_{1},\dots,\cal{B}_{n})\equiv\Bar H(
\imath_{\cal{B}_{1}},\dots,\imath_{\cal{B}_{n}})$ for the inclusion $
*$--homomorphisms $\imath_{\cal{B}_{k}}:\cal{B}_{k}\hookrightarrow\cal{A}$ of
finite--dimensional C$^{*}$--subalgebras $\cal{B}_{k}$ of $\cal{A}$.

\item Note that in (2.5,ii) above, $\Bar H(\gamma_{1},\dots,\gamma_{n})=
\log\Bar N(\beta)$, with $\beta\in\cal{O}(\cal{A})$ given by $\beta=
\gamma_{1}(\bar\alpha_{1})\Vec\vee\dots\Vec\vee\gamma_{n}(\bar\alpha_{n})$,
where $(\bar\alpha_{1},\dots,\bar\alpha_{n})$ maximizes to $\Bar H(\gamma_{1},
\dots,\gamma_{n})$ (note that this maximizing $n$--tuple of $\bar\alpha_{k}\in
\cal{O}^{+}(\cal{B}_{k})$ always exists, see (2.7,ii) below). By (2.2,iv), 
we know that for the
corresponding $\Hat\beta\in\cal{O}_{1}(\cal{A})$ as defined in (2.2,i), $\Bar
H(\gamma_{1},\dots,\gamma_{n})=\log\Bar N(\Hat\beta)$. --

On the other hand, we know that for $(\alpha_{1},\dots,\alpha_{n})$ with
$\alpha_{k}\in\cal{O}^{+}(\cal{B}_{k})\subset\cal{O}(\cal{B}_{k})$, 
there always exist
$\Check\alpha_{k}\in\cal{O}_{1}(\cal{B}_{k})$ defined as in (2.2,i) but even 
with $\Check\alpha_{k}\subset\cal{B}_{k}^{+}$, such that $\alpha_{k}\subseteq
\Check\alpha_{k}\enspace(k=1,\dots,n)$. 

Denoting generally by $\cal{O}_{2}^{+}(\cal{A})$ the set of all $\Check\beta
\in\cal{O}_{1}(\cal{A})$ such that $\Check\beta\subset\cal{A}^{+}$ (i.e.\
explicitly of the form $\Check\beta=\{B_{i}\in\cal{A}^{+}|\sum_{i}B_{i}^{2}=
\1\}$), we can thus equivalently replace in definition (2.5,ii) above the
$\max_{\{(\alpha_{1},\dots,\alpha_{n})\}}$ by the twofold optimization
$$\max_{\{(\Check\alpha_{1},\dots,\Check\alpha_{n})\}}\quad\max_{\{\alpha_{k}
\subseteq\Check\alpha_{k},\enspace k=1,\dots,n|\alpha_{k}\in\cal{O}^{+}(
\cal{B}_{k})\}},$$
where now $\Check\alpha_{k}\in\cal{O}_{2}^{+}(\cal{B}_{k})$, but again with
the additional restriction $\gamma_{k}\approxeq\gamma_{\ell},\enspace k<\ell
\Longrightarrow \Check\alpha_{\ell}\equiv\{\1_{\ell}\in\cal{B}_{\ell}\}$.

\item At this point, we should compare the definition (2.5,ii) with its
earlier versions in \cite{h1} respectively \cite{h2}: In both cases, we had
taken the maximum as in (2.5,ii) only over the smaller set $\{(\alpha_{1},
\dots,\alpha_{n})|\alpha_{k}\in\cal{O}_{2}^{+}(\cal{B}_{k})\}$ for rather
obvious ``physical'' reasons (cf.\ \cite{lin,kr} and the references in the
Introduction); where
we had still chosen a different restriction for repeated arguments of $\Bar
H(\gamma_{1},\dots,\gamma_{n})$ in \cite{h1} than the improved condition 
(2.5,i \& ii) in \cite{h2} and here. In addition, we had still tried in 
\cite{h1} to avoid the also ``physically'' less appealing fact that even then,
$\alpha=\gamma_{1}(\alpha_{1})\Vec\vee\dots\Vec\vee\gamma_{n}(\alpha_{n})\in
\cal{O}(\cal{A})$ will {\em not} be a generalized partition of unity in
general: $\alpha\not\in\cal{O}_{1}(\cal{A})$ (what we had denoted by $\cal{O
}_{2}(\cal{A})$ in \cite{h1,h2}), using the trick that we had replaced $\alpha
$ in (2.5,ii), as defined just before, by $\gamma_{1}[\alpha_{1}]_{2}\Vec\vee
\dots\Vec\vee\gamma_{n}[\alpha_{n}]_{2}\in\cal{O}_{1}(\cal{A})$ with the
deformed, non--linear ``modulus''--application $\gamma_{k}[\centerdot]_{2}:
\cal{B}_{k}\to\cal{A}^{+}$ as defined in \cite{h1}, where $\gamma_{k}[\cal{O
}_{2}^{+}(\cal{B}_{k})]_{2}\subset\cal{O}_{2}^{+}(\cal{A})$ holds true.

But the first part of (3) above shows that this unnatural deformation is not
necessary, as we can always ``renormalize'' the final $\alpha$ as above to
$\Hat\alpha\in\cal{O}_{1}(\cal{A})$ with the same entropy.

On the other hand, the second part of (3) shows that $\bar H(\gamma_{1},\dots,
\gamma_{n})$ as defined in (2.5,ii) here is generally greater (or equal) than
$\bar H(\gamma_{1},\dots,\gamma_{n})$ of \cite{h2}, although we are again
starting with the ``physically appealing'' $\Check\alpha_{k}\in\cal{O}_{2}^{+}
(\cal{B}_{k})$ in (3) above. Actually, for a {\em general} $\alpha\in\cal{O}
(\cal{A})$ as above (but with $\alpha_{k}\in\cal{O}^{+}(\cal{B}_{k})$) in
(2.5,ii), obviously $\alpha\subseteq\Tilde\alpha\equiv\gamma_{1}(\Check
\alpha_{1})\Vec\vee\dots\Vec\vee\gamma_{n}(\Check\alpha_{n})\in\cal{O}(
\cal{A})$ and hence $\Bar N(\alpha)\geqslant\Bar N(\Tilde\alpha)$ by (2.2,iv);
meaning that not only the maximum over the {\em larger set} in (2.5,ii)
increases $\Bar H(\gamma_{1},\dots,\gamma_{n})$ when compared to \cite{h2},
but even for fixed $(\Check\alpha_{1},\dots,\Check\alpha_{n})$ with $\Check
\alpha_{k}\in\cal{O}_{2}^{+}(\cal{B}_{k})$, we can always improve the 
resulting
``entropy'' by choosing {\em any} ``incomplete but still sufficient''
operations $\alpha_{k}\subset\Check\alpha_{k},\enspace\alpha_{k}\in\cal{O}^{+}
(\cal{B}_{k})$ (for $k=1,\dots,n$), in the twofold optimization as in (3)
above (cf.\ \cite{kr} for the notion of ``operation'').
\end{enumerate}
\end{fff}
\begin{gggg}The following properties of the entropy functional (2.5,ii) are
easy to deduce:
\begin{enumerate}
\item[(i)] For $\theta_{k}:\cal{B}_{k}\to\cal{A}_{k}$ and $\gamma_{k}:\cal{
A}_{k}\to\cal{A}$ both positive unital maps with finite--dimensional $C^{*}
$--algebras $\cal{A}_{k},\cal{B}_{k}\enspace(k=1,\ldots,n)$, such that for
$\gamma_{i}\approxeq\gamma_{j}$ (as defined in (1.5,i)) also $\gamma_{i}\circ
\theta_{i}\approxeq\gamma_{j}\circ\theta_{j}\enspace(\forall i,j=1,\dots,n)$,
we have: $$\Bar H(\gamma_{1}\circ\theta_{1},\dots,\gamma_{n}\circ\theta_{n})
\leqslant\Bar H(\gamma_{1},\dots,\gamma_{n}).$$ This follows from $\theta_{k}
(\cal{O}^{+}(\cal{B}_{k}))\subset\cal{O}^{+}(\cal{A}_{k})$, cf.\ (2.3,iii).

Note that this is the first main advantage  of the definition (2.5,ii) when
compared to that in \cite{h2}, as discussed in (2.6,4) above. 
\item[(ii)] For $\gamma_{k}:\cal{A}_{k}\to\cal{A}$ positive unital maps with
finite--dimensional $C^{*}$--algebras $\cal{A}_{k}$, we have
$$\Bar H(\gamma_{1},\dots,\gamma_{m})\leqslant\Bar H(\gamma_{1},\dots,
\gamma_{n})\leqslant\Bar H(\gamma_{1},\dots,\gamma_{m})+\Bar H(\gamma_{m+1},
\dots,\gamma_{n}),$$
$\forall m<n$ (where literally the order of $(1,\dots,n)$ has to be respected),
which follows from (2.3,i).

If in addition $\gamma_{i}\not\approxeq\gamma_{j}\enspace\forall i\ne j\in
\{1,\dots,n\}$, then even for $1\leqslant i_{1}<i_{2}<\dots<i_{\ell}
\leqslant n$ and $\ell<n$: $$\Bar H(\gamma_{i_{1}},\dots,\gamma_{i_{n}})
\leqslant\Bar H(\gamma_{1},\dots,\gamma_{n}),$$ as follows directly from Def.\
(2.5,ii). Note that the right hand side ``subadditivity'' above implies that
actually $\Bar H(\gamma_{1},\dots,\gamma_{n})$ is always finite, what we used
implicitely in (2.5,ii) with the $\max$.
\item[(iii)] 
For a unital $*$--endomorphism $\theta$ of $\cal{A}$ and $\gamma_{k}
\enspace(k=1,\dots,n)$ as in (ii), we have the inequality $\Bar H(\theta\circ
\gamma_{1},\dots,\theta\circ\gamma_{n})\leqslant\Bar H(\gamma_{1},\dots,
\gamma_{n})$, which follows from (2.3,iv).
\item[(iv)] If there exists a finite--dimensional $C^{*}$--subalgebra $
\cal{B}\subset{\cal A}$ such that the image Im$(\gamma_{k})\subseteq\cal{B}
\enspace\forall k=1,\dots,n$, then $\Bar H(\gamma_{1},\dots,\gamma_{n})
\leqslant\log D(\cal{B})\equiv\Bar H(\cal{B})$, generalizing (2.6,ii). This
follows again from Lemma (2.4).
\item[(v)] For $\gamma_{k}\approxeq\gamma_{k}'\enspace(k=1,\dots,n)$, we have
$\Bar H(\gamma_{1},\dots,\gamma_{n})=\bar H(\gamma_{1}',\dots,\gamma_{n}')$,
which follows directly from (2.5). 
\end{enumerate}
\end{gggg}
\begin{hhh}For a positive unital map $\gamma:\cal{B}\to\cal{A}$ from a
finite--dimensional $C^{*}$--algebra $\cal{B}$, and for a unital 
$*$--endomorphism $\theta$ of $\cal{A}$, the entropy of $\theta$ with respect
to $\gamma$ is defined by $\hbar(\theta,\gamma)=\lim_{n\to\infty}\frac{1}{n}
\Bar H(\gamma,\theta\circ\gamma,\dots,\theta^{n-1}\circ\gamma)$, where the
limit exists by (2.7,ii \& iii), and actually equals the {\em infimum} over
$n\in\Bbb{N}$ (by standard arguments).
\end{hhh}
\begin{iii}The entropy (2.8) has the following properties (as corollaries of
(2.7)): For $\gamma:\cal{B}\to\cal{A}$ and $\theta$ as in (2.8),
\begin{enumerate}
\item[(i)] $\hbar(\theta,\gamma)\leqslant\Bar H(\gamma)$, as follows from
(2.7,ii \& iii).
\item[(ii)] For positive unital maps $\gamma_{2}:\cal{B}_{2}\to\cal{A}$ and
$\gamma_{1}:\cal{B}_{1}\to\cal{B}_{2}$ with finite--dimensional $\cal{B}_{k}
\enspace(k=1,2)$, we have $\hbar(\theta,\gamma_{2}\circ\gamma_{1})\leqslant
\hbar(\theta,\gamma_{2})$, as follows from (2.7,i).
\item[(iii)] $\hbar(\sigma\circ\theta\circ\sigma^{-1},\sigma\circ\gamma)=
\hbar(\theta,\gamma)$ for a $*$--isomorphism $\sigma:\cal{A}\to\sigma(\cal{A}
)$, as follows from (2.7,iii).
\item[(iv)] If there exists a $k\in\Bbb{N}$ such that $\gamma\approxeq\theta^{
k}\circ\gamma$, i.e.\ $\theta$ acts periodically on $\gamma$ (up to pre-image
isomorphisms, to be precise) then $\hbar(\theta,\gamma)=0$. This follows
directly from definitions (2.5) and (2.8).
\item[(v)] $\hbar(\theta^{n},\gamma)\leqslant n\cdot\hbar(\theta,\gamma)
\enspace\forall n\in\Bbb{N}$. Note that if the right hand side is equal to
zero because of the situation (iv) above, then so is the left hand side; and
otherwise, the inequality follows from the second one in (2.7,ii).
\end{enumerate}
\end{iii}
\begin{jjj}Let $(X,d)$ be a {\em metric} compact space and $\cal{A}=C(X)$.
Let $(\cal{U}_{n})_{n\in\Bbb{N}}$ be a sequence of finite open covers $
\cal{U}_{n}=\{U_{i_{n}}^{(n)}\subset X|i_{n}=1,\dots,N_{n}:\bigcup_{
i_{n}}U_{i_{n}}^{(n)}=X\}$ of $X$, such that $d(\cal{U}_{n})\to 0$, where $d(
\cal{U}_{n})=\max_{i_{n}}d(U_{i_{n}}^{(n)})$ is the maximal $d$--diameter of
the sets in $\cal{U}_{n}$.
For each $n\in\Bbb{N}$, choose a partition of unity $\beta_{n}$ in $\cal{A}^{
+}$ such that $\beta_{n}=\{f_{i_{n}}^{(n)}\in\cal{A}^{+}|i_{n}=1,\dots,N_{n}:
\sum_{i_{n}}f_{i_{n}}^{(n)}=\1\}$ is ``supported'' exactly on the elements of
$\cal{U}_{n}$, i.e.\ $f_{i_{n}}^{(n)}(x)>0\Longleftrightarrow x\in U_{i_{n}}^{
(n)}\enspace\forall x\in X,\enspace\forall i_{n}=1,\dots,N_{n}$. Note that
we have $\beta_{n}\in\cal{O}^{+}(\cal{A})$ as defined in (2.1,ii), and define
new partitions of unity $\alpha_{n}=\beta_{1}\Vec\vee\beta_{2}\Vec\vee\dots
\Vec\vee\beta_{n}$ with $\Vec\vee$ as in (2.1,iii); but note that now for
$\alpha_{n}=\{g_{I_{n}}^{(n)}\equiv\prod_{k=1}^{n}f_{i_{k}}^{(k)}|\forall I_{n
}=(i_{1},\dots,i_{n})\}$, we have again $\alpha_{n}\in\cal{O}^{+}(\cal{A})$,
as $\cal{A}$ is Abelian.

Define positive unital maps $\tau_{n}:\cal{B}_{n}=\bigoplus_{I_{n}}(\Bbb{C})_{
I_{n}}\to\cal{A}$, on the Abelian direct sum $C^{*}$--algebra $\cal{B}_{n}
$ of as many copies of $\Bbb{C}$ as there are occurring multi--indices $I_{n}
$, by $\tau_{n}(e_{I_{n}}^{(n)})=g_{I_{n}}^{(n)}$ with the minimal projectors
$e_{I_{n}}^{(n)}\in\cal{B}_{n},\enspace\forall I_{n}=(i_{1},\dots,i_{n}),
\enspace\forall n\in\Bbb{N}$. Then the following is true:
\begin{enumerate}
\item[(i)] $(\tau_{n})_{n\in\Bbb{N}}$ is a ``cover--increasing'' sequence in
the sense that $\tau_{n}(\cal{O}^{+}(\cal{B}_{n}))\subseteq\tau_{n+1}(\cal{O
}^{+}(\cal{B}_{n+1}))\enspace(\subset\cal{O}^{+}(\cal{A}))$. This is obvious
from the fact that, by construction, $$g_{I_{n}}^{(n)}=\sum_{i_{(n+1)}}g_{I_{
(n+1)}}^{(n+1)}\qquad\forall I_{(n+1)}=(i_{1},\dots,i_{(n+1)}),\enspace\forall
n\in\Bbb{N}.$$

\item[(ii)] $(\tau_{n})_{n\in\Bbb{N}}$ is a sequence ``approximating for
nuclearity'' (of $\cal{A}$) in the following sense: There exists a sequence
$(\sigma_{n})_{n\in\Bbb{N}}$ of positive unital maps $\sigma_{n}:\cal{A}\to
\cal{B}_{n}$ with the Choi--Effros--Lance ``approximation property'' $\|
\tau_{n}\circ\sigma_{n}(A)-A\| \to 0,\enspace\forall A\in{\cal A}$ (see
\cite{ce} and the reference there to Lance's work, cf.\ also for example 
\cite{cnt}). See \cite{h1,h2} for the simple proof, left to the reader after
the essential hint that the maps $\sigma_{n}$ may be chosen to be $$\sigma_{n}
(A)=\sum_{I_{n}}A(x_{I_{n}}^{(n)})\cdot e_{I_{n}}^{(n)}\qquad\forall A\in
\cal{A}=C(X),$$ for any chosen sets of points $x_{I_{n}}^{(n)}\in X$ such that
$g_{I_{n}}^{(n)}(x_{I_{n}}^{(n)})>0 \enspace\forall I_{n},n\in\Bbb{N}$.
\bigskip
\item[(iii)] For any continuous map $T:X\to X$ from $X$ onto itself, we denote
by $\theta_{T}$ the unital $*$--endomorphism of $\cal{A}$ induced by $T$
(via $\theta_{T}(g)=g\circ T\enspace\forall g\in\cal{A}=C(X)$, which gives an
automorphism of $\cal{A}$ iff $T$ is a homeomorphism). Then the entropy (2.8)
gives exactly the classical entropy (1.1,ii) of the refining covers $\cal{V
}_{n}\equiv\bigvee_{k=1}^{n}\cal{U}_{k}$ of $X$: $\hbar(\theta_{T},\tau_{n})=
h(T,\cal{V}_{n})$, and thus the topological entropy \cite{akm} of $T$ can be
computed as $h(T)=\lim_{n\to\infty}h(\theta_{T},\tau_{n})$, where it follows
here from (i) that the sequence $(\hbar(\theta_{T},\tau_{n}))_{n\in\Bbb{N}}$
is monotonically non--decreasing in $\Bbb{R}^{+}$ (but possibly
divergent with $h(T)=\infty$). See again \cite{h1,h2} for the simple but 
tedious proof of the equivalence above.
\end{enumerate}
\end{jjj}
\begin{kkk}Again for a general C$^{*}$--algebra $\cal{A}$, we say that $
\cal{A}$ is ``appropriately approximated'' (AA for short) by a sequence $\tau
=(\tau_{n})_{n\in\Bbb{N}}$ of positive unital maps $\tau_{n}:\cal{B}_{n}\to
\cal{A}$ from (general) finite--dimensional C$^{*}$--algebras $\cal{B}_{n}$,
if {\em one} of the following two (independent) conditions is fulfilled:
\begin{enumerate}
\item[(i)] $\tau$ is a ``cover--increasing'' sequence, i.e.\ $\tau_{n}(\cal{O
}^{+}(\cal{B}_{n}))\subseteq\tau_{n+1}(\cal{O}^{+}(\cal{B}_{n+1})),\enspace
\forall n\in\Bbb{N}$. Note that this is the case whenever $\tau_{n}=\tau_{
n+1}\circ\sigma_{n,(n+1)}$ with positive unital maps $\sigma_{n,(n+1)}:\cal{B
}_{n}\to\cal{B}_{n+1},\enspace\forall n\in\Bbb{N}$ (for example if $\cal{B}
\subset\cal{A}$ is an AF algebra, i.e.\ $\cal{B}=\overline{\bigcup_{n\in\Bbb{N}
}\cal{B}_{n}}$ is the norm--closure of the increasing inductive limit of
finite--dimensional $C^{*}$--algebras $\cal{B}_{n}\subset\cal{B}_{n+1}\enspace
\forall n\in\Bbb{N}$, and $\tau_{n}=\imath_{\cal{B}_{n}}:\cal{B}_{n}
\hookrightarrow\cal{A}$ are the inclusion $*$--homomorphisms, where we can
choose $\sigma_{n,(n+1)}:\cal{B}_{n}\hookrightarrow\cal{B}_{n+1}$ to be the
inclusion maps, too).
\item[(ii)] $\cal{A}$ is separable and $\tau=(\tau_{n})_{n\in\Bbb{N}}$ is
``range approximating'' $\cal{A}$, i.e.\ $\overline{\bigcup_{n\in\Bbb{N}}
\tau_{n}(\cal{B}_{n})}=\cal{A}$. Note that this is the case in particular
whenever $\cal{A}$ is nuclear and $\tau$ is a sequence ``approximating for
nuclearity'' of $\cal{A}$ as in (2.10,ii), there for Abelian $\cal{A}$, above
(for example if $\cal{B}=\cal{A}$ in (i)).
\end{enumerate}
If $\cal{A}$ is AA, by the sequence $\tau$ (i.e.\ either (i) or (ii) is true),
we define the ``$\tau$--topological'' entropy of a unital $*$--endomorphism
$\theta$ of $\cal{A}$ by $\hbar_{\tau}(\theta)=\limsup_{n\to\infty}\hbar(
\theta,\tau_{n})$ with (2.8) on the right hand side. Note that if $\tau$ is
``cover--increasing'', that is (i) is realized, it follows that $\hbar_{\tau}
(\theta)=\lim_{n\to\infty}\hbar(\theta,\tau_{n})$ as an {\em increasing} 
limit.
\end{kkk}
\begin{llll} The following remarks should again eludicate the above definition
(1.11):
\begin{enumerate}
\item Again (cf.\ the remark in (2.6,1)), the definition (2.11) seems to be
not very ``canonical'' at first sight. The canonical approach to the
``topological'' entropy $\hbar(\theta)$ of a $*$--endomorphism $\theta$ of
$\cal{A}$, directly following our ``guiding'' theory \cite{cnt} of entropy
for automorphisms with respect to an invariant state, would be to define $
\hbar(\theta)=\sup_{\gamma}\hbar(\theta,\gamma)$, where the supremum is
taken over {\em all} completely positive unital maps $\gamma$ into $\cal{A}$
with finite rank (i.e.\ with finite--dimensional pre--image algebra). Let us
henceforth denote by $\cal{C}\cal{P}_{1}(\cal{A})$ the set of all such maps 
$\gamma$; and by $\cal{P}_{1}(\cal{A})$ we denote the subset of maps with
(finite--dimensional) {\em Abelian} pre--image algebra.

Then, however, we would have to {\em prove} an ``approximation theorem'' of 
the form $\hbar(\theta)=\lim_{n\to\infty}\hbar(\theta,\tau_{n})$, for example
in the case that $(\tau_{n})_{n\in\Bbb{N}}$ is a sequence ``approximating for
nuclearity'' of $\cal{A}$. But, as pointed out in \cite{h1,h2}, the entropy
$\hbar(\theta,\gamma)$ of (2.8) is {\em not} continuous in norm with respect
to the varying linear maps $\gamma:\cal{B}\to\cal{A}$, $\cal{B}$ fixed (for
example with
$\gamma\in\cal{C}\cal{P}_{1}(\cal{A})$); and this makes it impossible to
proof such a theorem along the lines of \cite{cnt}.

\item Let us briefly recall recall from \cite{h1,h2} that for a general C$^{
*}$--algebra $\cal{A}$ (also if it is {\em non\/}--Abelian, by a similar
argument as for Abelian $\cal{A}$), the set $\{\gamma\in\cal{P}_{1}(\cal{A})|
\gamma:\cal{B}\to\cal{A}\}$ with $\cal{B}$ fixed has even an {\em open, dense}
subset (in the norm of the linear maps $\gamma$) of $\gamma':\cal{B}\to\cal{A}
$, on which the entropy (2.8) vanishes for {\em any} $*$--endomorphism $
\theta$, i.e.\ $\hbar(\theta,\gamma')=0$.

In particular, for $\cal{A}=C(X)$ as in (2.10) above, one can easily construct
sequences $\tau=(\tau_{n})_{n\in\Bbb{N}}$ ``approximating for nuclearity'' as
in (2.10,ii), with $\tau_{n}:\cal{B}_{n}\to\cal{A}$ and $\cal{B}_{n}$ {\em
Abelian} (i.e.\ $\tau_{n}\in\cal{P}_{1}(\cal{A})$), but still in such a way
that $\hbar(\theta_{T},\tau_{n})\equiv 0\enspace\forall n\in\Bbb{N}$ (no
matter how $T$ respectively $\theta_{T}$ is chosen).

In other words, the ``approximation'' condition (2.10,ii) is by construction
necessary but not at all sufficient for the result $\hbar(\theta_{T},\tau_{n})
\to h(T)$ in (2.10,iii), which is rather due to the natural partial order on
the set of finite open covers of $X$, for which the sequence $\cal{V}_{n}=
\bigvee_{k=1}^{n}\cal{U}_{k}$ in (2.10) is cofinal, and even monotonically
``refining''.
\item But even the additional ``cover--increase'' condition (2.10,i), which
in a way represents the latter order monotonicity of the sequence $\cal{V}_{n}
$, is not sufficient together with (2.10,ii) to imply the result $\hbar(
\theta_{T},\tau_{n})\to h(T)$, as still the same counter--examples with
$\hbar(\theta,\tau_{n})\equiv 0$ as in (2) apply (again an easy exercise).

>From that algebraic point of view, this convergence in (2.10,iii) is rather
implied by the fact that the sequence $(\alpha_{n})_{n\in\Bbb{N}}$ with
$\alpha_{n}\in\cal{O}^{+}(\cal{A})$ as in (2.10) is again cofinal (and even
monotonic) in $\cal{O}^{+}(\cal{A})$ with respect to the natural partial
pre--order on $\cal{O}^{+}(\cal{A})$ ``inherited'' from that on the set of
finite open covers of $X$, see \cite{h1,h2}.

But for a {\em non\/}--Abelian C$^{*}$--algebra $\cal{A}$, this argument fails
for (at least) two different reasons: First, this mentioned partial pre--order
on $\cal{O}^{+}(\cal{A})$ relies on the usual partial order of $\cal{A}^{+}$
by positivity, and there is no way that this latter order is preserved in
(2.5,ii) when taking the operator products of the $\gamma_{k}(\alpha_{k})\in
\cal{O}^{+}(\cal{A})$ which would be order--related to other $\beta_{k}\in
\cal{O}^{+}(\cal{A})$, say $(k=1,\dots,n)$.

And secondly, for non--commutative (even separable) $\cal{A}$ there seems to
be generally
no {\em cofinality} possible for this ``natural'' partial pre--order on
$\cal{O}^{+}(\cal{A})$, when considering a fixed sequence of $\tau_{n}\in
\cal{C}\cal{P}_{1}(\cal{A})$ as used in (2.11) and in the Abelian case (2.10)
above (cf.\ \cite[Appendix]{h2}. In other words, we lack a ``non--commutative
Lebesgue's covering theorem'', whereas in (2.10) it follows from the basic
assumption $d(\cal{U}_{n})\to 0$ that the sequence $(\alpha_{n})_{n\in\Bbb{N}}
$, and already $(\beta_{n})_{n\in\Bbb{N}}$, is cofinal in $\cal{O}^{+}(\cal{A
})$ as mentioned; or there even equivalently, that $(\tau_{n})_{n\in\Bbb{N}}$
is increasing also ``cofinally'' in $\cal{P}_{1}(\cal{A})$ in a certain 
sense).
\item It should now be clear that our not very ``canonical'' definition
(2.11) is an (admittedly, provisional) attempt to retain as much of the
essential structure used for the classical computations of the topological
entropy, here in (2.10, iii), as is possible by now for non--Abelian C$^{*}
$--algebras, and to do so in the least restrictive way by imposing only one
of the two conditions (2.11,i \& ii) alternatively. Actually, with only the
first (2.11,i) being fulfilled, it seems to be an ``abuse of language'' to
call $\cal{A}$ A. {\em approximated} by $\tau$, and it would perhaps be more
appropriate to call $\cal{A}$ A. ``approached'' by $\tau$ in that case.

However, note that at least all AF algebras $\cal{A}$ are AA in the ``strong''
sense that {\em both} (2.11,i \& ii) are fulfilled, if $\tau$ and the 
AF--structure are chosen as noted in (2.11). But still then, and even ``more
so'' generally, we have first only the trivial inequality for an AA-sequence
$\tau\subset\cal{C}\cal{P}_{1}(\cal{A})$, i.e.\ with {\em completely}
positive maps $\tau_{n}$: $\hbar_{\tau}(\theta)\leq\hbar(\theta)$, defining 
the right hand side as in (1) above by $\hbar(\theta)=\sup_{\gamma\in\cal{C}
\cal{P}_{1}(\cal{A})}$. Note that for $\cal{A}$ an AF algebra as before, in
fact $\tau\subset\cal{C}\cal{P}_{1}(\cal{A})$.
\item It should be emphasized at this point, however, that this latter upper
bound is {\em not} necessarily a direct generalization of the classical
topological entropy $h(T)$ as in (2.10,iii) above:

Whereas it is easily seen that for $\cal{A}=C(X)$ generally $$h(T)=\sup_{
\gamma\in\cal{P}_{1}(\cal{A})}\hbar(\theta_{T},\gamma)\leqslant\hbar(\theta_{
T})$$ with the notations from (2.10,iii) and (1) above (cf.\ \cite{h1,h2} for
the left hand side equality), we have not been able yet to either prove or
disprove the equality $h(T)=\hbar(\theta_{T})$ in this $\cal{A}$--Abelian
case, where the problem comes from the additional non--commutative freedom
for $\alpha_{k}\in\cal{O}^{+}(\cal{B}_{k})$ in definition (2.5,ii) for
different $k\in\Bbb{N}$ (used for $\theta^{k}\circ\gamma:\cal{B}\to\cal{A}$ 
with $\cal{B}$ {\em non\/}--Abelian in definition (2.8) of $\hbar(\theta,
\gamma)$ with $\gamma\in\cal{C}\cal{P}_{1}(\cal{A})$). 

Although the analogous
equality in the case of Abelian $\cal{A}$ holds for the state--dependent
entropy of \cite{cnt}, identifying it with the classical measure--theoretic
entropy even when taking the analogous supremum over $\cal{C}\cal{P}_{1}
(\cal{A})$ for Abelian $\cal{A}$ (cf.\ \cite{h3}), we do not regard this
property as really essential; but on the other hand, we consider that
non--commutative freedom in Def.\ (2.5,ii) as indispensable for non--Abelian
$\cal{A}$ (cf.\ also section III below, after the proof of (3.1)).

While this ``hybrid'' $\cal{A}$--Abelian and $\cal{B}$--noncommutative case 
seems to be an interesting problem {\em per se}, we have now circumvented 
also this obstacle with our pragmatic definition (2.11), as we can always
choose the AA--sequence $\tau=(\tau_{n})_{n\in\Bbb{N}}$ for {\em Abelian}
$\cal{A}$ such that only $\tau_{n}\in\cal{P}_{1}(\cal{A})$ enters, $\forall
n\in\Bbb{N}$ (implying that we have in turn the general inequality $\hbar_{
\tau}
(\theta_{T})\leqslant h(T)$, where equality holds in the situation of (2.10)
for separable $\cal{A}$). We do not force this choice for Abelian $\cal{A}$
to be part of Definition (2.11), however, because there {\em are} some rather
obvious examples of AA--sequences $\tau$ with truly ``non--commutatively
mapping'' $\tau_{n}\in\cal{C}\cal{P}_{1}(\cal{A})$ for Abelian $\cal{A}$,
but still $\hbar_{\tau}(\theta_{T})=h(T)$; see section IV below (in particular
Example (4.13), where it should be an easy exercise for the reader to find 
such a sequence $\tau$).

Unfortunately, the possible {\em counter\/}--examples are not obvious enough,
if they exist at all. We leave this problem touched upon here for further 
study.
\end{enumerate}
\end{llll}
\begin{mmm}If $\cal{A}$ is AA with a sequence $\tau=(\tau_{n})_{n\in\Bbb{N}}$
as in (2.11), and $\sigma:\cal{A}\to\cal{B}$ is a $*$--isomorphism, then
rather obviously also $\cal{B}$ is AA by the sequence denoted as $\sigma(\tau)
\equiv(\sigma\circ\tau_{n})_{n\in\Bbb{N}}$ (i.e.\ both (2.11,i \& ii) are
preserved by $\sigma$). The entropy (2.11) of a $*$--endomorphism $\theta$ of
$\cal{A}$ has the following rather obvious properties (as corollaries of 
(2.9)):
\begin{enumerate}
\item[(i)] $\hbar_{\sigma(\tau)}(\sigma\circ\theta\circ\sigma^{-1})=\hbar_{
\tau}(\theta)\enspace\forall\sigma:\cal{A}\to\cal{B}$, $*$--isomorphisms.
\item[(ii)] $\hbar_{\tau}(\theta^{n})\leqslant n\cdot\hbar_{\tau}(\theta)
\enspace\forall n\in\Bbb{N}$.
\item[(iii)] For a {\em periodic} $*$--automorphism $\theta=\theta^{k}\enspace
(k\in\Bbb{N})$, we have $\hbar_{\tau}(\theta)=0$.
\end{enumerate}
\end{mmm}
Note that it is only at this point (iii) (and at the intermediate step 
(2.9,iv)) that the additional restriction in Def.\ (2.5) is really necessary,
as discussed already in (2.6,i). In other words, this restriction in (2.5) is
a necessary correction to exclude the additional ``quantum stochasticity''
which could otherwise lead to a positive $\hbar_{\tau}(\theta)$ also for
trivial or periodic $\theta$, due to the repeated ``operations'' $\gamma(
\alpha_{1}),\theta\circ\gamma(\alpha_{2}),\dots,\theta^{n-1}\circ\gamma(
\alpha_{n})$ with $\alpha_{k}\in\cal{O}^{+}(\cal{B})$ and $\cal{B}$ {\em
non\/}--Abelian finite--dimensional in (2.5,ii).
\begin{nnn} We define the following two notions for the later Theorem (2.15):
\begin{enumerate}
\item[(i)] We call a positive unital map $\gamma:\cal{B}\to\cal{A}$ from a
finite--dimensional C$^{*}$--algebra $\cal{B}$ {\em faithfully covering},
if there exists $\alpha'\in\cal{O}^{+}(\cal{B})$ such that $\alpha'$ at the
same time maximizes to $\Bar H(\gamma)$ as in (2.5), and is already 
``$\gamma$--minimal'', i.e.\ both equations
$$\exp\Bar H(\gamma)=\Bar N(\gamma(\alpha'))=\card\alpha'$$ are satisfied.
\item[(ii)] We call $n$ positive unital maps $\gamma_{k}:\cal{B}_{k}\to\cal{A}
\enspace(k=1,\dots,n)$ ($\cal{B}_{k}$ here not necessarily 
finite--dimensional) {\em independently covering}, if for $B_{k}\in\cal{B}_{k
}^{+}$ but $\gamma_{k}(B_{k})$ not invertible $(k=1,\dots,n)$ there exists a 
state $\omega\in S_{\cal{A}}$ which simultaneously annihilates all $\gamma_{k}
(B_{k})$: $\omega\circ\gamma_{k}(B_{k})=0\enspace\forall k=1,\dots
,n$. 

We say that two maps $\gamma_{1}$ and $\gamma_{2}$ are ``commuting''
(denoted symbolically $[\gamma_{1},\gamma_{2}]=0$) if $[\gamma_{1}(B_{1}),
\gamma_{2}(B_{2})]=0\enspace\forall B_{k}\in\cal{B}_{k}\enspace(k=1,2)$.
\end{enumerate}
\end{nnn}
\begin{ooo} If $\gamma_{k}:\cal{B}_{k}\to\cal{A}\enspace(k=1,\dots,n)$ are
positive unital maps from finite--dimensional $C^{*}$--algebras $\cal{B}_{k}$
into $\cal{A}$, which are {\em independently} and (each individually) {\em
faithfully} covering, and pairwise commuting, i.e.\ $[\gamma_{i},\gamma_{j}]=0
\enspace\forall i\ne j\in\{1,\dots,n\}$, then the optimal upper bound of
(2.7,ii) is attained: $\Bar H(\gamma_{1},\dots,\gamma_{n})=\sum_{k=1}^{n}
\Bar H(\gamma_{k})$.
\end{ooo}
\begin{pf}
Using the notation $\alpha'$ of (2.14,i), we choose $\alpha_{k}'\in\cal{O}^{+}
(\cal{B}_{k})$ such that by assumption also $\Bar N(\gamma_{k}(\alpha_{k}'))=
\card\alpha_{k}'\equiv N_{k}$ holds $(k=1,\dots,n)$. Clearly, $\beta=
\gamma_{1}(\alpha_{1}')\Vec\vee\dots\Vec\vee\gamma_{n}(\alpha_{n}')\in\cal{O}
(\cal{A})$ is a possible choice inside the maximum in definition (2.5,ii) of
$\Bar H(\gamma_{1},\dots,\gamma_{n})$. 

We now claim that $\Bar N(\beta)=\prod_{k=1}^{n}N_{k}$, which implies by
(2.5,ii) that $\Bar H(\gamma_{1},\dots,\gamma_{n})\geqslant\log\Bar N(\beta)=
\sum_{k=1}^{n}\Bar H(\gamma_{k})$ again by the faithfulness assumption on the
$\gamma_{k}$, and then (2.7,ii) gives the converse.

To prove that $\Bar N(\beta)=\prod_{k=1}^{n}N_{k}$, put $\alpha_{k}'=
\{A_{i_{k}}^{(k)}\in\cal{B}_{k}^{+}|i_{k}=1,\dots,N_{k}\}\enspace(k=1,\dots,n)
$ and use the following abbreviation:
$$B_{(i_{1},\dots,i_{n})}=\gamma_{1}(A_{i_{1}}^{(1)})\gamma_{2}(A_{i_{2}}^{(2)
})\cdot\dots\cdot[\gamma_{n}(A_{i_{n}}^{(n)})]^{2}\cdot\dots\cdot\gamma_{1}(
A_{i_{1}}^{(1)})\in\cal{A}^{+}$$
$\forall I_{n}\equiv(i_{1},\dots,i_{n})$, where $i_{k}\in\{1,\dots,N_{k}\}
\enspace\forall k$. The statement $\beta\in\cal{O}(\cal{A})$ means that the
following sum is strictly positive:
$$B\equiv\sum_{I_{n}}B_{I_{n}}=B_{<}+\sum_{k=1}^{n}\sum_{j_{k}\ne k}
B_{\leqslant}(k,j_{k})+B_{(N_{1},\dots,N_{n})}>0,$$
where we decomposed the sum using the notations
$$B_{<}=\sum\begin{Sb}\{(i_{1},\dots,i_{n})| \\ i_{k}\lneqq N_{k}\enspace
\forall k\} \end{Sb} B_{(i_{1},\dots,i_{n})},\qquad B_{\leqslant}(k,j_{k})=
\sum\begin{Sb}\{(i_{1},\dots,i_{n})| \\ i_{k}\lneqq N_{k}, \\
i_{j_{k}}=N_{j_{k}}\}\end{Sb} B_{(i_{1},\dots,i_{n})}$$
(note that in the summation defining $B_{\leqslant}(k,j_{k})$ for $k\ne j_{k}$ 
the other indices are free: $i_{j}\leqslant N_{j}\enspace\forall j\ne k,j_{k}
$).

That we have $\Bar N(\gamma_{k}(\alpha_{k}'))=\card\alpha_{k}'$ implies in
particular (but without loss of generality, by renumbering) that the following
elements $C_{k}\in\cal{A}$ are {\em not} strictly positive:
$C_{k}=\sum_{i_{k}=1}^{(N_{k}-1)}\gamma_{k}(A_{i_{k}}^{(k)})\ngtr 0\enspace
(\forall k=1,\dots,n)$: Remember from (2.2, ad (iv)) that for general $\alpha 
\in\cal{O}^{+}(\cal{A})$, equivalently $\Bar N(\alpha)=\min\{\card(\alpha'
\subset\alpha)|\alpha'\in\cal{O}^{+}(\cal{A})\}$.

By the assumption that the $\gamma_{k}$ are independently covering, there
exists a state $\omega\in S_{\cal{A}}$ such that $\omega(C_{k})=0\enspace
\forall k=1,\dots,n$. This is {\em a priori} not clear from definition 
(2.14,ii) for $D_{k}\equiv\sum_{i_{k}=1}^{(N_{k}-1)}[\gamma_{k}(A_{i_{k}}^{(k)
})]^{2}\ngtr 0\enspace(\forall k=1,\dots,n)$; but because of $A^{2}\leqslant
A\cdot\|A\|$ (for $A\in\cal{A}^{+}$) it follows obviously that also $\omega
(D_{k})=0\enspace\forall k$ and hence $\omega(D_{k}^{2})=0\enspace\forall k$.

Using the pairwise commutativity assumption $([\gamma_{i},\gamma_{j}]=0
\enspace\forall i\ne j)$, it is also obvious that the individual terms in the
decomposed sum above can be written as:
$$B_{<}=\prod_{k=1}^{n}D_{k},\qquad B_{\leqslant}(k,j_{k})=D_{k}\cdot(E_{j_{k}
}\cdot\prod_{\ell\ne k,j_{k}}F_{\ell}) \quad\forall k\ne j_{k},$$
where we used the notations $E_{k}=[\gamma_{k}(A_{N_{k}}^{(k)})]^{2}$ and
$F_{k}=\sum_{i_{k}=1}^{N_{k}}[\gamma_{k}(A_{i_{k}}^{(k)})]^{2}>0$ (the {\em
full} sum), $\forall k=1,\dots,n$.

>From the Cauchy-Schwarz inequality for the state $\omega$ it follows in
particular that $|\omega(D_{k}\cdot A)|^{2}\leqslant\omega(D_{k}^{2})\cdot
\omega(A^{*}A)=0\enspace\forall A\in\cal{A},\enspace\forall k$, and hence
by summation that $\omega(B-B_{(N_{1},\dots,N_{n})})=0$, but on the other hand
we know $\omega(B)>0$ because of $B>0$. This implies both $\Bar N(\beta)=
\card\beta$ {\em and} $B_{(N_{1},\dots,N_{n})}\ne 0$, hence $B_{I_{n}}\ne 0
\enspace\forall I_{n}$ (by renumbering), so $\card\beta=\prod_{k=1}^{n}N_{k}$
and the claim is proved.
\end{pf}
\begin{ppp} The following remarks and simple results refer to (2.14) and 
(2.15):
\begin{enumerate}
\item The first condition (2.14,i) of ``faithfully covering'' $\gamma$ (which
was still missing in \cite{h1,h2} for reasons to become clear in (4) below)
is always fulfilled for $\cal{B}\subset\cal{A}$ a finite--dimensional unital
(C$^{*}$--)subalgebra and $\gamma=\imath_{\cal{B}}:\cal{B}\hookrightarrow
\cal{A}$ the 
unital inclusion $*$--homomorphism: This is obvious from (2.4).

Furthermore, in the Abelian case $\cal{A}=C(X)$ as in (2.10), this condition
means the following: Starting with an {\em arbitrary} open cover $\cal{U}_{1}=
\{U_{i}\subset X|i=1,\dots,n:\bigcup_{i}U_{i}=X\}$ and choosing {\em any}
corresponding positive unital map $\tau_{1}:\cal{B}_{1}=\bigoplus_{i=1}^{N}
(\Bbb{C})_{i}\to\cal{A}$ as in (2.10) (where we take $n=1$ to avoid the 
additional refinements and multi--indices), the condition that $\tau_{1}\in
\cal{P}_{1}(\cal{A})$ (with the notation (2.12,i)) be {\em faithfully}
covering means that there exists a ``cover'' $\{\frak{X}_{1},\dots,\frak{X}_{
M}\}\subset 2^{\frak{X}}\equiv \cal{P}(\frak{X})$ of the index set $\frak{X}=
\{1,\dots,N\}$ of $\cal{U}_{1}$ (i.e.\ $\bigcup_{j=1}^{M}\frak{X}_{j}=\frak{X}
$), such that for the resulting {\em coarser} cover (than $\cal{U}\equiv\cal{
U}_{1}$) $\cal{W}$ of $X$ given by the resulting unions:
$$\cal{W}=\{W_{j}=\bigcup_{i_{j}\in\frak{X}_{j}}U_{i_{j}}|\forall j=1,\dots,M
\},$$ we have the corresponding {\em classical} relation $N(\cal{U})=N(\cal{W}
)=\card\cal{W}$ (note that generally we know only the inequalities $N(\cal{U})
\geqslant N(\cal{W})\leqslant\card\cal{W}$, see (1.2,iv)). This is obviously a
very natural condition on $\cal{U}$ for requiring that $\cal{U}$ has not {\em
too} much ``redundancy'' in covering $X$, although it need {\em not} be itself
``minimal'' at all (i.e.\ $N(\cal{U})\leqslant\card\cal{U}$ is well possible).
Also in this classical context, we call $\cal{U}$ {\em faithfully} covering in
this case.
\item But also more generally (now again for non--Abelian $\cal{A}$), there
are very natural {\em non\/}--subalgebra examples of faithfully covering maps
$\gamma\in\cal{C}\cal{P}_{1}(\cal{A})$, i.e.\ with non--commutative 
(pre--)image: Choose any concrete C$^{*}$--algebra (i.e.\ in a Hilbert space 
representation) $\cal{A}\subset\cal{B}(\cal{H})$ with two unitaries $U,V\in
\cal{A}$ such that $[U,V]\ne 0$ but $U$ and $V$ have two common eigenvectors
$\xi_{\pm}\in\cal{H}$ with eigenvalues $\pm 1$ (thus belonging to the pure 
point
spectrum of $U,V$), respectively: $U\xi_{\pm}=V\xi_{\pm}=\pm\xi_{\pm}$. Such
examples almost obviously exist, and then the ``formal almost--Mathieu
Hamiltonian'' (just to give it a name, cf.\ \cite{arveson}) $h=\frac{1}{4}
(U+V+U^{*}+V^{*})$ has of course again $\pm 1$ in its (pure point) spectrum:
$h\xi_{\pm}=\pm\xi_{\pm}$.

On the other hand, we can always define a {\em completely} positive unital map
$\gamma:\cal{B}=M_{2}(\Bbb{C})\to\cal{A}$ by $\gamma(e_{ij})=A_{ij}$, where
$e_{ij}$ are the canonical matrix units in $M_{2}=\cal{B}$ and $A_{ij}\in\cal
{A}$ are the elements of a {\em positive} matrix $A=(A_{ij})$ in the 
C$^{*}$--algebra 
$M_{2}(\cal{A})$ of $(2\times 2)$--matrices over $\cal{A}$, given by:
$$A=\frac{1}{2}\begin{pmatrix} \1& U+V^{*} \\ V+U^{*}&\1 \end{pmatrix}
\in M_{2}(\cal{A})^{+}.$$
Note that with $B=\bigl( \begin{smallmatrix} \1& U \\ V& \1 \end{smallmatrix}
\bigr)$, and unitary $U,V$, we have $A=B^{*}B\in M_{2}(\cal{A})$; and thus 
$\gamma$ is a {\em completely} positive map by \cite{ce} (see the further
reference in the introduction there).

Now define $\beta'\in\cal{O}^{+}(\cal{B})$ by $\beta'=\{p_{+},p_{-}\}$ where
$p_{\pm}=\frac{1}{2}\bigl( \begin{smallmatrix}1&\pm 1 \\ \pm 1&1 
\end{smallmatrix} \bigr)$; note that $\beta'$ is even a (square) partition of 
unity: $p_{+}^{(2)}+p_{-}^{(2)}=\1_{2}$ (i.e.\ $\beta\in\cal{O}_{2}^{+}(
\cal{B})$ as defined in (2.6,3)). 

Then $\gamma(p_{\pm})=\frac{1}{2}(\1\pm h)$, and hence by the above choice
$\gamma(p_{\pm})\ngtr 0$, which implies that $\Bar N(\gamma(\beta'))=\log 2$.
Thus obviously $\Bar H(\gamma)=\log 2$ by (2.6,2), and so, as said before, 
this $\gamma\in\cal{C}\cal{P}_{1}(\cal{A})$ is a {\em faithfully} covering
map (with $\beta'\in\cal{O}^{+}(\cal{B})$ as the defining entity of (2.14,i),
which is even uniquely determined in this simple example). 

The important point here is of course that $\gamma(\cal{B})=\text{span}\{\1,
(U+V^{*}),(V+U^{*})\}\subset\cal{A}$ is generally {\em not} contained in any
finite--dimensional C$^{*}$--subalgebra of $\cal{A}$, nor will C$^{*}(U+
V^{*})$ be Abelian (more or less by assumption) regardless that in this simple
example with $\beta'\in\cal{O}_{2}^{+}(M_{2})$ necessarily $[\gamma(p_{+}),
\gamma(p_{-})]=0$.
\item The second condition in (2.14,ii) of ``independently covering'' maps
$\gamma_{1},\dots,\gamma_{n}$ has at least a natural {\em terminology}, as it
is obviously equivalent to the following reformulations: For $B_{k}\in\cal{B
}_{k}^{+}$ but $\gamma_{k}(B_{k})\ngtr 0\enspace(k=1,\dots,n)$, there exists
an $\omega\in S_{\cal{A}}$ such that $\omega(\sum_{k=1}^{n}\gamma_{k}(B_{k}))=
0$; which is in turn equivalent to its negation: For $B_{k}\in\cal{B}_{k}^{+}
\enspace(k=1,\dots,n)$ such that $\{\gamma_{k}(B_{k})|k=1,\dots,n\}\in\cal{O}^
{+}(\cal{A})$, there exists already a {\em single} $\gamma_{k}(B_{k})>0$.

But the condition (2.14,ii) is also equivalent to the ``natural'' notion of
C$^{*}$--independence \cite{sum} for {\em two} C$^{*}$--subalgebras $\cal{A}_{
1},\cal{A}_{2}\subset\cal{A}$ which are commuting ($[\cal{A}_{1},\cal{A}_{2}]=
0$), but still not necessarily finite--dimensional: As extensively reviewed in
\cite{sum}, generally $\cal{A}_{1}$ and $\cal{A}_{2}$ are $C^{*}$--{\em 
independent} if {\em one} of the following two equivalent conditions is
fulfilled (among other possible characterizations): For all pairs of states
$\phi_{i}\in S_{\cal{A}_{i}}\enspace(i=1,2)$ there exists an extending state
$\varphi\in S_{\cal{A}}$ with $\varphi\upharpoonright_{\cal{A}_{i}}=\phi_{i}
\enspace\forall i=1,2$; or also equivalently: $0\ne A_{i}\in\cal{A}_{i}
\enspace(i=1,2)$ implies $0\ne A_{1}A_{2}(\in\cal{A})$. 

It is easy to see that, given $[\cal{A}_{1},\cal{A}_{2}]=0$, the inclusion
$*$--homorphisms $\imath_{\cal{A}_{1}}$ and $\imath_{\cal{A}_{2}}$ are
independently covering {\em iff} $\cal{A}_{1}$ and $\cal{A}_{2}$ are
C$^{*}$--independent; see \cite{h2} for the simple proof of the one
non--trivial implication, running along the line of the proof of \cite{sum}:
(3.2,1) $\Longrightarrow$ (3.2,2) there.

If $\cal{A}_{1}$ and $\cal{A}_{2}$ are in addition {\em 
finite\/}--dimensional, it is clear (cf.\ \cite{sum}) that C$^{*}(\cal{A}_{1},
\cal{A}_{2})\cong\cal{A}_{1}\otimes\cal{A}_{2}$; and consequently, if
$\cal{B}_{1},\dots,\cal{B}_{n}\subset\cal{A}$ are {\em finite\/}--dimensional
and pairwise commuting, it is again equivalent that $\imath_{\cal{B}_{1}},
\dots,\imath_{\cal{B}_{n}}$ are independently covering, to that the natural
$*$--homomorphism mapping $\bigotimes_{k=1}^{n}\cal{B}_{k}\to\text{C}^{*}
(\cal{B}_{1},\dots,\cal{B}_{n})$ is an isomorphism. More generally, all
conditions (2.14) of the additivity result (2.15) are met if the $\gamma_{k}:
\cal{B}_{k}\to\cal{A}$ ($\cal{B}_{k}$ finite--dimensional, $k=1,\dots,n$) are
faithfully covering maps to $\cal{A}=\cal{A}_{1}\otimes\dots\otimes\cal{A}_{n}
$, for example with nuclear $\cal{A}_{k}\ni\1_{k}$, and $\gamma_{k}:\cal{B}_{
k}\to\1_{1}\otimes\dots\otimes\cal{A}_{k}\otimes\dots\otimes\1_{n}\enspace
(\forall k=1,\dots,n)$.
\item The reason why the first condition (2.14,i) was still missing in the
earlier preliminary versions \cite{h1,h2} of the theory presented here is the
following: It had escaped the attention of the author that there had been a
gap in the very first paper on (``classical'') topological entropy by Adler,
Konheim and McAndrew \cite{akm}, namely the claim that for a product cover
$\cal{U}_{1}\times\cal{U}_{2}=\{U\times V|U\in\cal{U}_{1},\enspace V\in\cal{U
}_{2}\}$ of a product space $X=X_{1}\times X_{2}$ (with product topology), the
covering counting functional $N(\centerdot)$ would be multiplicative:
$N_{X}(\cal{U}_{1}\times\cal{U}_{2})=N_{X_{1}}(\cal{U}_{1})\cdot N_{X_{2}}(
\cal{U}_{2})$, not only (rather trivially) {\em sub\/}--multiplicative.

This seemingly innocent statement would have been again analogous to the
additivity of the measure--theoretic partition entropy for independent
partitions (in particular, product partitions of product probability spaces,
cf.\ \cite{me}); but at this point the simple analogy fails: It was then
apparently first published by Goodwyn \cite{goo}, that ``without much
difficulty one can construct counter--examples where this equality does not 
hold''; and with the right intuition, this is indeed true.

We leave it to the reader as a simple exercise to construct a 
counter--example for two three--point spaces $X_{1}=X_{2}=\{p_{1},p_{2},p_{3}
\}$, both with discrete topology (or, rather obviously ``equivalently'', for
$X_{1}=X_{2}=\bold{S}^{1}$ and $X=\Bbb{T}^{2}$ with usual topology and a
product cover by $9=3^{2}$ {\em connected} open sets); and actually, this flaw
in the argument of \cite{akm} had been immediately pointed out by R. Bowen
\cite{roy}, leading to his alternative definition of topological entropy 
(see \cite{dgs,me} and the references there), but unfortunately the latter 
seems to be not (yet) apt for non--commutative generalization. 

Anyway, the proof of the earlier version of (2.15) in \cite{h2} (implicit in
\cite{h1}) had still contained the same old mistake, which was kindly brought
to the author's attention by Klaus Thomsen (cf.\ section III below).

Note that the fact that now this gap in the proof can be easily avoided, by
the additional condition (2.14,i) of {\em faithfully} covering maps, is the
second main advantage of the definition (2.5,ii) when compared to that in
\cite{h2} (i.e.\ taking the maximum with $\alpha_{k}\in\cal{O}^{+}(\cal{B}_{k}
)$ instead of $\cal{O}_{2}^{+}(\cal{B}_{k})$; cf.\ (2.7,i)). Of course, the
result (2.15) holds in particular for Abelian $\cal{A}=C(X)$, where it can be
retranslated in the classical situation $X=X_{1}\times X_{2}$ of (1) above as
follows: If both $\cal{U}_{1}$ and $\cal{U}_{2}$ are even {\em faithfully}
covering (in the sense of (1)), then $N_{X}(\cal{U}_{1}\times\cal{U}_{2})=
N_{X_{1}}(\cal{U}_{1})\cdot N_{X_{2}}(\cal{U}_{2})$.

Let us repeat in simpler terms the proof of (2.15) in this classical case:
\begin{pf}By definition, $N(\cal{U}_{i})=\card\cal{W}_{i}=N(\cal{W}_{i})$
where by construction $\cal{U}_{i}$ {\em refines} $\cal{W}_{i}$ in the natural
partial (pre--)order of open covers ($i=1,2$), hence obviously $\cal{U}_{1}
\times\cal{U}_{2}$ refines $\cal{W}_{1}\times\cal{W}_{2}$, and by (1.2,iv) we
know $N(\cal{U}_{1}\times\cal{U}_{2})\geq N(\cal{W}_{1}\times\cal{W}_{2})=
N(\cal{W}_{1})\cdot N(\cal{W}_{2})$, where the factorization for the $\cal{W
}_{i}$ is almost obvious here. The reverse inequality is rather trivial (see
(1.2,i)), as already mentioned before.
\end{pf}
\item Unfortunately, however, the ``superadditivity'' of the entropy (2.8)
repectively (2.11) for a tensor product $\cal{A}=\cal{A}_{1}\otimes\cal{A}_{
2}$ with factorizing $*$--endomorphism $\theta=\theta_{1}\otimes\theta_{2}$
($\theta_{i}:\cal{A}_{i}\to\cal{A}_{i}$), with respect to a product map
$\gamma=\gamma_{1}\otimes\gamma_{2}$ (respectively a product sequence
$\tau=\tau_{1}\otimes\tau_{2}$), cannot be proven any more, as the gap in
that proof of \cite{h2} still remains also here. 
\end{enumerate}
\end{ppp}

\section{Additional properties and Thomsen's modified approach}
We first prove additional properties of the entropy (2.11) for 
endomorphisms of AA C$^{*}$--algebras (with respect to a corresponding 
sequence 
$\tau$). Again, generally $\cal{A}\in\1$ denotes a unital C$^{*}$--algebra.
\begin{qqq}The following are non--commutative generalizations of additional
properties of the classical topological entropy (cf.\ in the Introduction I):
\begin{enumerate}
\item[(i)] Let $\1\in\cal{B}\subset\cal{A}$ be an AA $C^{*}$--subalgebra with
corresponding sequence $\tau=(\tau_{n})_{n\in\Bbb{N}}$, and $\theta$ be a
unital $*$--endomorphism of $\cal{A}$ such that $\theta(\cal{B})\subset\cal{B
}$. If there exists a sequence $\pi=(\pi_{k})_{k\in\Bbb{N}}$ of positive
unital maps $\pi_{k}:\cal{C}_{k}\to\cal{A}$ ($\cal{C}_{k}$ finite--dimensional
$C^{*}$) which renders also $\cal{A}$ AA, in such a way that
$$\exists N\in\Bbb{N}:\forall n\geqslant N,\enspace\tau_{n}(\cal{O}^{+}
(\cal{B}_{n}))\subseteq\pi_{n}(\cal{O}^{+}(\cal{C}_{n})),$$
then $\hbar_{\tau}(\theta\upharpoonright_{\cal{B}})\leqslant\hbar_{\pi}(\theta
)$.

Note that this is the case in particular if (2.11,i) is valid for $\tau$, 
where for $\pi\equiv\tau$ even equality holds.
\item[(ii)] Let $\cal{A}$ be AA with corresponding sequence $\tau=(\tau_{n})$ 
as in (2.11), $q:\cal{A}\to\cal{C}$ a surjective $*$--homomorphism 
onto a unital
$C^{*}$--algebra $\cal{C}$ and $\sigma:\cal{C}\to\cal{C}$ a unital 
$*$--endomorphism such that $\sigma\circ q=q\circ\theta$ for $\theta$ as in 
(i). 

Then also $\cal{C}$ is AA with the sequence $q(\tau)\equiv(q\circ\tau_{n}
)_{n\in\Bbb{N}}$, and $\hbar_{q(\tau)}(\sigma)\leqslant\hbar_{\tau}(\theta)$.

\item[(iii)] Let $\cal{A}_{1}\ni\1_{1}$ and $\cal{A}_{2}\ni\1_{2}$ be unital 
AA
$C^{*}$--algebras, with corresponding sequences $\tau^{1}=(\tau_{n}^{1})_{n\in
\Bbb{N}}$ and $\tau^{2}=(\tau_{n}^{2})_{n\in\Bbb{N}}$, respectively, where we
assume that {\em one} of the two conditions (2.11,i \& ii) is true for {\em
both} $(\cal{A}_{1},\tau^{1})$ and $(\cal{A}_{2},\tau^{2})$.

Let $\theta_{i}:\cal{A}_{i}\to\cal{A}_{i}$ be unital $*$--endomorphisms ($i=
1,2$) and $\theta_{1}\oplus\theta_{2}$ the direct sum endomorphism of $\cal{A
}_{1}\oplus\cal{A}_{2}$ (considered as unital C$^{*}$--algebra with unit
$\1_{1}\oplus\1_{2}$); then the latter is again AA with respect to the
sequence $\tau=\tau^{1}\oplus\tau^{2}\equiv(\tau_{n}^{1}\oplus\tau_{n}^{2})_{
n\in\Bbb{N}}$, and we have that $\hbar_{\tau}(\theta_{1}\oplus\theta_{2})=
\max\{\hbar_{\tau^{1}}(\theta_{1}),\hbar_{\tau^{2}}(\theta_{2})\}$.
\end{enumerate}
\end{qqq}
\begin{rrr}The respective proofs of (i) and (ii) are followed by some remarks:
\begin{enumerate}
\item[(i)] is obvious; and correspondingly, this result is not very powerful,
although not meaningless. Note, however, that the conditions (2.11,i \& ii)
on a sequence $\pi$ (as here, where in the case (2.11,ii) we have to restrict
to {\em increasing} range spaces: $\pi_{n}(\cal{C}_{n})\subseteq\pi_{n+1}
(\cal{C}_{n+1})$, at this point) are stable with respect to taking a {\em
sub\/}sequence, hence we can always relate $\tau_{n}$ to $\pi_{n}$ for the
same $n\geqslant N$ (if at all) as already done in (i). 

In the Abelian case $\cal{A}=C(X)$ and $\tau_{n},\pi_{n}\in\cal{P}_{1}(\cal{A}
)\enspace\forall n\in\Bbb{N}$ (cf. (2.12,1)), (i) means the following: Given
a factor system $(Y,S)$ of $(X,T_{\theta}\equiv T)$ as in (2.10), i.e.\ a
continuous surjection $\Phi:X\to Y$ such that $\Phi\circ T=S\circ\Phi$, and
calculating the entropy of $(Y,S)$ with respect to a sequence of open covers
$(\cal{U}_{n})_{n\in\Bbb{N}}$ of $Y$ which should be {\em cofinal} or at
least monotonically ``refining'' (cf.\ (2.12,2 \& 3)), then if we choose a
sequence $(\cal{V}_{n})_{n\in\Bbb{N}}$ of open covers of $X$ which should
again fulfill one of these two requirements, by putting $\cal{V}_{n}=\Phi^{-1}
(\cal{U}_{n})\vee\cal{W}_{n}\enspace(\forall n\geqslant N)$
for some sequence $(\cal{W}_{n})_{n\in\Bbb{N}}$ of $X$--covers, it follows
that $\limsup_{n\to\infty}h_{Y}(S,\cal{U}_{n})\leqslant\limsup_{n\to
\infty}h_{X}(T,\cal{V}_{n})$.

Clearly, this implies in particular that $h_{Y}(S)\leqslant h_{X}(T)$ (for the
full topological entropies, cf.\ \cite{dgs}), by choosing {\em cofinal} (and
refining) sequences for {\em both} $(\cal{U}_{n})_{n\in\Bbb{N}}$ in $Y$ and
$(\cal{W}_{n})_{n\in\Bbb{N}}$ in $X$; but for non--Abelian $\cal{A}$, we do
not (yet?) have these tools at hand, remember (2.12,3).

\item[(ii)] First, it is easy to see that both (2.11,i) or (2.11,ii) are 
preserved
by $q$: As $q$ is a $*$--homomorphism, clearly again $q\circ\tau_{n}(\cal{O}^{
+}(\cal{B}_{n}))\subset\cal{O}^{+}(\cal{C})$, and trivially $q\circ\tau_{n}
(\cal{O}^{+}(\cal{B}_{n}))\subset q\circ\tau_{n+1}(\cal{O}^{+}(\cal{B}_{n+1}))
\enspace\forall n\in\Bbb{N}$, by assumption (2.11,i).

If on the other hand $\overline{\bigcup_{n\in\Bbb{N}}\tau_{n}(\cal{B}_{n})}=
\cal{A}$ by (2.11,ii), it follows obviously from the automatic continuity
and the surjectivity of $q$ that also $\overline{\bigcup_{n\in\Bbb{N}}q\circ
\tau_{n}(\cal{B}_{n})}=\cal{C}$. So, the result follows from the definitions
involved in $\hbar_{\tau}(\theta)$ as in (2.12) and the fact that $\Bar N(q( 
\alpha))\leqslant\Bar N(\alpha)\enspace\forall\alpha\in\cal{O}(\cal{A})$, cf.\
(2.3,iv).\qed

Again back in the Abelian case $\cal{A}=C(X)$, the meaning of (ii) is that
for a closed, $T$--invariant subset $Y\subseteq X$ such that $T(Y)\subseteq 
Y$, and for a cofinal(--refining) sequence of open covers $(\cal{U}_{n})_{n
\in\Bbb{N}}$ for $X$ with resulting sequence $(\cal{V}_{n})_{n\in\Bbb{N}}$
of open covers $\cal{V}_{n}=\cal{U}_{n}\cap Y$ (symbolically, but clear) for
$Y$, we have $\lim(\sup)_{n\to\infty}h_{Y}(T\restriction_{Y},\cal{V}_{n})
\leqslant\lim(\sup)_{n\to\infty}h_{X}(T,\cal{U}_{n})$.

\item[(iii)] It is clear that each one of (2.11,i) and (2.11,ii) is preserved
by the direct sum; and it follows from (ii) that $\max\{\hbar_{\tau^{1}}(
\theta_{1}),\hbar_{\tau^{2}}(\theta_{2})\}\leqslant\hbar_{\tau}(\theta_{1}
\oplus\theta_{2})$, where the $q_{i}$ as used in (ii) is given by the 
respective canonical projection onto the respective direct summand $\cal{A}_{i
}$. For completeness it should be noted for the purists that here actually
$q_{i}(\tau)=(q_{i}\circ(\tau_{n}^{1}\oplus\tau_{n}^{2}):\cal{B}_{n}^{1}
\oplus\cal{B}_{n}^{2}\to\cal{A}_{i}$), but of course $q_{i}\circ
(\tau_{n}^{1}\oplus\tau_{n}^{2})(\cal{B}_{n}^{1}\oplus\cal{B}_{n}^{2})=
\tau_{n}^{i}(\cal{B}_{n}^{i})$ anyway ($i=1,2$).

To prove the reverse inequality, let us assume for the sake of notational
simplicity that (2.11,i) is fulfilled for both $(\cal{A}_{i},\tau^{i})\enspace
(i=1,2)$, such that the general $\limsup$ in definition (2.11) of $\hbar_{
\tau^{i}}(\theta_{i})$ is actually an increasing limit. It will then be clear
at the end of the proof that in the other case (2.11,ii) it is sufficient
to choose suitable subsequences in the subsequent.

Now fix $n\in\Bbb{N}$ and also $k\in\Bbb{N}$ and use Def.\ (2.5,ii) for
$\theta=\theta_{1}\oplus\theta_{2},\enspace\tau=(\tau_{n})_{n\in\Bbb{N}}$ to
get 
$$\frac{1}{k}\Bar H(\tau_{n},\theta\circ\tau_{n},\dots,\theta^{k-1}\circ
\tau_{n})=\frac{1}{k}\log\Bar N(\tau_{n}(\alpha_{1})\Vec\vee\theta\circ\tau_{n
}(\alpha_{2})\Vec\vee\dots\Vec\vee\theta^{k-1}\circ\tau_{n}(\alpha_{k})),$$
where $\alpha_{j}=\alpha_{j}(k,n)\in\cal{O}^{+}(\cal{B}_{n}^{1}\oplus
\cal{B}_{n}^{2})$ with corresponding direct summands $\alpha_{j}^{i}\in\cal{O
}^{+}(\cal{B}_{n}^{i}):\alpha_{j}=\alpha_{j}^{1}\oplus\alpha_{j}^{2}$
(symbolically, but clear). It is easily seen that from the right hand side we
get:\begin{multline*}
\Bar N(\tau_{n}(\alpha_{1})\Vec\vee\theta\circ\tau_{n}(\alpha_{2})\Vec\vee
\dots\Vec\vee\theta^{k-1}\circ\tau_{n}(\alpha_{k})) \leqslant \\ \leqslant
\Bar N(\tau_{n}^{1
}(\alpha_{1}^{1})\Vec\vee\theta_{1}\circ\tau_{n}^{1}(\alpha_{2}^{1})\Vec\vee
\dots\Vec\vee\theta_{1}^{k-1}\circ\tau_{n}^{1}(\alpha_{k}^{1}))+ \\ + 
\Bar N(\tau_{n}^{2}(\alpha_{1}^{2})\Vec\vee\theta_{2}\circ\tau_{n}^{2}(
\alpha_{2}^{2})\Vec\vee\dots\Vec\vee\theta_{2}^{k-1}\circ\tau_{n}^{2}(
\alpha_{k}^{2}))\equiv\end{multline*}
$\equiv N_{1}(k,n)+N_{2}(k,n).$ On the other hand, we know, again by (2.5,ii),
that each
$$\log N_{i}(k,n)\leqslant \Bar H(\tau_{n}^{i},\theta_{i}\circ\tau_{n}^{i},
\dots,\theta_{i}^{k-1}\circ\tau_{n}^{i})\equiv H_{i}(k,n),$$
for $i=1,2$. Together it follows by Definition (2.8) as $k\to\infty$ on the
very first left hand side above (making a choice, without loss of generality):
$$\hbar(\theta,\tau_{n})\leqslant\frac{1}{k}H_{1}(k,n)+\frac{1}{k}\log(1+
e^{[H_{2}(k,n)-H_{1}(k,n)]}),$$
where we used that the limit $k\to\infty$ in (2.8) is actually the infimum.
To repeat it, by (2.8) and (2.12) we know $\hbar_{\tau^{i}}(\theta_{i})=
\lim_{n\to\infty}\lim_{k\to\infty}\frac{1}{k}H_{i}(k,n)$; and first assuming 
that $\hbar_{\tau^{1}}(\theta_{1})\ne\hbar_{\tau^{2}}(\theta_{2})$, we choose
(again without loss of generality) that $\hbar_{\tau^{1}}(\theta_{1})\gneqq
\hbar_{\tau^{2}}(\theta_{2})$. Then it is easily seen that there exists $N\in
\Bbb{N}$ such that $\forall n\geqslant N,\enspace\forall k\geqslant K(n)$ we
have $H_{1}(k,n)\geqslant H_{2}(k,n)$. Thus $\forall n\geqslant N$ the right 
hand side above tends to $\hbar(\theta_{1},\tau_{n}^{1})$ as $k\to\infty$.
In the other case $\hbar_{\tau^{1}}(\theta_{1})=\hbar_{\tau^{2}}(\theta_{2})$
it follows from $\log(1+x)\leqslant\log x+1\enspace\forall x\geqslant 1$
that we can still bound $\hbar(\theta,\tau_{n})$ by $\max\{\hbar(\theta_{1},
\tau_{n}^{1}),\hbar(\theta_{2},\tau_{n}^{2})\}$ (using both choices for the
stronger inequality before, alternatively). In both cases, performing the 
limit $n\to\infty$ first on the right hand, then on the left hand side yields
the result $\hbar_{\tau}(\theta)\leqslant\hbar_{\tau^{1}}(\theta_{1})$.\qed
\end{enumerate}
\end{rrr}
As already noted in the Introduction I, the three additional properties (3.1)
are the direct analogues of the (remaining) basic properties proved by
Thomsen \cite{t} for his slightly modified topological entropy for local
C$^{*}$--algebras; in particular (i), (ii) respectively (iii) of (3.1)
corresponds to (ii), (iii) of Theorem (1.2) in \cite{t} respectively to
Proposition (1.3) in \cite{t}.

At this point the reader will ask also for the analogues here of the
``modest'' coninuity properties proved by Thomsen for his modified entropy
(section 2 in \cite{t}), and in the following we will answer this question
to the affirmative; however, we are able to do so only with a severe
restriction of the theory discussed so far, which is of great independent
interest, though (particularly concerning the physical interpretation): 

It turns out (see (3.5,iv) and the subsequent remarks below) that the 
non--commutative freedom in Def.\ (2.5,ii), admitting {\em different} and
{\em general} $\alpha_{k}\in\cal{O}^{+}(\cal{B}_{k})$ for $k=1,\dots,n$ (cf.\
the discussion in (2.12,5)) is too much when trying to control $\hbar(\theta,
\gamma)$ as in (2.8) via any norm--continuity along the lines of \cite{t}.
But on the other hand, we have to consider it as inadequate to completely
abandon that freedom in the $\cal{B}_{k}$'s, what is done in \cite{t} by
taking the joint entropy of the images (under the iterated $*$--endomorphism
$\theta$) of one and the same ``partition'' $\alpha$, i.e.\ something of the
form $\alpha\vee\theta(\alpha)\vee\dots\vee\theta^{n}(\alpha)$.

Both for (mathematically) conceptual and for physical reasons, we want to
maintain the ``non--commutative open covers'' represented by the positive
unital maps $\gamma:\cal{B}\to\cal{A}$, with the finite--dimensional
C$^{*}$--algebra $\cal{B}$ representing some ``operational'' 
quantum--mechanical (e.g.\ spin) degrees of freedom being transformed by
$\theta$ (in the ``macroscopic realization'' represented by $\gamma$).

Surprisingly, now, it is {\em exactly} the ``traditional'' (or more
conventional) notion of a finite ``measurement'' performed on this degree of
freedom $\cal{B}$, i.e.\ a partition of the unit $\1_{\cal{B}}$ by mutually
orthogonal projections, which has to be substituted for the more general
$\alpha\in\cal{O}^{+}(\cal{B})$ used so far (but again, for {\em each} $k$
in (2.5,ii) separately), in order to make the continuity arguments work.
As remarked in more detail below (3.3), not too much of the other desirable
properties of the entropy functionals gets lost due to this restriction; and
for the hitherto {\em computable} examples, that modification makes not any
difference (see section IV). 

In this sense, the following can be viewed as an alternative ``theory within
the theory''; and at this stage we can still leave it to the personal taste
of the reader as to which definitions or properties should be preferred.
\begin{sss}We define the following alternatives (i) for (2.1,ii), (ii) for
(2.5) and (iii) for (2.8):
\begin{enumerate}
\item[(i)] For a finite--dimensional unital C$^{*}$--algebra $\cal{B}\ni
\1_{\cal{B}}$, we denote by $\cal{O}\cal{P}_{1}(\cal{B})\subset\cal{O}_{2}^{+}
(\cal{B})$ the set of all partitions of the unit $\1_{\cal{B}}$ by mutually
orthogonal projections: 
$$\cal{O}\cal{P}_{1}(\cal{B})\ni\beta=\{p_{i}=p_{i}^{*}=p_{i}^{2}|p_{i}p_{j}=
0\enspace\forall i\ne j,\enspace\sum_{i=1}^{\card\beta}p_{i}=\1_{\cal{B}}\}
,$$ where it follows trivially that $\card\beta\leqslant 
D(\cal{B})$ as defined in (1.4).
\item[(ii)] In the situation of (2.5,ii), the ``partition'' (or 
``projection'') entropy of $(\gamma_{1},\dots,\gamma_{n})$ is defined as
$$\Bar H_{P}(\gamma_{1},\dots,\gamma_{n})=\max_{\{(\beta_{1},\dots,\beta_{n})
\}}\log\Bar N(\gamma_{1}(\beta_{1})\Vec\vee\dots\Vec\vee\gamma_{n}(\beta_{n}))
,$$ where $\beta_{k}\in\cal{O}\cal{P}_{1}(\cal{B}_{k})\enspace\forall k=1,
\dots,n$; and with the {\em same} additional restriction as in (1.5,ii): For
$\gamma_{k}\approxeq\gamma_{\ell}\enspace(k<\ell)$, fix $\beta_{\ell}\equiv
\{\1_{\ell}\}$.
\item[(iii)] In the situation of (2.8), the P--entropy of $\theta$ with 
respect to $\gamma$ is $\hbar_{P}(\theta,\gamma)=\lim_{n\to\infty}\frac{1}{n} 
\Bar H_{P}(\gamma,\theta\circ\gamma,\dots,\theta^{n-1}\circ\gamma)$, which is
again the infimum of the sequence.
\end{enumerate}
\end{sss}
\begin{ttt} Note first that obviously $\Bar H_{P}(\gamma_{1},\dots,\gamma_{n})
\leqslant\Bar H(\gamma_{1},\dots,\gamma_{n})$ and consequently $\hbar_{P}(
\theta,\gamma)\leqslant\hbar(\theta,\gamma)$. Nervertheless, the following
properties of the right hand side functionals remain valid for the 
P--entropies: (2.6,2), (2.7;ii, iii, v) and also (2.7,iv), where for the
latter we still have to use (2.4) (with $\alpha\in\cal{O}(\cal{A})$!) to get
the inequality; hence also (2.9;i, iii, iv, v) and all of (2.13) still hold. 
Unfortunately, (2.7,i) respectively (2.9,ii) hold only for 
$*$--{\em homomorphisms} $\theta_{k}$ respectively $\gamma_{1}$, instead of
general positive unital maps (without the P--restriction).

In (2.11,i) respectively (2.14,i), the $\cal{O}^{+}(\cal{B}_{n})$ respectively
$\cal{O}^{+}(\cal{B})$ have to be replaced by $\cal{O}\cal{P}_{1}(\cal{B}_{n})
$ respectively $\cal{O}\cal{P}_{1}(\cal{B})$, rendering the respective
condition more restrictive. Note, however, that the non--trivial example
(2.16,2) for (2.14,i) is of the form with $\beta'\in\cal{O}\cal{P}_{1}(
\cal{B})$. Apart from that, (2.15) and the following discussion remain fully
valid. Finally, (3.1) remains valid with suitable changes of $\cal{O}^{+}(
\cal{B})$'s into $\cal{O}\cal{P}_{1}(\cal{B})$'s.
\end{ttt}
\begin{uuu} The following definitions are adaptions of the definitons in
\cite{t} for our needs:
\begin{enumerate}
\item[(i)] For $\varepsilon\in[0,1),\enspace\varepsilon\ne 1$, we define the
set of ``$\varepsilon$--operator covers'' $\cal{O}(\cal{A},\varepsilon)=\{
\alpha\in\cal{O}(\cal{A})|\forall\omega\in S_{\cal{A}}\enspace\exists A\in
\alpha:\omega(A^{*}A)>\varepsilon\}$.

Note that $\cal{O}(\cal{A})=\cal{O}(\cal{A},0)\supseteq\cal{O}(\cal{A},
\varepsilon)\supseteq\cal{O}(\cal{A},\delta)$ for $\varepsilon\leqslant\delta
<1$.
\item[(ii)] For $\alpha\in\cal{O}(\cal{A},\varepsilon)$, define
$\Bar N(\alpha,\varepsilon)=\min\{\card\alpha'|\alpha'\subseteq\alpha,\enspace
\alpha'\in\cal{O}(\cal{A},\varepsilon)\}$.

Note that $\Bar N(\alpha)=\Bar N(\alpha,0)\leqslant \Bar N(\alpha,\delta) 
\leqslant\Bar N(\alpha,\varepsilon)$ for
$0<\delta<\varepsilon<1$ and $\alpha\in\cal{O}(\cal{A},\varepsilon)$.
\item[(iii)] Again (cf.\ definition (2.1)), we can define also $\cal{O}^{+}
(\cal{A},\varepsilon)=\{\alpha\in\cal{O}^{+}(\cal{A})|\forall\omega\in S_{
\cal{A}}\enspace\exists A\in\alpha:\omega(A)>\varepsilon\}$.

Note that it follows easily from the Schwartz inequality for the state 
$\omega$
(in particular, from $\omega(A^{2})\geqslant\omega(A)^{2}\enspace\forall A\in
\cal{A}^{+}$) that $\cal{O}^{+}(\cal{A},\varepsilon)\subset\cal{O}(\cal{A},
\varepsilon^{2})\enspace\forall\varepsilon<1$ (note that there is no $^{+}$ on
the right hand side $\cal{O}$!).
\end{enumerate}
\end{uuu}
\begin{vvv} The following properties are, in the case of (i) and (ii), again
adaptions of the corresponding properties proved in \cite{t}, and in the case
of (iii) and (iv) additional properties important in our context here.
\begin{enumerate}
\item[(i)] $\Bar N(\alpha\Vec\vee\beta,\varepsilon\delta)\leqslant\Bar N(
\alpha,\varepsilon)\cdot\Bar N(\beta,\delta)$ for $\alpha\in\cal{O}(\cal{A},
\varepsilon),\enspace\beta\in\cal{O}(\cal{A},\delta)$.
\item[(ii)] $\Bar N(\theta(\alpha),\varepsilon)\leqslant\Bar N(\alpha,
\varepsilon)$ for a unital $*$--endomorphism $\theta$ of $\cal{A}$ and $\alpha
\in\cal{O}(\cal{A},\varepsilon)$, whence also $\theta(\alpha)\in\cal{O}(
\cal{A},\varepsilon)$.
\item[(iii)] For a positive unital map $\gamma:\cal{B}\to\cal{A}$, and $\beta
\in\cal{O}^{+}(\cal{B},\varepsilon)$, obviously also $\gamma(\beta)\in
\cal{O}^{+}(\cal{A},\varepsilon)$, and $\Bar N(\gamma(\beta),\varepsilon^{2})
\leqslant\Bar N(\beta,\varepsilon^{2})$.
\item[(iv)] Let $\cal{B}$ be a finite--dimensional unital $C^{*}$--algebra.
For $\varepsilon<D(\cal{B})^{-1}$ with $D(\cal{B})$ as in (1.4), the set of
all ``projection--partitions'' $\cal{O}\cal{P}_{1}(\cal{B})\subset\cal{O}^{+}
(\cal{B},\varepsilon)$ is included as shown (where for these {\em 
projections\/}, actually also $\cal{O}\cal{P}_{1}(\cal{B})\subset\cal{O}(
\cal{B},\varepsilon)$; and in other words, more explicitly: for $\beta\in
\cal{O}\cal{P}_{1}(\cal{B}),\enspace\beta=\{p_{i}\},\enspace\forall\omega\in
S_{\cal{B}}\enspace\exists p_{i}:\omega(p_{i})>\varepsilon$).
\end{enumerate}
\end{vvv}
\begin{pf} The proof of (ii) and (iii) is rather trivial, and also (iv) is
left as an easy exercise for the reader. For the sake of completeness, we
reproduce the simple proof of (i) from \cite{t}:

For $\alpha'\subset\alpha$ respectively $\beta'\subset\beta$ such that
$\Bar N(\alpha,\varepsilon)=\card\alpha$ respectively $\Bar N(\beta,\delta)=
\card\beta$, obviously $\alpha'\Vec\vee\beta'$ is a subset of $\alpha\Vec\vee
\beta$; but it is even such that $\alpha'\Vec\vee\beta'\in\cal{O}(\cal{A},
\varepsilon\delta)$: For $\omega\in S_{\cal{A}},\enspace A\in\alpha'$ such 
that $\omega(A^{*}A)>\varepsilon$, we get the state $\varphi_{A}=\omega(A^{*}
\centerdot A)\omega(A^{*}A)^{-1}\in S_{\cal{A}}$  and thus there exists a
$B\in\beta':\varphi_{A}(B^{*}B)>\delta$. Then $(BA)\in\alpha'\Vec\vee
\beta'$ and $\omega((BA)^{*}BA)>\varepsilon\delta$.
\end{pf}

Note, however, that for $\cal{O}^{+}(\cal{B})$ instead of $\cal{O}\cal{P}_{1}
(\cal{B})$ (as we had used it in section II and (3.1)), we could not prove
anything like (iv) again: $\forall\varepsilon>0$, it is {\em not} true that
$\cal{O}^{+}(\cal{B})\subset\cal{O}^{+}(\cal{B},\varepsilon)$ (only the 
trivial converse inclusion holds), but even also {\em not} that $\cal{O}_{1}^{
+}(\cal{B})\subset\cal{O}^{+}(\cal{B},\varepsilon)$: The {\em trivial}
example is $\beta\in\cal{O}_{1}^{+}(\cal{B})$ given by $\beta=\{\delta_{i}
\cdot\1_{\cal{B}}|\delta_{i}\ne\delta_{j},\enspace\delta_{i}<\varepsilon
\enspace\forall i,\enspace\sum_{i}\delta_{i}=\1_{\cal{B}}\}\not\in\cal{O}^{+}
(\cal{B},\varepsilon)$; but of course, this would give only zero entropies
((2.5,ii), for example). 

It is not hard to realize, however, that for any $\alpha\in\cal{O}^{+}_{(1)}
(\cal{B})$ maximizing ``some'' entropy (as in (2.5,ii)), and particularly for
one argument $\gamma:\cal{B}\to\cal{A}$ where we have to have an analogue of
Lemma (2.4) (as we will get it in Remark (3.7) below, here by the automatic
bound on $\card[\beta\in\cal{O}\cal{P}_{1}(\cal{A})]$ by definition), there
would be no guaranty that $\alpha\in\cal{O}^{+}(\cal{B},\varepsilon)$ for any
fixed $\varepsilon>0$.

This is the ``obstruction'' to using the norm--continuity arguments as in
\cite{t} with the definition from section II, as we had announced it before
(3.2) above.
\begin{www} The following are again the analogous definitions to (2.5) and
(2.8), now using the ``$\varepsilon$--covering'' definitions from above:
\begin{enumerate}
\item[(i)] Again in the situation of (2.5,ii), let $\varepsilon<\min\{D(\cal{B
}_{k})^{-1}|k=1,\dots,n\}$ be fixed, then we can define (by (3.5))
$$\Bar H_{P}^{\varepsilon}(\gamma_{1},\dots,\gamma_{n})=\max_{\{(\beta_{1},
\dots,\beta_{n})\}}\log\Bar N(\gamma_{1}(\beta_{1})\Vec\vee\dots\Vec\vee
\gamma_{n}(\beta_{n}),\varepsilon^{(2n)}),$$
where $\beta_{k}\in\cal{O}\cal{P}_{1}(\cal{B}_{k})\enspace\forall k$, with the
same restriction as in (2.5,ii) or (3.2,ii).

Note that we have
$\Bar H_{P}(\gamma_{1},\dots,\gamma_{n})=\Bar H_{P}^{0}(\gamma_{1},
\dots,\gamma_{n})\leqslant \Bar H_{P}^{\delta}(\gamma_{1},\dots,\gamma_{n})
\leqslant\Bar H_{P}^{\varepsilon}(\gamma_{1},\dots,\gamma_{n})$ for $0<\delta
<\varepsilon$ (using the $\max$ in above definition, this is really obvious).

\item[(ii)] $\hbar_{P}^{\varepsilon}(\theta,\gamma)=\lim_{n\to\infty}
\frac{1}{n}\Bar H_{P}^{\varepsilon}(\gamma,\theta\circ\gamma,\dots,
\theta^{n-1}\circ\gamma)$ for a $*$--endomorphism $\theta$ of $\cal{A}$
with respect to $\gamma$ as in (2.8) has again the analogous properties. In
particular, the limit is the {\em infimum}.
\end{enumerate}
\end{www}
\begin{xxx} For a single map $\gamma:\cal{B}\to\cal{A}$ as before, it follows
from (3.5,iii) that $\Bar H_{P}^{\varepsilon}(\gamma)\leqslant\Bar H_{P}^{
\varepsilon}(\cal{B})=\log D(\cal{B})$, where the right hand side equality
is rather trivial for $\varepsilon<D(\cal{B})^{-1}$ as assumed (and where we
used again the general notation as in (2.6,2)).

In the following results (3.8)--(3.10), we can now in turn ``imitate''
Thomsen's results on continuity properties (section 2 in \cite{t}).
\end{xxx}
\begin{yyy} For $\gamma:\cal{B}\to\cal{A}$ with $\cal{B}$ finite--dimensional,
the map $[0,D(\cal{B})^{-1})\ni\varepsilon\mapsto\hbar_{P}^{\varepsilon}(
\theta,\gamma)$ is non--decreasing and continuous from the {\em right}; 
in particular, $\lim_{\varepsilon\to 0+}\hbar_{P}^{\varepsilon}(\theta,\gamma)
=\hbar_{P}(\theta,\gamma)$.
\end{yyy}
\begin{pf} It follows from the remark in (3.6,i) above that $\hbar_{P}^{
\varepsilon}(\theta,\gamma)$ is non--decreasing in $\varepsilon$. To prove
that $\lim_{\varepsilon\to\varepsilon_{0}+}\hbar_{P}^{\varepsilon}(\theta,
\gamma)=\hbar_{P}^{\varepsilon_{0}}(\theta,\gamma)$ for $0\leqslant
\varepsilon_{0}<\varepsilon<D(\cal{B})^{-1}$, let $\delta>0$ and choose
$n\in\Bbb{N}$ such that $\frac{1}{n}\Bar H_{P}^{\varepsilon_{0}}(\gamma,
\theta\circ\gamma,\dots,\theta^{n-1}\circ\gamma)\leqslant\hbar_{P}^{
\varepsilon_{0}}(\theta,\gamma)+\delta$. For all $\varepsilon\geqslant
\varepsilon_{0}$ sufficiently close to $\varepsilon_{0}$, 
\begin{multline*}\Bar H_{P}^{\varepsilon}(\gamma,\theta\circ\gamma,\dots,
\theta^{n-1}\circ\gamma)=\log\Bar N(\gamma(\beta_{1})\Vec\vee\dots\Vec\vee
\theta^{n-1}\circ\gamma(\beta_{n}),\varepsilon^{(2n)})=\\=\log\Bar N(\gamma
(\beta_{1})\Vec\vee\dots\Vec\vee\theta^{n-1}\circ\gamma(\beta_{n}),
\varepsilon_{0}^{(2n)})\leqslant\Bar H_{P}^{\varepsilon_{0}}(\gamma,\theta
\circ\gamma,\dots,\theta^{n-1}\circ\gamma)\end{multline*}
(and hence also equality on the right; with some $\beta_{k}\in\cal{O}\cal{P}_{
1}(\cal{B}_{k}),\enspace k=1,\dots,n$). Thus $\hbar_{P}^{\varepsilon}(\theta,
\gamma)\leqslant\hbar_{P}^{\varepsilon_{0}}(\theta,\gamma)+\delta$.
\end{pf}
\begin{zzz} Let $\gamma:\cal{B}\to\cal{A}$ be a positive unital map, and let
$(\gamma_{n})_{n\in\Bbb{N}}$ be a sequence of such maps $\gamma_{n}:\cal{B}
\to\cal{A}$ ($\cal{B}$ finite--dimensional) such that $\lim_{n\to\infty}
\frac{1}{n}\log\|\gamma-\gamma_{n}\|=-\infty$ (in other words, $\|\gamma-
\gamma_{n}\|^{\frac{1}{n}}\to 0$); and such that $\theta$ does {\em not} act
periodically on any of the $\gamma,\gamma_{n}\enspace(n\in\Bbb{N})$. Then
(\/$\forall\varepsilon,\delta_{n}<D(\cal{B})^{-1}$):
\begin{enumerate}
\item[(i)] $\sup_{\delta<\varepsilon}\hbar_{P}^{\delta}(\theta,\gamma)
\leqslant\liminf_{n\to\infty}\frac{1}{n}\Bar H_{P}^{\varepsilon}(\gamma_{n},
\theta\circ\gamma_{n},\dots,\theta^{n-1}\circ\gamma_{n})$, \em{and}\newline
$\limsup_{n\to\infty}\frac{1}{n}\Bar H_{P}^{\varepsilon}(\gamma_{n},\theta
\circ\gamma_{n},\dots,\theta^{n-1}\circ\gamma_{n})\leqslant\hbar_{P}^{
\varepsilon}(\theta,\gamma)$. 

In particular, $\hbar_{P}(\theta,\gamma)=\lim_{\varepsilon\to 0}(\limsup_{
n\in\Bbb{N}}\frac{1}{n}\Bar H_{P}^{\varepsilon}(\gamma_{n},\theta\circ\gamma_{
n},\dots,\theta^{n-1}\circ\gamma_{n}))$.
\item[(ii)] $\hbar_{P}(\theta,\gamma)=\lim_{n\to\infty}\frac{1}{n}\Bar H_{P}^{
\delta_{n}}(\gamma_{n},\theta\circ\gamma_{n},\dots,\theta^{n-1}\circ\gamma_{n}
)$ for $\delta_{n}=c\cdot\|\gamma-\gamma_{n}\|^{\frac{1}{(2n)}}$, for any 
$c>1,\enspace c\ne 1$.
\end{enumerate}
\end{zzz}
\begin{pf} Define the generalized Hausdorff distance (metric) on $\cal{O}(
\cal{A})\ni\alpha,\beta$ by
$$D(\alpha,\beta)=\max\{\max_{A\in\alpha}\min_{B\in\beta}\|A-B\|,\enspace
\max_{B\in\beta}\min_{A\in\alpha}\|A-B\| \}.$$
\begin{enumerate}
\item[(i)] The following general inequality in the situation of (i), with
{\em arbitrary} $\alpha_{k}\in\cal{O}^{+}(\cal{B})\enspace(\forall
k=1,\dots,n)$ is easy to deduce:\begin{multline*}
D(\gamma(\alpha_{1})\Vec\vee\theta\circ\gamma(\alpha_{2})\Vec\vee\dots\Vec
\vee\theta^{n-1}\circ\gamma(\alpha_{n}),\gamma_{n}(\alpha_{1})\Vec\vee\theta
\circ\gamma_{n}(\alpha_{2})\Vec\vee\dots\Vec\vee\theta^{n-1}\circ\gamma_{n}
(\alpha_{n})) \\ \leqslant n\cdot\|\gamma-\gamma_{n}\|. 
\end{multline*} 
Now let $0<\delta_{1}<\varepsilon<\delta_{2}<D(\cal{B})^{-1}$, all 
inequalities strict. By assumption, $\|\gamma-\gamma_{n}\|\leqslant
\frac{1}{2n}\delta_{1}^{(2n)}$ for all sufficiently large $n$. It follows that
for $\omega\in S_{\cal{A}}$ a state, and a suitably chosen (see below)
$n$--tuple $(\alpha_{1},\dots,\alpha_{n})\in\cal{O}\cal{P}_{1}(\cal{B})^{n}$,
in particular, for any $A\in\gamma_{n}(\alpha_{1})\Vec\vee\dots\Vec\vee
\theta^{n-1}\circ\gamma_{n}(\alpha_{n})$ such that $\omega(A^{*}A)>
\varepsilon^{(2n)}$, there exists $B\in\gamma(\alpha_{1})\Vec\vee\dots\Vec\vee
\theta^{n-1}\circ\gamma(\alpha_{n})$ such that $\omega(B^{*}B)>\varepsilon^{
(2n)}-\delta_{1}^{(2n)}>\delta_{1}^{(2n)}$ for sufficiently large $n$.

Hence, by the above definitions:\begin{multline*}
\frac{1}{n}\Bar H_{P}^{\delta_{1}}(\gamma,\theta\circ\gamma,\dots,\theta^{n-1}
\circ\gamma)= \\ =
\frac{1}{n}\log\Bar N(\gamma(\alpha_{1})\Vec\vee\theta\circ
\gamma(\alpha_{2})\Vec\vee\dots\Vec\vee\theta^{n-1}\circ\gamma(\alpha_{n}),
\delta_{1}^{(2n)})\leqslant \\ \leqslant\frac{1}{n}\log\Bar N(\gamma_{n}
(\alpha_{1})\Vec\vee\theta\circ\gamma_{n}(\alpha_{2})\Vec\vee\dots\Vec\vee
\theta^{n-1}\circ\gamma_{n}(\alpha_{n}),\varepsilon^{(2n)})\leqslant \\
\leqslant\frac{1}{n}
\Bar H_{P}^{\varepsilon}(\gamma_{n},\theta\circ\gamma_{n},\dots,\theta^{n-1}
\circ\gamma_{n})\end{multline*}
for all sufficiently large $n$, where the last inequality uses the 
non--periodicity assumption on $\theta$ with respect to $\gamma_{n}$.

Similarly, we can choose a tuple $(\beta_{1},\dots,\beta_{n})\in\cal{O}\cal{P
}_{1}(\cal{B})^{n}$ such that \begin{multline*}
\frac{1}{n}\Bar H_{P}^{\varepsilon}(\gamma_{n},\theta\circ\gamma_{n},\dots,
\theta^{n-1}\circ\gamma_{n})= \\ =
\frac{1}{n}\log\Bar N(\gamma_{n}(\beta_{1})
\Vec\vee\theta\circ\gamma_{n}(\beta_{2})\Vec\vee\theta^{n-1}\circ\gamma_{n}
(\beta_{n}),\varepsilon^{(2n)})\leqslant \\ \leqslant\frac{1}{n}\Bar H_{P}^{
\delta_{2}}(\gamma,\theta\circ\gamma,\dots,\theta^{n-1}\circ\gamma),
\end{multline*} for all sufficiently large $n$, now by the non--periodicity
assumption on $\theta$ with respect to $\gamma$.

Hence we have $\hbar_{P}^{\delta_{1}}(\theta,\gamma)\leqslant\liminf_{n\to
\infty}\frac{1}{n}\Bar H_{P}^{\varepsilon}(\gamma_{n},\theta\circ\gamma_{n},
\dots,\theta^{n-1}\circ\gamma_{n})$, and on the other hand, $\limsup_{n\to
\infty}\frac{1}{n}\Bar H_{P}^{\varepsilon}(\gamma_{n},\theta\circ\gamma_{n},
\dots,\theta^{n-1}\circ\gamma_{n})\leqslant\hbar_{P}^{\delta_{2}}(\theta,
\gamma)$. Since $\delta_{1}<\varepsilon$ was arbitrary, and on the other hand
$\lim_{\delta_{2}\to\varepsilon+}\hbar_{P}^{\delta_{2}}(\theta,\gamma)=
\hbar_{P}^{\varepsilon}(\theta,\gamma)$ by Lemma (3.8), we get the desired
inequalities.
\item[(ii)] Let $\varepsilon>0$ be fixed. By Lemma (3.8), there exists $\delta
>0$ such that $\hbar_{P}^{\delta}(\theta,\gamma)\leqslant\hbar_{P}(\theta,
\gamma)+\varepsilon$. For a chosen $n$--tuple $(\alpha_{1},\dots,\alpha_{n})
\in\cal{O}\cal{P}_{1}(\cal{B})^{n}$, a state $\omega\in S_{\cal{A}}$ and
$A\in\gamma_{n}(\alpha_{1})\Vec\vee\theta\circ\gamma_{n}(\alpha_{2})\Vec\vee
\dots\Vec\vee\theta^{n-1}\circ\gamma_{n}(\alpha_{n})$ such that $\omega(A^{*}
A)>\delta_{n}^{(2n)}$, there exists $B\in
\gamma(\alpha_{1})\Vec\vee\theta\circ\gamma(\alpha_{2})\Vec\vee\dots\Vec\vee
\theta^{n-1}\circ\gamma(\alpha_{n})$ such that $\omega(B^{*}B)>\delta_{n}^{
(2n)}-2n\|\gamma-\gamma_{n}\|>0$ for all sufficiently large $n$, by definition
of $\delta_{n}$ (note at this point that (ii) still holds for $\delta_{n}=
c_{n}\|\gamma-\gamma_{n}\|^{\frac{1}{(2n)}}$ with a {\em sequence} $(c_{n})_{
n\in\Bbb{N}},\enspace c_{n}\to 1$ but $c_{n}>(2n)^{\frac{1}{(2n)}}$ for all
sufficiently large $n$; for example $c_{n}=(2n)^{\frac{1}{(2n)}}+\left( 
\frac{1}{2n} \right)^{2n}$!). 

On the other hand, for a chosen tuple $(\beta_{1},\dots,\beta_{n})\in\cal{O}
\cal{P}_{1}(\cal{B})^{n}$ and a state $\omega\in S_{\cal{A}}$, take $A\in
\gamma(\beta_{1})\Vec\vee\theta\circ\gamma(\beta_{2})\Vec\vee\dots\Vec\vee
\theta^{n-1}\circ\gamma(\beta_{n})$ such that $\omega(A^{*}A)>(2\delta_{n})^{
2n}$; then there exists $B\in\gamma_{n}(\beta_{1})\Vec\vee\theta\circ\gamma_{
n}(\beta_{2})\Vec\vee\dots\Vec\vee\theta^{n-1}\circ\gamma_{n}(\beta_{n})$
such that $\omega(B^{*}B)>(2\delta_{n})^{2n}-2n\|\gamma-\gamma_{n}\|>
\delta_{n}^{(2n)}+(\delta_{n}^{(2n)}-2n\|\gamma-\gamma_{n}\|)>\delta_{n}^{(2n)
}$ for all sufficiently large $n$. Thus \begin{multline*}
\Bar H_{P}(\gamma,\theta\circ\gamma,\dots,\theta^{n-1}\circ\gamma)\equiv \\
\equiv\Bar H_{P}^{0}(\gamma,\theta\circ\gamma,\dots,\theta^{n-1}\circ\gamma)
\leqslant \Bar H_{P}^{\delta_{n}}(\gamma_{n},\theta\circ
\gamma_{n},\dots,\theta^{n-1}\circ\gamma_{n})\leqslant \\ \leqslant
\Bar H_{P}^{(2\delta_{n})}
(\gamma,\theta\circ\gamma,\dots,\theta^{n-1}\circ\gamma)\leqslant
\Bar H_{P}^{\delta}(\gamma_{n},\theta\circ\gamma_{n},\dots,\theta^{n-1}\circ
\gamma_{n})\end{multline*}
for all sufficiently large $n$, such that also $2\delta_{n}<\delta$ from
above (chosen to be $\delta<D(\cal{B})^{-1}$).

Consequently, $\hbar(\theta,\gamma)\leqslant\liminf_{n\to\infty}\frac{1}{n}
\Bar H_{P}^{\delta_{n}}(\gamma_{n},\theta\circ\gamma_{n},\dots,\theta^{n-1}
\circ\gamma_{n})$ {\em and} 
\begin{multline*} \limsup_{n\to\infty}\frac{1}{n}\Bar H_{P}^{
\delta_{n}}(\gamma_{n},\theta\circ\gamma_{n},\dots,\theta^{n-1}\circ\gamma_{n})
\leqslant \\ \leqslant\lim_{n\to\infty}\frac{1}{n}\Bar H_{P}^{\delta}(\gamma,
\theta\circ\gamma,\dots,\theta^{n-1}\circ\gamma)=\hbar_{P}^{\delta}(\theta,
\gamma)\leqslant\hbar_{P}(\theta,\gamma)+\varepsilon. \qed \end{multline*}
\end{enumerate}
\renewcommand{\qed}{}
\end{pf}
\begin{zzza}The following are important consequences of Theorem (3.9):
\begin{enumerate}
\item[(a)] Under the assumptions of the Theorem (3.9) we get that $\hbar_{P}(
\theta,\gamma)\geqslant\limsup_{n\to\infty}\Bar H_{P}(\gamma_{n},\theta\circ
\gamma_{n},\dots,\theta^{n-1}\circ\gamma_{n})$.
\item[(b)] If, furthermore, the following limit exists as an {\em infimum}
for {\em all} $\varepsilon$ small enough (i.e.\ also for $\varepsilon=0$):
$\lim_{n\to\infty}\frac{1}{n}\Bar H_{P}^{\varepsilon}(\gamma_{n},\theta\circ
\gamma_{n},\dots,\theta^{n-1}\circ\gamma_{n})$, then we have that $\hbar_{P}
(\theta,\gamma)=\lim_{n\to\infty}\frac{1}{n}\Bar H_{P}(\gamma_{n},\theta\circ
\gamma_{n},\dots,\theta^{n-1}\circ\gamma_{n})$.
\end{enumerate}
\end{zzza}
\begin{pf} We give the very simple proof, for the sake of completeness: 
\begin{enumerate}
\item[(a)] By (i) of (3.9), $\hbar_{P}^{\varepsilon}(\theta,\gamma)\geqslant
\limsup_{n\to\infty}\frac{1}{n}\Bar H_{P}(\gamma_{n},\dots,\theta^{n-1}\circ
\gamma_{n})$, and for $\varepsilon\to 0$ the left hand side tends to $\hbar_{
P}(\theta,\gamma)$ by (3.8).
\item[(b)] Assuming this, we can prove that (cf.\ the proof of (3.8)):
\begin{multline*}
\lim_{\varepsilon\to 0+}(\lim_{n\to\infty}\frac{1}{n}\Bar H_{P}^{\varepsilon}
(\gamma_{n},\theta\circ\gamma_{n},\dots,\theta^{n-1}\circ\gamma_{n}))= \\ =
\lim_{n\to\infty}\Bar H(\gamma_{n},\theta\circ\gamma_{n},\dots,\theta^{n-1}
\circ\gamma_{n}).\end{multline*}
The equality therefore follows by the $\varepsilon\to 0$ part of (3.9,i). \qed
\end{enumerate}
\renewcommand{\qed}{}
\end{pf}
\begin{zzzb} The following remarks and simple results should clarify the
relation between this ``P--entropy'' approach in section III, our original
approach as in section II, and Thomsen's approach \cite{t} as cited above.
\begin{enumerate}
\item Theorem (3.9) and Corollary (3.10) are particularly ``natural'' for a
separable {\em nuclear} C$^{*}$--algebra $\cal{A}$ with a Choi--Effros--Lance
sequence $(\sigma_{n},\tau_{n})_{n\in\Bbb{N}}$ of completely positive unital
maps $\sigma_{n}:\cal{A}\to\cal{B}_{n},\enspace\tau_{n}:\cal{B}_{n}\to\cal{A}$
with finite--dimensional (multi--)matrix algebras $\cal{B}_{n}$, such that
$\|\tau_{n}\circ\sigma_{n}(A)-A\|\to 0\enspace\forall A\in\cal{A}$ \cite{ce}:

Defining, for given $\gamma\in\cal{C}\cal{P}_{1}(\cal{A})$ say, $\gamma_{n}=
\tau_{n}\circ\sigma_{n}\circ\gamma\enspace(\forall n\in\Bbb{N})$, it is
obvious that for any such $\gamma\in\cal{C}\cal{P}_{1}(\cal{A})$ there exists 
a
suitable subsequence $(n_{k})_{k\in\Bbb{N}}$ for which the condition of (3.9)
is met: $\|\gamma-\gamma_{n_{k}}\|^{\frac{1}{k}}\to 0$. Thus it follows from
(3.9,ii) with $\delta_{k}=c\|\gamma-\gamma_{n_{k}}\|^{\frac{1}{k}}$ that
$\hbar_{P}(\theta,\gamma)=\lim_{k\to\infty}\frac{1}{k}\Bar H_{P}^{\delta_{k}}
(\tau_{n_{k}}\circ\sigma_{n_{k}}\circ\gamma,\theta\circ\tau_{n_{k}}\circ
\sigma_{n_{k}}\circ\gamma,\dots,\theta^{k-1}\circ\tau_{n_{k}}\circ\sigma_{n_{
k}}\circ\gamma)$.

Note, however, that here we can {\em not} conlude that the right hand side
is dominated by $\frac{1}{k}\Bar H_{P}^{\delta_{k}}(\tau_{n_{k}},\theta\circ
\tau_{n_{k}},\dots,\theta^{k-1}\circ\tau_{n_{k}})$ as $k\to\infty$, because
$\Bar H_{P}^{\varepsilon}$ (and $\Bar H_{P}$) is known to be monotonic only
with respect to compositions with $*$--homomorphisms (which $\sigma_{n_{k}}
\circ\gamma$ are definitely {\em not\/}). We {\em could} use such an 
argument if by some ``miracle'' part (b) of (3.10) could be verified for
these $\gamma_{n_{k}}$, with a ``minimally spaced'' subsequence $(n_{k})_{k\in
\Bbb{N}}$, implying that: \begin{multline*}
\hbar_{P}(\theta,\gamma)=\lim_{k\to\infty}\frac{1}{k}\Bar H_{P}(\tau_{n_{k}}
\circ\sigma_{n_{k}}\circ\gamma,\theta\circ\tau_{n_{k}}\circ\sigma_{n_{k}}
\circ\gamma,\dots,\theta^{k-1}\circ\tau_{n_{k}}\circ\sigma_{n_{k}}\circ\gamma)
\leqslant \\ \leqslant \limsup_{k\to\infty}\frac{1}{k}\Bar H(\tau_{n_{k}}
\circ\sigma_{n_{k}}\circ\gamma,\theta\circ\tau_{n_{k}}\circ\sigma_{n_{k}}
\circ\gamma,\dots,\theta^{k-1}\circ\tau_{n_{k}}\circ\sigma_{n_{k}}\circ\gamma)
\leqslant \\ \leqslant \limsup_{k\to\infty}\frac{1}{k}\Bar H(\tau_{n_{k}},
\theta\circ\tau_{n_{k}},\dots,\theta^{k-1}\circ\tau_{n_{k}})\leqslant\infty,
\end{multline*} now by (2.7,i) and the first trivial inequality.

This non--trivial upper bound (if finite) seems to be most interesting for
an AF algebra $\cal{A}=\overline{\bigcup_{n\in\Bbb{N}}\cal{A}_{n}}\equiv
\overline{\cal{A}_{\infty}}$, where $\tau_{n}=\imath_{\cal{A}_{n}}:\cal{A}_{n}
\hookrightarrow\cal{A}$ is the inclusion, with an ``$\cal{A}_{\infty}
$--shifty'' automorphism $\theta$ as in (4.5) below. 

In that case we could use that by assumption on $\theta$, we have $\Bar H(
\cal{A}_{n_{k}},\theta(\cal{A}_{n_{k}}),\dots,\theta^{k-1}(\cal{A}_{n_{k}}))
\leqslant\Bar H
(\cal{A}_{N_{k}})$ for $N_{k}=n_{k}+k\enspace(\forall k\in\Bbb{N})$; and
at least for sufficiently ``$\cal{A}_{\infty}$--smooth'' $\gamma:\cal{B}\to
\cal{A}$ (where this is a not defined notion, in contrast to ``shifty'' above)
but with $\gamma(\cal{B})\nsubseteq\cal{A}_{\infty}$, the resulting
even larger upper bound $\limsup_{k\to\infty}\frac{1}{k}\Bar H(\cal{A}_{N_{k}
})$ could turn out to be {\em finite}, although
on the other hand this might be incompatible with the assumption that (3.10,b)
holds for $(\gamma_{n_{k}})_{k\in\Bbb{N}}$.  
\item In any case, this ``mere'' speculation (1) shows that, also if we (or 
the reader) would now choose Definition (3.2) instead of the corresponding
definitions in section II -- either for reasons of the even more concrete
physical interpretation of (3.2), or because it enables the modest continuity
results (3.9 \& 10) above -- also then, our still ``more non--commutative''
definitions (2.5,ii) etc.\ of section II would not be obsolete at all: 
Not only in the situation of the ``diagonalization trick'' above (of the 
double sequence $(\frac{1}{k}\Bar H(\tau_{n},\theta\circ\tau_{n},\dots,
\theta^{k-1}\circ\tau_{n}))_{k,n}$ in the last line of (1), for example), but
{\em generally}, the definitions with $\cal{O}^{+}$ instead of $\cal{O}\cal{P
}_{1}$ give natural upper bounds for the latter ``P--entropies'' (cf.\ Remark
(3.3)), and the former might be easier to calculate.
\item Although we know by (2.2,i) that for any $\alpha\in\cal{O}^{+}(\cal{A})$
as defined in (2.1,ii), there exists a map $\gamma\in\cal{P}_{1}(\cal{A})
\subset\cal{C}\cal{P}_{1}(\cal{A})$ as in (2.12,i) (i.e.\ $\gamma:\cal{B}\to
\cal{A}$ with $\cal{B}$ {\em Abelian} finite--dimensional) together with a
$\beta\in\cal{O}\cal{P}_{1}(\cal{B})$ as above such that $\alpha\subseteq
\gamma(\beta)$ (this is obvious), we can {\em not} relate our modified version
(3.2) here to Thomsen's modified definition in \cite{t} (which clearly then
would have to be a {\em lower} bound, as \cite{t} uses the same $\alpha$ for
all $\theta^{k}$; cf.\ before (3.2)), for {\em two} (related) reasons:

First, Thomsen uses right away instead of {\em positive} sets $\alpha\in\cal{
O}^{+}(\cal{A})$ more general ``operator covers'' $\alpha\in\cal{O}(
\cal{A})$ for the individual ``non--commutative open covers'', but in turn
with the restriction that $\alpha\in\cal{O}(\cal{A})$ be {\em 
self--adjoint\/}: $\alpha=\alpha^{*}$. This additional condition has then the
main advantage that the analogue (see below) of (2.3,ii) holds for {\em
arbitrary} such $\alpha$, not only for the positive $\alpha\in\cal{O}^{+}
(\cal{A})$.

Secondly, however, to get again an inner operation on these ``parti\-ti\-ons''
$\cal{P}(\cal{A})=\{\alpha\in\cal{O}(\cal{A})|\alpha=\alpha^{*}\}$ (in the
same way that we have here $\Vec\vee:\cal{O}(\cal{A})^{2}\to\cal{O}(\cal{A})$,
cf.\ also (2.2,iii)), Thomsen \cite{t} consequently has to replace $\Vec\vee$
by the operation leading to the {\em totally symmetrized} set of operator
products of the individual ``factors'', denoted by $\vee:\cal{P}(\cal{A})^{2}
\to\cal{P}(\cal{A})$.

On the {\em one} hand, this latter modification has the formal disadvantage
that this operation $\vee$ is not associative any more; but still the
analogue of the subadditivity (2.3,i) (there actually submultiplicativity)
holds. And because Thomsen takes the supremum in the final definition (for the
entropy of a $*$--endomorphism $\theta$) really over this set $\cal{P}
(\cal{A})$, he can use the {\em same} simple proof for the generalization
of the classical inequality $h(T^{n})\geqslant n\cdot h(T)\enspace\forall n\in
\Bbb{N}$ as used already in the very original paper by Adler et al.\
\cite{akm} (such as if we would define here an $\hbar(\theta)$ by a supremum
over $\cal{O}(\cal{A})$, with our ``join'' $\Vec\vee$); whereas this
inequality is (still) missing in our approach.
\item On the other hand, however, this totally symmetrized join $\vee:\cal{P}
(\cal{A})^{2}\to\cal{P}(\cal{A})$ has the really {\em serious} drawback that
Thomsen \cite{t} can {\em not} prove the monotonicity $\Bar N(\alpha)\leqslant 
\Bar N(\alpha\vee\beta)$ (or $\Bar N(\beta)\leqslant \Bar N(\alpha\vee\beta)
$) for all
$\alpha,\beta\in\cal{P}(\cal{A})$ any more, in contrast to our (2.3,i) and
even more importantly (2.7,ii) here. This forces \cite{t} to define the
``unscaled'' entropy of an endomorphism $\theta$ by $\hslash\hslash(\theta)=
\sup_{k\in\Bbb{N}}\frac{1}{k}\hslash(\theta^{k})$ (where we denote here as 
$\hslash(\theta)=\sup_{\alpha\in\cal{P}(\cal{A})}\hslash(\theta,\alpha)$; in 
the notation of \cite{t}: $\hslash(\theta,\alpha)=\hslash_{1}(\theta,\alpha)
$), in order to maintain the essential converse inequality $\hslash\hslash
(\theta^{n})\leqslant n\cdot\hslash\hslash(\theta)\enspace(\forall n\in\Bbb{N}
)$.

This second supremum (taken over the scaled entropies of the endomorphism's
powers) is obviously a rather ``cheap trick''; whereas in our approach here, 
the corresponding inequality (2.9,v) respectively (2.13,ii) follows from
(2.7,ii) by the also simple but {\em natural} argument analogous to the 
non--commutative {\em state\/}--dependent entropy theory, cf.\ \cite{cnt}.

>From the physical point of view, it is clear that the totally {\em 
symmetrized} operation $\vee$ as in \cite{t} completely destroys the natural
interpretation of the ordered operator products in our $\Vec\vee:\cal{O}_{(1)}
(\cal{A})^{2}\to\cal{O}_{(1)}(\cal{A})$, from (2.1,iii), as successive
quantum--mechanical ``measurements'' (or so-called ``operations'', cf.\
\cite{kr,lin}), see the references in the Introduction I and particularly 
also the forthcoming publication \cite{h?}.

Then it is not surprising that the remaining partial monotonicity (2.3,i)
of the entropy with respect to additional such arguments gets also lost,
which is actually against any ``physical'' intuition if the entropy $\Bar H$
respectively $\Bar N$ should be a sensible measure of ``information'' in some
sense (side--remark: the {\em non\/}--monotonicity of the quantum--mechanical
von Neumann entropy $S(\omega)$, cf.\ \cite{fredl,op}, of a state $\omega\in
S_{\cal{A}}$ with respect to restriction to subalgebras $\cal{B}\subset\cal{A}
$ is not a good excuse at this point, as it refers to a rather different
aspect of ``quantum information''; see again also \cite{h?} in preparation).
\item Finally, we should add the following remarks: Although Thomsen's entropy
\cite{t} or actually the corresponding functional $\Bar N(\alpha_{1}\vee
\alpha_{2}\vee\dots\vee\alpha_{n})$ is clearly {\em symmetric} in its
arguments $\alpha_{k}\in\cal{P}(\cal{A})$, it is not (yet?) much better suited
for more general (semi--)group actions (than $\Bbb{Z}$-- respectively $\Bbb{N}
$--actions here) by $*$--auto--(respectively endo--)morphisms of $\cal{A}$,
than our above approach with the {\em ordered} operation $\Vec\vee$, which
seems to be not suited at all for that purpose. A bit more explicitly: To
proceed again along the lines of \cite{h4} (for example), Thomsen's
approach \cite{t} would still need (at least) the invariance of $\Bar N(
\alpha_{1}\vee\dots\vee\alpha_{n})$ with respect to repetitions of $\alpha_{k}
=\alpha_{\ell}\enspace(k<\ell\leqslant n)$, which is not provable in this
general form (contrast with our {\em built--in} invariance in (2.5,ii) etc.).

But of course, one could use this symmetrized functional to define {\em
independently} some formal ``entropy densities'' of $\Bbb{Z}^{n}$--actions
by automorphisms, which seems to be not possible with our {\em ordered} 
operation $\Vec\vee$ any more. In other words, the theory presented here is
really dealing with {\em dynamical} entropy (in a physical sense) of {\em
single} $*$--endomorphisms $\theta$ of $\cal{A}$ only.
\end{enumerate}
\end{zzzb}
\section{AF algebras and related examples}
We first reproduce the following example from \cite{t} with the entropy
definitions of section II, noting that throughout this section {\em all}
examples would give the same final results if we would use the restricted
``P--entropy'' definitions of section III instead. This first example due
to Thomsen ``pretends'' to be more general than AF algebras, although the
{\em positive} and {\em finite} entropy values result from the AF algebra
case only.
\begin{zzzc} Let $\cal{B}$ be a (unital) AA C$^{*}$--algebra as defined in
(2.11) with corresponding sequence $\tau=(\tau_{n})_{n\in\Bbb{N}}$, where we
assume that either (2.11,ii) is fulfilled with {\em increasing} range spaces
$\tau_{n}(\cal{B}_{n})\subseteq\tau_{n+1}(\cal{B}_{n+1})$ $(n\in\Bbb{N}$), 
or that $\tau_{n}:\cal{B}_{n}\hookrightarrow\cal{B}$ are even
the inclusions of unital {\em sub\/}algebras $\cal{B}_{n}\subset
\cal{B}$ which are {\em increasing} ($\cal{B}_{n}\subseteq\cal{B}_{n+1}$) such
that {\em a fortiori} (2.11,i) is fulfilled. The general case of (2.11) is
apparently less tractable in this example.

Let then $\cal{A}=\bigotimes_{k\in\Bbb{N}}(\cal{B})_{k}$ be the infinite
tensor product C$^{*}$--algebra of a countable number of copies of $\cal{B}$.
If $\cal{B}$ is nuclear, this tensor product is uniquely determined; and
otherwise we choose for example the injective or projective tensor product
(cf.\ \cite{tak}). For each $k\in\Bbb{N}$, the simple tensors of the form
$b_{1}\otimes b_{2}\otimes\dots\otimes b_{k}\otimes\1_{k+1}\otimes\dots$
($b_{k}\in(\cal{B})_{k}$) generate a unital C$^{*}$--subalgebra $\cal{A}_{k}$ 
of $\cal{A}$ which is $*$--isomorphic to the tensor product $\cal{B}\otimes
\cal{B}\otimes\dots\otimes\cal{B}$ of $k$ copies of $\cal{B}$.

We assume that the AA maps $\tau_{n}:\cal{B}_{n}\to\cal{B}$ are even
{\em completely} positive (which is automatically fulfilled in the special
case of (2.11,i) as above), i.e.\ $\tau_{n}\in\cal{C}\cal{P}_{1}(\cal{B})$
with our notation from (2.12,1), $\forall n\in\Bbb{N}$. Then we can define
maps $\pi_{n}:\bigotimes_{k=1}^{n}(\cal{B}_{n})_{k}\to\cal{A}_{n}$ by the 
$n$--fold
product map $\bigotimes_{k=1}^{n}(\tau_{n})_{k}$ on the tensor product of $n$
copies of $\cal{B}_{n}$ (which is again completely positive, for example by
Stinespring's original representation of completely positive maps \cite{st})
composed still with the $*$--isomorphism $\bigotimes_{k=1}^{n}(\cal{B})_{k}
\cong\cal{A}_{n}\enspace(\forall n\in\Bbb{N})$.

We first claim that $\cal{A}$ is again AA with respect to the sequence
$\pi=(\pi_{n})_{n\in\Bbb{N}}$.

Now, let $\sigma:\Bbb{N}\to\Bbb{N}$ be an injective map. For each $b\in\cal{B
}$, let $b(i)$ denote the element $\1_{\cal{B}}\otimes\dots\otimes\1_{\cal{B}}
\otimes b \otimes\1_{\cal{B}}$, where $b$ occurs as the $i$--th tensor factor,
$i\in\Bbb{N}$. There is a unique unital $*$--endomorphism $\theta_{\sigma}$
of $\cal{A}$ given by $\theta_{\sigma}(b(i))=b(\sigma(i)),\enspace b\in\cal{B}
\enspace(\forall i\in\Bbb{N})$. Then the entropy $\hbar_{\pi}(\theta_{\sigma})
$ depends on $\sigma,\tau,\cal{B}$ as follows:
\begin{enumerate}
\item[(i)] $\hbar_{\pi}(\theta_{\sigma})=0$ if $\sigma$ has no infinite orbit
in $\Bbb{N}$.
\item[(ii)] $\hbar_{\pi}(\theta_{\sigma})\geqslant\limsup_{n\to\infty}\Bar H
(\tau_{n})$ if $\sigma$ has an infinite orbit in $\Bbb{N}$, and where we
assume in addition that in case (2.11,ii) $\tau_{n}$ is {\em faithfully}
covering as defined in (2.14,i), $\forall n\in\Bbb{N}$. In particular, if
$\cal{B}$ is infinite--dimensional and $\Bar H(\tau_{n})\to\infty$, we have
$\hbar_{\pi}(\theta_{\sigma})=\infty$.
\item[(iii)] If $\cal{B}$ is finite--dimensional, we have in the (rather
useless) general case the inequality $\hbar_{\pi}(\theta_{\sigma})\leqslant
r\cdot\Bar H(\cal{B})$, where $r$ is the number of infinite orbits of $\sigma$
in $\Bbb{N}$. For the obvious choice $\tau\equiv(\text{Id}_{\cal{B}})_{n\in
\Bbb{N}},
\enspace\text{Id}_{\cal{B}}:\cal{B}\to\cal{B}$ the identity map, equality 
follows.
\end{enumerate}
\end{zzzc}
\begin{pf} We first show that $\cal{A}$ is again AA with respect to $\pi$.
This is obvious if the special case (2.11,i) is fulfilled as assumed above,
because then $\cal{C}_{n}\equiv\bigotimes_{k=1}^{n}(\cal{B}_{n})_{k}$ is
$*$--isomorphic to a {\em subalgebra} $\pi_{n}(\cal{C}_{n})$ of $\cal{A}_{n}$
and these are again increasing, by assumption: $\pi_{n}(\cal{C}_{n})\subseteq
\pi_{n+1}(\cal{C}_{n+1})$, such that we have again {\em a fortiori} also
$\pi_{n}(\cal{O}^{+}(\cal{C}_{n}))\subseteq\pi_{n+1}(\cal{O}^{+}(\cal{C}_{n+1}
))$.

And in the other case (2.11,ii), it clearly follows from $\cal{A}=\overline{
\bigcup_{k\in\Bbb{N}}\cal{A}_{k}}$ and the assumption $\overline{\bigcup_{n\in
\Bbb{N}}\tau_{n}(\cal{B}_{n})}=\cal{B}$ that also $\overline{\bigcup_{n\in
\Bbb{N}}\pi_{n}(\cal{C}_{n})}=\cal{A}$.

Now we compute the entropy of $\theta_{\sigma}$ with respect to $\pi$:
\begin{enumerate}
\item[(i)] If $\sigma$ has no infinite orbit in $\Bbb{N}$, it is clear that
for each $n\in\Bbb{N}$, there is an integer $N(n)$ such that $\theta_{\sigma
}^{N(n)}$ is the identity on $\cal{A}_{n}$, hence $\theta_{\sigma}^{N(n)}
\circ\pi_{n}\approxeq\pi_{n}$ by (2.5,i), and by (2.9,iv), $\hbar(\theta_{
\sigma},\pi_{n})=0$.
\item[(ii)] For a finite subset $F\subset\Bbb{N}$, we denote by $\cal{A}(F)$
the $*$--algebra generated by elements of the form $b(i),\enspace i\in F$.
Let $m\in\Bbb{N}$ be an integer such that $\{\sigma^{n}(m)|n\in\Bbb{N}\}$ is
infinite. Then for all $n\geqslant m$, \begin{multline*}
\hbar(\theta_{\sigma},\pi_{n})=\lim_{k\to\infty}\frac{1}{k}\Bar H(\pi_{n},
\theta_{\sigma}\circ\pi_{n},\dots,\theta_{\sigma}^{k-1}\circ\pi_{n})\geqslant
\\ \geqslant \lim_{k\to\infty}\frac{1}{k}\Bar H(\tau_{n,m},\theta_{\sigma}
\circ\tau_{n,m},\dots,\theta_{\sigma}^{k-1}(\tau_{n,m})),\end{multline*}
where we denote by $\tau_{n,m}:\cal{B}_{n}\to\cal{A}(\{m\})$ the map given by
the composition of $\pi_{n}$ {\em after} the inclusion homomorphism of the
$m$--th tensor factor $(\cal{B})_{m}\hookrightarrow\cal{C}_{n}\equiv
\bigotimes_{k=1}^{n}(\cal{B}_{n})_{k}$, such that the inequality follows by
(2.7,i), using the fact that $\theta_{\sigma}^{k}\circ\pi_{n}\not\approxeq
\pi_{n}\enspace\forall k\in\Bbb{N}$.

By the assumption that $\tau_{n}$ (and hence $\tau_{n,m}$) is faithfully
covering, it follows from (2.15) that $$\Bar H(\tau_{n,m},\theta_{\sigma}
\circ(\tau_{n,m}),\dots,\theta_{\sigma}^{k-1}\circ(\tau_{n,m}))=k\cdot\Bar H(
\tau_{n,m}),$$ because by construction the arguments on the left hand side are
independently covering and commuting (cf.\ (2.16,3)). Thus it follows that
$\hbar(\theta_{\sigma},\pi_{n})\geqslant\Bar H(\tau_{n,m})\equiv\Bar H(\tau_{
n})\enspace\forall n\geqslant m$.
\item[(iii)] We use the following expression for the number $r$ of infinite 
periodic
orbits of $\sigma$, derived in \cite{t}: $$r=\lim_{n\to\infty}\lim_{k\to
\infty}\frac{1}{k}\card(F_{n}\cup\sigma(F_{n})\cup\sigma^{2}(F_{n})\cup\dots
\cup\sigma^{k}(F_{n})),$$
where $F_{n}=\{1,2,\dots,n\}$, and it is shown in \cite{t} if not obvious
that the first limit is a supremum, the second one an infimum (from left to
right).

Note that for a general finite subset $F\subset\Bbb{N}$, with $\cal{A}(F)$ 
as defined in (4.1,ii) above, we have that $\Bar H(\cal{A}(F))=\card F\cdot
\bar H(\cal{B})$, and that the notations in (i) and (ii) are related by
$\cal{A}_{n}=\cal{A}(F_{n})$. Now use that $$\hbar(\theta_{\sigma},\pi_{n})
=\lim_{k\to\infty}\Bar H(\pi_{n},\theta_{\sigma}\circ\pi_{n},\dots,\theta_{
\sigma}^{k-1}\circ\pi_{n})\leqslant\lim_{k\to\infty}\frac{1}{k}\Bar H(\cal{A}
(G_{k,n})),$$
where $G_{k,n}=F_{n}\cup\sigma(F_{n})\cup\sigma^{2}(F_{n})\cup\dots\cup
\sigma^{k-1}(F_{n})$ as used for $r$ above ($k,n\in\Bbb{N}$), and where the
right hand side inequality follows from (2.7,iv). To repeat it, we know that
$\Bar H(\cal{A}(G_{k,n}))=\card G_{k,n}\cdot\Bar H(\cal{B})$, and together
it follows that $\hbar(\theta_{\sigma},\pi_{n})\leqslant r\cdot\Bar H(\cal{B})
,\enspace\forall n\in\Bbb{N}$, as the first limit for $r$ is the supremum.

To prove the reverse inequality for $\tau_{n}\equiv\text{Id}_{\cal{B}}
:\cal{B}\to\cal{B}
\enspace\forall n$, let $\beta=\{e_{i}|i=1,\dots,D(\cal{B})\}\subset\cal{B}$
be a partition of unity $\1_{\cal{B}}$ by mutually orthogonal minimal
projections $e_{i}$ (i.e.\ $\beta\in\cal{O}\cal{P}_{1}(\cal{B})$ as defined
in (3.2,i) above). For each $m\in\Bbb{N}$, let $\alpha_{m}=\bigotimes_{k=1}^{
m}(\beta)_{k}$ (symbolically, but clear) be the $m$--th tensor power of
$\beta$, explicitly $\alpha_{m}=\{e_{i_{1}}\otimes_{i_{2}}\otimes\dots\otimes
e_{i_{m}}|i_{k}=1,\dots,D(\cal{B})\enspace\forall k=1,\dots,m\}$, which is
a $\cal{O}\cal{P}_{1}$--partition of unity in $\bigotimes_{k=1}^{m}(\cal{B})_{
k}\cong\cal{A}_{m}$; and $\pi_{m}(\alpha_{m})\in\cal{O}^{+}(\cal{A})$ is the
image under that latter isomorphism. Then 
$$\hbar(\theta_{\sigma},\pi_{m})\geqslant\lim_{k\to\infty}\frac{1}{k}\log\Bar
N(\pi_{m}(\alpha_{m})\Vec\vee\theta_{\sigma}\circ\pi_{m}(\alpha_{m})\Vec\vee
\dots\Vec\vee\theta_{\sigma}^{k-1}\circ\pi_{m}(\alpha_{m})),$$
and it is easy to see that the right hand side gives
$$\Bar N(\pi_{m}(\alpha_{m})\Vec\vee\theta_{\sigma}\circ\pi_{m}(\alpha_{m})
\Vec\vee\dots\Vec\vee\theta_{\sigma}^{k-1}\circ\pi_{m}(\alpha_{m}))=
D(\cal{B})^{\card G_{k,m}}.$$
Together we get that $\hbar(\theta_{\sigma},\pi_{m})\geqslant\log D(\cal{B})
(\lim_{k\to\infty}\frac{1}{k}\card G_{k,m})$, and by letting $m\to\infty$ it
follows that $\hbar_{\pi}(\theta_{\sigma})\geqslant r\cdot\Bar H(\cal{B})$.
\qed
\end{enumerate}
\renewcommand{\qed}{}
\end{pf}
In the rest of this final section, we briefly review the theory and examples
on the entropy calculation for AF algebras, following the earlier versions
\cite{h1,h2} but now including the rather trivial extension of these results
to $*$--{\em endo\/}morphisms of AF algebras (not only automorphisms), again
inspired by Choda's results \cite{choda} for the Connes--Stormer entropy.
This is just meant to provide the reader with something at least formally new,
as it is also the case in \cite{t}. For this reason, we do not repeat the
detailed proofs from \cite{h1,h2}, where in the published version \cite{h1}
the proofs for the general calculation methods are already included
(for $*$--automorphisms, not yet $*$--endomorphisms), whereas
the calculations for the examples reviewed below are given in full detail only
in the unpublished thesis \cite{h2}. We can leave it to the reader to repeat
these calculations as exercises, and to extend the general results for
$*$--endomorphisms as stated below, following \cite{choda}.

Throughout the remainder of this section, $\cal{A}=\overline{\bigcup_{n\in
\Bbb{N}}\cal{A}_{n}}$ is a (unital) AF algebra with finite--dimensional
$\cal{A}_{n}\subseteq\cal{A}_{n+1}$ (all with the same unit) and their
algebraic inductive limit $\cal{A}_{\infty}=\bigcup_{n\in\Bbb{N}}\cal{A}_{n}$.

\begin{zzzz} For the canonical ``AA structure'' on $\cal{A}$ as in (2.11),
i.e.\ $\tau=(\tau_{n})_{n\in\Bbb{N}}$ with $\tau_{n}:\cal{A}_{n}
\hookrightarrow\cal{A}$ the inclusions, we have that $\hbar_{\tau}(\theta)
\equiv\lim_{n\to\infty}\hbar(\theta,\cal{A}_{n})=\sup_{\cal{B}\subset\cal{A}_{
\infty}}\hbar(\theta,\cal{B})$, where the supremum is taken over all
finite--dimensional $C^{*}$--subalgebras $\cal{B}\subset\cal{A}_{\infty}$.
Thus we can use the notation $\hbar_{\cal{A}_{\infty}}(\theta)$ for the above
AF--entropy.
\end{zzzz}
\begin{pf} This is a direct consequence of (2.9,ii), as any $\cal{B}\subset
\cal{A}_{\infty}$ is contained in some $\cal{A}_{n}\supset\cal{B}$.
\end{pf}
\begin{zzzd} We say with Choda \cite{choda} that the sequence
$(\cal{A}_{n})_{n\in\Bbb{N}}$ as above is {\em periodic} with period $p$, if
$\exists n_{0}\in\Bbb{N}$ such that $\forall j\geq n_{0}$:
\begin{enumerate}
\item[(i)] The inclusion matrices (cf.\ \cite{choda,jones}) are periodic:
$[\cal{A}_{j}\hookrightarrow
\cal{A}_{j+1}]=[\cal{A}_{j+p}\hookrightarrow\cal{A}_{j+p+1}]$.
\item[(ii)] The (hence necessarily square--) matrix $T_{j}=[\cal{A}_{j}
\hookrightarrow
{\cal A}_{j+p}]$ is {\em primitive} ($\Longleftrightarrow\exists\ell\in\Bbb{N}
:$
$(T_{j}^{\ell})_{ik}>0\enspace\forall i,k\Longleftrightarrow$ by the inclusion
$\cal{A}_{j}\subset\cal{A}_{j+\ell p}$ each simple direct summand of $\cal{
A}_{j}$ is ``contained'' in every simple direct summand of $\cal{A}_{j+\ell p}
\Longrightarrow$ the correpsonding
Bratteli diagram \cite{bratt,jones} of $\cal{A}$ is {\em connected\/}), 
which implies that $T_{j}$ has a unique Perron--Frobenius eigenvalue
$\beta_{j}>0$ (which, together with (i), is actually independent of $j
\geqslant n_{0}$!).
\end{enumerate}
\end{zzzd}
\begin{zzze} For a periodic sequence $(\cal{A}_{n})_{n\in\Bbb{N}}$ as above
with period $p$ and Perron--Frobenius eigenvalue $\beta$ of $T_{j}=[\cal{A}_{
j}\hookrightarrow\cal{A}_{j+p}]$ for $j\geqslant n_{0}$, we have $\lim_{n\to
\infty}\frac{1}{n}\Bar H(\cal{A}_{n})=\frac{1}{p}\log\beta$ (see \cite{h1}
for the correct proof, and cf.\ \cite{choda} for the Connes--St\o rmer 
entropy). 
\end{zzze}
\begin{zzzf} A $*$--endomorphism $\theta$ of $\cal{A}$ is said to be
``$\cal{A}_{\infty}$--shifty'', if the following conditions are fulfilled
(cf.\ \cite{choda,t}):
\begin{enumerate}
\item[(i)] For all $j,m\in\Bbb{N}$, the C$^{*}$--algebra generated by
$\cal{A}_{j},\theta(\cal{A}_{j}),\dots,\theta^{m-1}(\cal{A}_{j})$ is
finite--dimensional (although these need not be pairwise commuting), and
$\Bar H(\cal{A}_{j},\theta(\cal{A}_{j}),\dots,\theta^{m-1}(
\cal{A}_{j})) \leqslant\Bar H(\cal{A}_{j+m})$.
\item[(ii)] There is a sequence $(n_{j}\in\Bbb{N})_{j\in\Bbb{N}}$ such that
$\forall k\in\Bbb{N}:
\cal{A}_{j},\theta^{n_{j}}(\cal{A}_{j}),\dots,\theta^{kn_{j}}(\cal{A}_{j})$
are pairwise commuting {\em and} independently covering as defined in 
(2.14,ii) and in the sense of (2.16,3) (i.e., here: of tensor product form);
and such that $\lim_{j\to\infty}\frac{n_{j}-j}{j}=0$.
\end{enumerate}
\end{zzzf}
\begin{zzzg} For an ``$\cal{A}_{\infty}$--shifty'' $*$--endomorphism $\theta$
of $\cal{A}$, the AF--entropy (4.2) is $\hbar_{\cal{A}_{\infty}}(\theta)=
\lim_{n\to\infty}\frac{1}{n}\Bar H(\cal{A}_{n})$ (see \cite{h1} for the  
correct proof; cf.\ also \cite{choda,t}, and be prepared to use the fact that
the limit in our basic definition (2.5,ii) is actually the infimum).

If $\theta$ is even a 
$*$--automorphism of $\cal{A}$, then $\hbar_{\cal{A}_{\infty}}(\theta^{-1})=
\hbar_{\cal{A}_{\infty}}(\theta)$ (because obviously by (4.5) above, also 
$\theta^{-1}$ is ``$\cal{A}_{\infty}$--shifty'').
\end{zzzg}
\begin{zzzh} If $(\cal{A}_{n})_{n\in\Bbb{N}}$ is a {\em periodic} sequence
with period $p$ and Perron--Frobenius eigenvalue $\beta$ of the inclusion
matrix $[\cal{A}_{j}\hookrightarrow\cal{A}_{j+p}]\enspace\forall j\geqslant
n_{0}$, then for any ``$\cal{A}_{\infty}$--shifty'' $*$--endomorphism $\theta$
of $\cal{A}$, the AF--entropy (4.2) is $\hbar_{\cal{A}_{\infty}}(\theta)=
\frac{1}{p}\log\beta$.
\end{zzzh}
\begin{zzzi} We consider the $n^{\infty}$--UHF algebra (cf.\ \cite{black})
$\cal{A}=\bigotimes_{k\in\Bbb{Z}}(M_{n}(\Bbb{C}))_{k}$ as the {\em bilaterally 
}infinite C$^{*}$--tensor product of copies of the $(n\times n)$--matrix 
algebra $M_{n}(\Bbb{C})$, and we choose the AF structure for $\cal{A}$ with
the following notation (in contrast to \cite{h1}, where the notation had been 
``too short'' here): $\forall j\geqslant 0$,
$$\cal{A}_{2j+1}=\1_{n}^{\otimes\infty}\otimes\bigotimes_{k=-j}^{j}(M_{n}(
\Bbb{C}))_{k}\otimes\1_{n}^{\otimes\infty},\qquad\cal{A}_{2j}=\1_{n}^{
\otimes\infty}\otimes\bigotimes_{k=-j+1}^{j}(M_{n}(\Bbb{C}))_{k}\otimes\1_{n
}^{\otimes\infty},$$ where we still use a short notation, denoting by the 
symbolic infinite tensor power of $\1_{n}\in M_{n}(\Bbb{C})$ the corresponding
unital inclusions into $\cal{A}$. The unit shift on $\Bbb{Z}$ determines a 
$*$--automorphism $\theta_{n}$ of $\cal{A}$ by $$\theta_{n}(\1_{n}^{\otimes
\infty}\otimes(M_{n}(\Bbb{C}))_{k}\otimes\1_{n}^{\otimes\infty})=
\1_{n}^{\otimes(\infty+1)}\otimes(M_{n}(\Bbb{C}))_{k+1}\otimes\1_{n}^{\otimes
(\infty-1)}\qquad\forall k\in\Bbb{N},$$ and from (4.7) with $p=1$ and
$[\cal{A}_{m}\hookrightarrow\cal{A}_{m+1}]=n\in\Bbb{N}\enspace(\forall m\in
\Bbb{N})$ it follows that $\hbar_{\cal{A}_{\infty}}(\theta_{n})=\log n$.
\end{zzzi}
Note that this is the same result as in example (4.1) with $\cal{B}=M_{n}
(\Bbb{C})$ and $\sigma(k)=k+1\enspace(\forall k\in\Bbb{N})$, choosing $\tau$
as the constant identity map on $\cal{B}$ in (4.1,iii). This can be cast in
words and formulae more generally, in the following complement of (3.1,i\&ii):
\begin{zzzj} Let $\theta$ be an ``$\cal{A}_{\infty}$--shifty'' 
$*$--automorphism
of $\cal{A}$, and assume that there exists an AF--subalgebra $\cal{B}\subset
\cal{A}$ (included with the same unit $\1\in\cal{B}$) such that $\theta(\cal{B
})\subset\cal{B}$, and furthermore that it can be represented as $\cal{B}=
\overline{\bigcup_{k\in\Bbb{N}}\cal{B}_{k}}\equiv\overline{\cal{B}_{\infty}}$
such that $\theta\restriction_{\cal{B}}$ is still ``$\cal{B}_{\infty}
$--shifty'' as a $*$--endomorphism of $\cal{B}$, and at the same time $\lim_{
n\to\infty}\frac{1}{n}\Bar H(\cal{A}_{n})=\lim_{k\to\infty}\frac{1}{k}\Bar H
(\cal{B}_{k})$. Then it follows obviously from (4.6) above that we have
$\hbar_{\cal{A}_{\infty}}(\theta)=\hbar_{\cal{B}_{\infty}}(\theta
\restriction_{\cal{B}})$.
\end{zzzj}
\begin{zzzk}Let $\cal{A}$ be the unital AF algebra generated by the sequence 
of Jones projections \cite{jones} $(e_{i})_{i\geqslant 0}$ (in the hyperfinite
II$_{1}$ factor with trace $\operatorname{tr}$), which satisfy the relations 
$e_{i}=e_{i}^{*}=e_{i}^{2}$, $e_{i}e_{i\pm 1}e_{i}=\lambda e_{i}$ for some 
$\lambda\leqslant
1$, $e_{i}e_{j}=e_{j}e_{i}$ for $|i-j|\geqslant 2$, and the additional 
relation $\operatorname{tr}(w\cdot e_{i})=\lambda\cdot\operatorname{tr}(w)$ 
when $w$ is a word in $\1,e_{1},\dots,e_{i-1}$, $\forall i\geqslant 0$. 
As shown in \cite{jones}, such a sequence exists
exactly iff $\lambda\in(0,\frac{1}{4}]\cup\{(4\cos^{2}\frac{\pi}{m})^{-1}|
m\in\Bbb{N},\enspace m\geqslant 3\}$. 

Using the same notation as in \cite{jones}, we denote by $\cal{A}_{n}$
the finite--dimensional C$^{*}$--algebra generated by $\1,e_{0},e_{1},\dots,
e_{n-1}$
and by $\cal{A}_{\infty}$ the corresponding inductive limit as used before,
then the AF--algebra $\cal{A}$ is again defined by $\cal{A}=\overline{\cal{A
}_{\infty}}$ (norm closure within the hyperfinite II$_{1}$--factor).

The translation on the index set $\Bbb{N}$ of the Jones sequence determines
a ($\operatorname{tr}$--preserving) unital $*$--endomorphism $\theta_{\lambda}
$ of $\cal{A}$ by $\theta_{\lambda}(e_{i})=e_{i+1}\enspace\forall i\geqslant 0
$. Then it follows again from (4.7) that for $\lambda>\frac{1}{4}$ the entropy
(4.2) is given by $\hbar_{\cal{A}_{\infty}}(\theta_{\lambda})=-\frac{1}{2}
\log\lambda$, whereas for $\lambda\leqslant\frac{1}{4}$ it follows from (4.6)
itself, using \cite{jones}, that $\hbar_{\cal{A}_{\infty}}(\theta_{\lambda})
\equiv\log 2\enspace\forall\lambda\leqslant\frac{1}{4}$, as explicitly shown
in \cite{t}.

In \cite{h1,h2} we had actually used the automorphic version of this Jones
shift, following the results of \cite{pp,choda} for the Connes--St\o rmer 
entropy, where the sequence $(e_{i})_{i\geqslant 0}$ is extended to a
bilateral sequence $(e_{i})_{i\in\Bbb{Z}}$ (being the reason that above the
sequence is indexed by $\Bbb{N}\cup\{0\}$), again generating the hyperfinite
II$_{1}$--factor and satisfying the analogous relations. Replacing all the
above $\cal{A}$'s in (4.10) here by $\cal{B}$'s and defining instead $\forall
j\geqslant 0$:
$$\cal{A}_{2j+1}=\text{C}^{*}(\{e_{i}|i=-j,\dots,j\}),\qquad\cal{A}_{2j}=
\text{C}^{*}(\{e_{i}|i=-j+1,\dots,j\}),$$
with corresponding inductive limit $\cal{A}_{\infty}$ and $\cal{A}=\overline{
\cal{A}_{\infty}}$, it follows from \cite{choda} and results cited there that
the assumptions of (4.9) above are met for the extension $\theta_{\lambda}^{
\pm}(e_{i})=e_{i+1}\enspace(\forall i\in\Bbb{Z})$, which implies that
$\hbar_{\cal{A}_{\infty}}(\theta_{\lambda}^{\pm})=\hbar_{\cal{B}_{\infty}}
(\theta_{\lambda})$ for all possible values of $\lambda$ (as shown directly in
\cite{h1,h2} for $\lambda>\frac{1}{4}$).
\end{zzzk}
\begin{zzzl} Let $S\subset\Bbb{N}$ be any finite subset, and choose $n\in
\Bbb{N}$. There exists then a sequence $(u_{i})_{i\in\Bbb{N}}$ of unitaries,
generating the hyperfinite II$_{1}$--factor, with relations $u_{i}^{n}=\1
\enspace\forall i\in\Bbb{N}$, $u_{i}u_{j}=\exp(\frac{2\pi i}{n})u_{j}u_{i}$
when $|i-j|\in S$, and $[u_{i},u_{j}]=0$ when $|i-j|\not\in S$ (besides the
relation $u_{i}^{*}=u_{i}^{-1}\enspace\forall i\in\Bbb{N}$). The generated
C$^{*}$--algebras $\cal{A}_{n}=\text{C}^{*}(\{u_{i}|i=1,\dots,n\})$ are all
finite--dimensional, and hence again $\cal{A}=\overline{\cal{A}_{\infty}}$ is
an AF algebra (unitally included in the hyperfinite II$_{1}$--factor).

Again, the index translation on $\Bbb{N}$ induces a unital $*$--endomorphism
$\theta_{n}$ of $\cal{A}$, explicitly $\theta_{n}(u_{i})=u_{i+1}\enspace
\forall i\in\Bbb{N}$. Then it follows from (4.7) that $\hbar_{\cal{A}_{\infty}
}(\theta_{n})=\frac{1}{2}\log n$, see again \cite{choda} with the 
Connes--St\o rmer entropy and references there, cf.\ \cite{t}. Again,
$\theta_{n}$ can be extended to an automorphism $\theta_{n}^{\pm}$ of the AF
algebra generated by the bilaterally extended sequence of $u_{i}$'s, and (4.9)
applies to give the same AF--entropy for $\theta_{n}^{\pm}$, see again
\cite{choda}.
\end{zzzl}
\begin{zzzm} The following example and the ideas for the method of calculation
are due to Narnhofer and Thirring \cite{nt}, see there for further references
to its origins in the work of Powers and Price.

Let $\cal{A}_{n}=\text{C}^{*}(\{e_{i}|i=-n,\dots,n\})$ be the (universal, and
existing) C$^{*}$--algebras generated by the ``generalized Pauli--matrices''
$e_{i}\enspace(i\in\Bbb{Z})$ with the relations $e_{i}=e_{i}^{*}$, $e_{i}^{2}
=\1\enspace\forall i\in\Bbb{Z}$ and $e_{i}e_{j}=e_{j}e_{i}(-1)^{g(|i-j|)}
\enspace\forall i\ne j$ with $g:\Bbb{N}\to\{0,1\}$. Again $\cal{A}=\overline{
\cal{A}_{\infty}}$ is a unital AF algebra and $\theta_{g}(e_{i})=e_{i+1}
\enspace(\forall i\in\Bbb{Z})$ induces a $*$--automorphism $\theta_{g}$ of
$\cal{A}$. Then if {\em either} $g(n)\equiv 1\enspace\forall n\in\Bbb{N}$,
{\em or} if $g(n)=1\Longleftrightarrow n\in S\subset\Bbb{N}$ on a {\em finite} 
subset
$S\ne\emptyset$ of $\Bbb{N}$ (where in the second case we still have to add
the -- possibly redundant -- assumption that for the {\em centers} $\frak{C}
(\cal{A}_{n})$, it follows $\frac{1}{n}\log\dim\frak{C}(\cal{A}_{n})\to 0$),
we get $\hbar_{\cal{A}_{\infty}}(\theta_{g})\equiv\frac{1}{2}\log 2$.

See \cite{h2} for the correct proof, and cf.\ \cite{fabio} for a published
version of it in the first case $g\equiv 1$. 
We leave it to the reader as an interesting problem to relate
this example to (4.3--5) above, so as to deduce the result directly from 
(4.7),
again in special cases (and as a second problem, to find out if the additional
assumption in the second case for $g$ above is redundant or not).
\end{zzzm}
\begin{zzzn} Let $\theta_{A}$ be the shift on an AF algebra $\cal{A}$ 
associated with a topological Markov chain (also called subshift of finite
type) as treated by Evans \cite{evans} with his ``AF--imitation'' of the
topological entropy (via the Connes--St\o rmer entropy), and following the
constructions of Cuntz and Krieger (see the references in \cite{evans}).

Without repeating this lengthy example here (see \cite{h2}), we just compare
with Evans' notation: Our $(\cal{A},\theta_{A})=(\cal{C}^{A},\sigma_{0})$ of
Evans, where $A$ is a primitive $(n\times n)$--matrix with entries in $\{0,1\}
$ defining the subshift, and our $\cal{A}_{n}=N_{n}\enspace(n\in\Bbb{N})$
respectively $\cal{A}_{\infty}=\bigcup_{n\in\Bbb{N}}N_{n}$ of Evans. 

Then, repeating the proof of the main theorem in \cite{evans} with our $\Bar 
H$ instead of Evans' $H$ and using the properties (2.7,i \& ii) and (2.6,2) of
$\Bar H$ (and also its definition (2.5,ii)), we immediately get the same
result: $\hbar_{\cal{A}_{\infty}}(\theta_{A})=\log\lambda$, where $\lambda$
is the spectral radius (i.e.\ the Perron--Frobenius eigenvalue) of $A$. See
\cite{h2} for the details of the proof, left as an exercise here: 
Note that we do not need at all
even the {\em ingredients} of Evans' Proposition 2 \cite{evans} for the second
part of the rewritten proof, as the latter amounts in our case to the {\em 
same} estimate by ``$\log|\cal{M}_{n,n+k}|$'' in Evans' notation as the first
part, only from below instead from above.

Actually, the same result holds also for the larger AF algebra $\cal{B}$
containing the above $\cal{A}\subset\cal{B}$ as defined in \cite{evans} (see
\cite{h2} for the correspondence of notations) with $\theta_{A}$ naturally 
extended to $\cal{B}$, and with a corresponding sequence $\cal{B}_{n}\subset
\cal{B}_{n+1}$ for $\cal{B}=\overline{\cal{B}_{\infty}}$ such that $\cal{A}_{n
}\subset\cal{B}_{n}\enspace\forall n\in\Bbb{N}$ holds true: $\hbar_{\cal{B}_{
\infty}}(\theta_{A})=\log\lambda$. The proof is exactly the same as for the
``gauge--invariant observable'' algebra $\cal{A}$ before, see \cite{h2,evans}.

On the other hand, however, the {\em classical} subshift is even contained in
$\cal{A}$ as the restriction of $\theta_{A}$ to a canonical Abelian AF 
subalgebra
$\cal{D}\subset\cal{A}$ which is {\em diagonal} in $\cal{A}$ (and also in 
$\cal{B}$) in the sense that $\cal{D}=\overline{\bigcup_{n\in\Bbb{N}}\cal{D}_{
n}}$ with finite--dimensional (Abelian) $\cal{D}_{n}\subset\cal{A}_{n},
\enspace\forall n\in\Bbb{N}$. The classical topological entropy of the 
original Markov chain is then again given by $\hbar_{\cal{D}_{\infty}}(
\theta_{A}\restriction_{\cal{D}})=\log\lambda$; and as pointed out also in
\cite{t}, it follows here from (3.1,i) that for every AF subalgebra $\cal{C}
\subset\cal{B}$ such that $\cal{D}_{n}\subseteq\cal{C}_{n}\subseteq\cal{B}_{n}
\enspace\forall n\in\Bbb{N}$ (for example, take $\cal{A}=\cal{C}$), we get 
still $\hbar_{\cal{C}_{\infty}}(\theta_{A}\restriction_{\cal{C}})=\log\lambda
$.\end{zzzn}
\begin{zzzo} In all the AF examples considered above,
{\em except} in the ``new'' case $\lambda<\frac{1}{4}$ of (4.10), we have at
least the partial {\em analogue} of the variational principle (1.2,ix) in the
following sense: $\hbar_{\cal{A}_{\infty}}(\theta)=h_{\text{tr}}(\theta)$,
where $\text{tr}\in S_{\cal{A}}$ is a canonically given, $\theta$-invariant
trace state on $\cal{A}$, respectively; and where $h_{\omega}(\theta)$, here
with $\omega=\text{tr}$, is the state--dependent C$^{*}$--dynamical entropy
of Connes, Narnhofer and Thirring \cite{cnt}, actually also naturally
extended for $*$--endomorphisms $\theta$ (along the lines of \cite{choda} for
the Connes--St\o rmer entropy). See the respective references in the 
respective examples for hints how to show this claim.

The desirable but generally ``utopic'' non--commutative {\em generalization}
of (1.2,ix) in the form $\hbar_{\cal{A}_{\infty}}(\theta)=\sup_{\omega\in
S_{\cal{A}}^{\theta}}h_{\omega}(\theta)$, with $S_{\cal{A}}^{\theta}=\{
\omega\in S_{\cal{A}}|\omega\circ\theta=\omega\}$, is beyond the scope of this
paper, but it will be treated together with the detailed proofs of the above
claim for the mentioned examples in the computational part of the forthcoming
publication \cite{h?}.

We should remark still here, however, that for the case $\lambda<\frac{1}{4}$
in (4.10), the problem remains open: Choosing a fixed $\lambda\in(0,\frac{1}{
4})$, the algebra $\cal{A}=\cal{A}_{\lambda}$ of (4.10), first in the final
automorphic version of $\theta_{\lambda}$ there, has the canonical trace state
$\text{tr}_{\lambda}\in S_{\cal{A}_{\lambda}}^{\theta_{\lambda}}$ equal to
the restriction of the unique trace state $\text{tr}$ on the generated
hyperfinite II$_{1}$--factor containing $\cal{A}_{\lambda}$; and by 
\cite[(VII.2)]{cnt} it follows that $h_{\text{tr}_{\lambda}}(\theta_{\lambda})
\restriction_{\cal{A}_{\lambda}}$ as introduced above is equal to the
Connes--St\o rmer entropy of the extended II$_{1}$--factor automorphism
$\Bar\theta_{\lambda}$ with respect to $\text{tr}$. The latter entropy was
shown in \cite{pp,choda} to be $h_{\text{tr}}(\Bar\theta_{\lambda})=
-t\log t-(1-t)\log(1-t)$, where $\lambda=t(1-t)$; and as pointed out in
\cite{t}, for all $\lambda\in(0,\frac{1}{4})$ this gives the strict
inequality $h_{\text{tr}_{\lambda}}(\theta_{\lambda})<\hbar_{\cal{A}_{\infty}}
(\theta_{\lambda})\equiv\log 2$ from (4.10), which violates the above claim
for this canonical trace state $\text{tr}_{\lambda}\in S_{\cal{A}_{\lambda}}^{
\theta_{\lambda}}$.

But this still leaves open the possibility that the {\em strong} form above
of the non--commutative generalization of (1.2,ix) could be true, i.e.\ that
the supremum over $h_{\omega}(\theta_{\lambda})$ for $\omega\in S_{\cal{A}_{
\lambda}}^{\theta_{\lambda}}$, invariant states 
on the {\em norm\/}--completed inductive limit $\overline{\bigcup_{
n\in\Bbb{N}}\cal{A}_{n}}=\cal{A}_{\lambda}$, would in fact be equal to
$\log 2\equiv \hbar_{\cal{A}_{\infty}}(\theta_{\lambda})$ from (4.10), for
all $\lambda\in(0,\frac{1}{4})$.

It follows easily from the analysis in \cite{pp} and from \cite{cnt}
respectively \cite{connes}, that $h_{\omega}(\theta_{\lambda})\leqslant\log 2
\enspace\forall\omega\in S_{\cal{A}_{\lambda}}^{\theta_{\lambda}}$, for any
fixed $\lambda<\frac{1}{4}$: By the explicit construction in \cite[(5.5)]{pp},
$\cal{A}_{\lambda}$ is isomorphically represented as C$^{*}$--subalgebra of
the $2^{\infty}$--UHF algebra (cf.\ \cite{black} and see (4.8) above), such
that $\theta_{\lambda}$ is given by the restriction of the unit shift
$\theta_{2}$ of (4.8) to $\cal{A}_{\lambda}$. For the latter $\theta_{2}$ it
follows from \cite{cnt,connes} that $\sup_{\phi}h_{\phi}(\theta_{2})=\log 2$,
with supremum over all $\theta_{2}$--invariant states $\phi$ on the $2^{\infty
}$--UHF algebra; and by standard arguments this implies that $h_{\omega}(
\theta_{\lambda})\leqslant\log 2\enspace\forall\omega\in S_{\cal{A}_{\lambda}
}^{\theta_{\lambda}}$.

But to show that this upper bound is attained by the supremum over $S_{\cal{A
}_{\lambda}}^{\theta_{\lambda}}$, it is necessary to compute $h_{\omega}(
\theta_{\lambda})$ for states $\omega$ other than the ``canonical'' trace
$\text{tr}_{\lambda}$ on $\cal{A}_{\lambda}$, and that is not as easy a task
as it might seem at first sight. Note from \cite{jones,choda} that all the
$\cal{A}_{\lambda}$ for $\lambda<\frac{1}{4}$ are isomorphic (as AF algebras),
and one could be tempted to use the explicit construction as in \cite{pp} for
computing $h_{\omega}(\theta_{\lambda})$ with other $\theta_{\lambda}
$--invariant states $\omega\in S_{\cal{A}_{\lambda}}^{\theta_{\lambda}
}$: There \cite{pp}, the GNS representation of the $2^{\infty}$--UHF algebra
with the faithful (infinite tensor product) Powers states with eigenvalues
$\{t,1-t\}$ of the individual $M_{2}(\Bbb{C})$--state factors, giving rise
to the respective Powers factor (cf.\ \cite{br}) of type III$_{t/(1-t)}$ for
$t\in(0,\frac{1}{2})$, was shown to have as the centralizer of that respective
Powers state exactly the hyperfinite II$_{1}$--factor generated by $\cal{A}_{
\lambda}$ in this particular representation, for the respective $\lambda=
t(1-t)$.

But although $\theta_{\lambda}$ is always given by the respective restriction
of the $2$--shift $\theta_{2}$ to $\cal{A}_{\lambda}$ in the respective
representation of \cite{pp}, and although $\hbar_{\cal{A}_{\infty}}(\theta_{
\lambda})\equiv
\log 2$ for $\lambda\in(0,\frac{1}{4})$ (which might suggest that the
sufficient condition (2.13,i) is actually fulfilled), the $\theta_{\lambda}$
are ``unfortunately'' {\em not} necessarily C$^{*}$--algebraically conjugate
via those $*$--isomorphisms of the $\cal{A}_{\lambda}$ for pairs of different
$\lambda<\frac{1}{4}$. If they {\em were} all pairwise conjugate, we could
simply use that for any fixed $\theta_{\lambda}$, and for all other 
possible $\lambda'\ne\lambda$, we would have the
conjugacy--invariance from \cite{cnt}: $h_{\omega_{\lambda'}}(\theta_{\lambda}
)=h_{\text{tr}_{\lambda'}}(\theta_{\lambda'})$, where $\omega_{\lambda'}$
denotes the ``pullback'' to $\cal{A}_{\lambda}$ of the respective Powers
state for $\lambda'$ (restricted to $\cal{A}_{\lambda'}$ in the respective
construction of \cite{pp}). By the results for the Connes--St\o rmer entropy
cited before we know that $\lim_{\lambda'\to 1/4-}h_{\text{tr}_{\lambda'
}}(\theta_{\lambda'})=\log 2$, which would then imply that also $\lim_{
\lambda'\to 1/4-}h_{\omega_{\lambda'}}(\theta_{\lambda})=\log 2$ holds.

So this ``cheap trick'' cannot be applied here; but on the other hand, at 
least on the face of it, the different $\theta_{\lambda}$ are also {\em not}
necessarily pairwise {\em non\/}--conjugate via the $*$--isomorphisms of the
different $\cal{A}_{\lambda}$, although the analogous non--conjugacy is in 
fact realized for their respective natural extensions $\Bar\theta_{\lambda}$ 
to the respective hyperfinite II$_{1}$--factor $\cal{R}$ generated by the
GNS representation of $\cal{A}_{\lambda}$ with the respective canonical trace
state $\text{tr}_{\lambda}$: The important point of the construction in
\cite{pp} is just the aforementioned fact that $\Bar\theta_{\lambda}$ on
$\cal{R}$ is the Connes-Krieger-St\o rmer ``non--commutative Bernoulli shift''
(cf.\ \cite{stoe}) with weights $\{t,1-t\}$ for $\lambda=t(1-t)$, and the 
natural
extension of $\text{tr}_{\lambda}$ is always the {\em unique} trace $\text{tr}
$ on $\cal{R}$. To repeat it now, the Connes--St\o rmer entropy (CNT entropy
with the trace) $h_{\text{tr}}(\Bar\theta_{\lambda})=-t\log t-(1-t)\log(1-t)$
shows that all the automorphisms $\Bar\theta_{\lambda}$ of $\cal{R}$ are
pairwise non--conjugate, but this does {\em not} yet imply the analogous
non--conjugacy for the respective restrictions $\theta_{\lambda}$ on $
\cal{A}_{\lambda}$: Not any of the $*$--isomorphisms between $\cal{A}_{\lambda
}\subset\cal{R}$ and $\cal{A}_{\lambda'}\subset\cal{R}$ for $\lambda\ne
\lambda'$ can be extended to an isomorphism of $\cal{R}$ onto itself, 
because none of those C$^{*}$--isomorphisms preserves the unique trace $
\text{tr}$ on $\cal{R}$ (restricted to the $\cal{A}_{\lambda}$ respectively
$\cal{A}_{\lambda'}$, where it gives the C$^{*}$--traces $\text{tr}_{\lambda}$
respectively $\text{tr}_{\lambda'}$ with {\em different} trace vectors on the
Bratteli diagrams). By the way, if $\theta_{\lambda}$ and $\theta_{\lambda'}$ 
for $\lambda\ne\lambda'$ are non--conjugate via $\cal{A}_{\lambda}\cong
\cal{A}_{\lambda'}$, this shows that the ``$\cal{A}_{\infty}$--topological''
entropy $\hbar_{\cal{A}_{\infty}}(\theta_{\lambda})$ is a ``weaker'' conjugacy
invariant than the Connes--St\o rmer entropy $h_{\text{tr}}(\Bar\theta_{
\lambda})$.

But still, the opening question remains: Can one compute $h_{\omega}(\theta_{
\lambda})\to\log 2$ for a family of states $\omega\in S_{\cal{A}_{\lambda}}^{
\theta_{\lambda}}$ (with $\omega\ne\text{tr}_{\lambda}$, throughout, $\forall
\lambda<\frac{1}{4}$)? Put differently, the problem is that for general
$\theta_{2}$--invariant faithful states $\phi_{\lambda}$ (the subscript
$\lambda$ here indicates the possible adaptation of $\phi_{\lambda}$ for the
fixed $\lambda=t(1-t)$, not any concrete construction of $\phi_{\lambda}$) on
the $2^{\infty}$--UHF algebra, {\em different} from the respective Powers
state with weights $\{t,1-t\}$, the relation between the generated von 
Neumann algebra in the GNS construction with $\phi_{\lambda}$ on the one 
hand, and the enveloping von Neumann algebra of $\cal{A}_{\lambda}$ (the
latter constructed as in \cite{pp} inside the $2^{\infty}$--UHF algebra) 
within
this GNS representation for $\phi_{\lambda}$ on the other hand, seems 
generally not to be obtainable by exactly the same method as in \cite{pp}, 
even
if $\phi_{\lambda}$ is tracial on $\cal{A}_{\lambda}$ and one takes again
the $\phi_{\lambda}$--centralizer of the full GNS--bicommutant of the $2^{
\infty}$--UHF algebra (so for example with $\phi_{\lambda}$ given by the
unique C$^{*}$--trace on the latter simple algebra, for which the CNT entropy 
of the full shift $\theta_{2}$ is $\log 2$, as desired). Some modified method
seems necessary to relate the entropy to be calculated,
$h_{\phi_{\lambda}\restriction
\cal{A}_{\lambda}}(\theta_{\lambda})$, to the entropy $h_{\phi_{\lambda
}}(\theta_{2})$ of the full shift as in (4.8), whereupon one could use the
results of \cite{cnt,connes}. The other, direct way would be to find 
sufficiently close
{\em lower} bounds (or even exact values) for $h_{\phi_{\lambda}\restriction
\cal{A}_{\lambda}}(\theta_{\lambda})>0$ by explicit calculation methods for
the CNT entropy \cite{cnt}, and then to show that these bounds still tend to 
$\log 2$
for suitably chosen $\phi_{\lambda}\restriction\cal{A}_{\lambda}$.

\bigskip
As stated already in the corresponding concluding remark in \cite{h1}, it will
presumably be still much harder in the case of any computable {\em non}--AF
example $(\cal{A},\theta)$ to find a corresponding AA--sequence $\tau$ as in
(2.11) such that again $\hbar_{\tau}(\theta)=\sup_{\omega\in S_{\cal{A}}^{
\theta}} h_{\omega}(\theta)$ holds. In the meantime, since \cite{h1} had been
submitted for publication, Heide Narnhofer \cite{n} has succeeded in
``estimating'' the Connes--St\o rmer entropy (i.e.\ CNT entropy with the trace
state) for the SL$(2,\Bbb{Z})$--action (``CAT map'') by single automorphisms
on the irrational rotation C$^{*}$--algebra $\cal{A}_{\theta}$ ($\theta$ now
the rotation parameter!):

For ``almost all'' values of the irrational rotation parameter $\theta$, 
\cite{n}
finds entropy {\em zero} for {\em any} CAT map (and for the ``exceptional''
values $\theta\in\Bbb{R}\smallsetminus\Bbb{Q}$, still the {\em upper} bound 
given by the corresponding classical CAT--entropy, i.e.\ the logarithm of the
larger SL$(2,\Bbb{Z})$--eigenvalue).

Despite some serious efforts, we have not been able to estimate the ``$\tau
$--topological'' entropy (2.11) of this non--commutative CAT map for any
{\em non\/}--trivial sequence $\tau\equiv(\tau_{n})_{n\in\Bbb{N}},\enspace
\tau_{n}:\cal{B}_{n}\to\cal{A}_{\theta}$ (even with {\em Abelian} 
finite--dimensional $\cal{B}_{n}$, or $\tau_{n}\in\cal{P}_{1}(\cal{A}_{\theta}
)$ with our notation from section II). The problem comes from the fact
that invertibility in $\cal{A}_{\theta}$ is very hard to control, see for
example \cite{put}, and cf.\ Arveson's recent C$^{*}$--algebraic numerical 
work \cite{arveson}. --

We recall that Watatani \cite{w} had used the ``Pontriagin dual'' of the {\em
classical} topological entropy on compact Abelian {\em groups} (such as
$\Bbb{T}^{2}$) to compute the ``K$_{1}$--entropy'' of the $\cal{A}_{\theta}
$--induced SL$(2,\Bbb{Z})$--action by group automorphisms on the discrete
Abelian K$_{1}$--group K$_{1}(\cal{A}_{\theta})$ (cf.\ \cite{black}), where
using the explicit construction for K$_{1}(\cal{A}_{\theta})\cong\Bbb{Z}\oplus
\Bbb{Z}$ due to Pimsner and Voiculescu \cite[(2.5)]{pv} it then follows rather
trivially that the result is again the classical topological (group) entropy
of the $\Bbb{T}^{2}$--CAT map (for {\em all} values of $\theta\in\Bbb{R}
\smallsetminus\Bbb{Q}$!).

Clearly, this ``K$_{1}$--entropy'' of \cite{w} is not suited at all for
AF--algebras $\cal{A}$ where $K_{1}(\cal{A})\equiv 0$ (see \cite{black}),
for which in turn the analogous ``K$_{0}$--entropy'' \cite{w} might give some
non--trivial results. --

On the other hand, Thomsen's approach \cite{t} as discussed and cited before,
defining the entropy by the supremum over $\cal{P}(\cal{B})$ (see (3.11, 3 \&
4)) for an invariant
{\em local $C^{*}$--algebra} $\cal{B}\subset\cal{A}$ (cf.\ also 
\cite{black}) might not be well suited for $\cal{A}=\cal{A}_{\theta}$ instead
of an AF algebra $\cal{A}$ as in \cite{t}, as SL$(2,\Bbb{Z})$--action 
invariant local C$^{*}$--subalgebras of
$\cal{A}_{\theta}$ (for example, subalgebras $\cal{B}$ of ``smooth'' elements)
might be equally hard to control as all of $\cal{A}$ concerning questions of
invertibility. Further work should show if the more pragmatic ``AA entropy''
proposed here is better suited for that purpose.
\end{zzzo}
\begin{ack} I thank
Professors Walter Thirring and Heide Narnhofer for their guidance during
the Ph.D. Thesis \cite{h2,h1} (and already during earlier work for my
first degree, resulting in \cite{h4,h3}).

More recently, I want to thank the Mathematics Department, University of
California at Berkeley, for the hospitality extended to me as Visiting 
Scholar, providing me with very pleasant working conditions for this year.

Last but not least, financial support by {\em Fonds zur F\"orderung der
wissenschaftlichen Forschung in \"Osterreich} for me as Erwin Schr\"odinger
Fellow (J0852-Phy), enabling this post--doc year in Berkeley, is gratefully
acknowledged.
\end{ack}
\begin{ded} This work is dedicated to the memory of Alfred Wehrl, whose
untimely death struck me while being here in Berkeley. Fredl had been my
first guiding teacher in mathematical physics, then my co--operative colleague
and helpful friend. 

In his most frequently cited, excellent review \cite{fredl}, he wrote in
section IV (Related concepts) after part A on dynamical entropies (not
topological, but only measure--theoretic; also mentioning Emch, Lindblad and
Connes--St\o rmer in the ``quantum case'') at the end of the short section B:
\begin{quotation}
{\em Some concepts measuring the amount of information have been described.
The list is not exhaustive and it is left to everyone to invent new such
quantities. However, it will be very hard to establish their physical
meaning.}
\end{quotation}
I hope that the present work can cope with this latter sentence 
in the affirmative.
\end{ded}

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