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\let\REF\txtref
\REF AKLT \AKLT \Jref
I. Affleck, T. Kennedy, E.H. Lieb, H. Tasaki
"Valence bond ground states in isotropic quantum
antiferromagnets"
Commun.Math.Phys. @115(1988) 477--528
\REF AF \AFri \Jref
L. Accardi, A. Frigerio "Markovian Cocycles"
\hfill\break
Proc.R.Ir.Acad. @83A{(2)}(1983) 251--263
\REF FNWb \FCS \Jref
M. Fannes, B. Nachtergaele, R.F. Werner
"Finitely correlated states of quantum spin chains"
Commun.Math.Phys. @144(1992) 443--490
\REF FNWf \FCD \Gref
M. Fannes, B. Nachtergaele, R.F. Werner
"Abundance of translation invariant pure states on quantum
spin chains"
to appear in {\it Lett.Math.Phys.}
\REF CE \CE \Jref
M.D. Choi and E. Effros
"Injectivity and operator spaces"
J. Funct. Anal. @24(1977) 156--209
\REF FNWg \FSY \Gref
M. Fannes, B. Nachtergaele, R.F. Werner
"Symmetries of finitely correlated states"
In preparation
\REF BR \BraRo \Bref
O. Bratteli, D.W. Robinson
"Operator algebras and quantum statistical mechanics"
2 volumes, Springer Verlag, Berlin, Heidelberg, New York
1979 and 1981
\REF Cho \CHOI \Jref
M.D. Choi
"A Schwarz inequality for positive linear maps on C*-algebras"
Illinois J. Math. @18(1974)565--574
\REF AHK \Albev \Jref
S. Albeverio, R. H\o egh-Krohn
"Frobenius theory of positive maps of von Neumann algebras"
Commun.Math.Phys. @64(1978) 83--94
\REF FNWc \FCSE \Jref
M. Fannes, B. Nachtergaele, R.F. Werner
"Entropy estimates for finitely correlated states"
Ann. Inst. H. Poincar\'e @57(1992) 1--19
\REF Dav \DAV \Bref
E.B. Davies
"Quantum theory of open systems"
Academic Press, London 1976
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% \headline{\it \hfil Finitely Correlated Pure States
% \dots Draft Version of \today}
\voffset=2\baselineskip
\hrule height 0pt
\vskip 80pt plus80pt
\centerline{{\BF Finitely Correlated Pure States}$\ ^\bigdagger$}
\vskip 30pt plus30pt
\centerline{
M.~Fannes$^{1,2}$, B.~Nachtergaele$^3$, and R.F.~Werner$^4$}
\vskip 12pt
\centerline{\tt fgbda20@blekul11.bitnet\quad bxn@math.princeton.edu
\quad reinwer@dosuni1.bitnet}
\vskip 80pt plus80pt
\noindent {\bf Abstract}\hfill\break
We study a w*-dense subset of the translation invariant states on an
infinite tensor product algebra $\bigotimes_\Ir\A$, where $\A$ is a
matrix algebra. These ``finitely correlated states'' are explicitly
constructed in terms of a finite dimensional auxiliary algebra $\B$
and a \cp\ map $\E:\A\otimes\B\to\B$. We show that such a state
$\om$ is pure if and only if it is extremal periodic and its entropy
density vanishes. In this case the auxiliary objects $\B$ and $\E$ are
uniquely determined by $\om$, and can be expressed in terms of an
isometry between suitable tensor product Hilbert spaces.
\noindent {\bf Mathematics Subject Classification (1991):}
\class 46L60 Applications of selfadjoint operator algebras to
physics*
\class 82B10 Quantum equilibrium statistical mechanics
(general)*
\class 46L30 Selfadjoint operator algebras: States*
\class 82B20 Lattice systems (Ising, dimer, Potts, etc.)*
\vfootnote{$\dagger$}
{Dedicated to Professor F. Cerulus on the occasion of his
65th birthday}
\vfootnote1
{Inst. Theor. Fysica, Universiteit Leuven, B-3001 Leuven, Belgium}
\vfootnote2
{Bevoegdverklaard Navorser, N.F.W.O. Belgium}
\vfootnote3
{Dept. of Physics, Princeton University, NJ-08544-708, USA;\nl
\vrule width 0pt \qquad on leave from Universiteit Leuven, B-3001
Leuven, Belgium}
\vfootnote4
{Fachbereich Physik, Universit\"at Osnabr\"uck, Pf. 4469,
Osnabr\"uck, Germany}
\vfill\eject
%%%%%%%%%%%%%%%%%
% file fcp1.tex %
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\beginsection 0. Introduction
In recent years impressive progress has been made in the statistical
mechanics of classical systems. In comparison, the statistical
mechanics of quantum systems has turned out to be surprisingly
difficult. Even for one-dimensional lattice systems (``spin
chains'') important basic questions remain unanswered, whereas all
classical models in one dimension are essentially trivial. These
difficulties are intimately connected to the fundamental differences
between quantum theory and classical probability in the description
of composite systems. For example, it is a standard procedure in
classical probability to condition a state of a composite system
with respect to the configuration in a subsystem, i.e.\ to represent
it as an integral over states whose restrictions to one subsystem
are pure states. Many states in quantum theory, such as non-product
pure states of a composite system, simply do not allow such
representations. An example in the context of spin chains, first
constructed by \cite\AKLT, is a state on a spin-$1$ chain which is
translation invariant and pure, but has only impure local
restrictions. This state occurs as the ground state of an
antiferromagnetic interaction.
Due to the higher complexity of correlations in quantum systems it
is often impossible to obtain an explicit representation of quantum
states of large systems. Exceptions are convex combinations of
product states, which play a crucial role in the mean-field
approximation, and quasi-free states.
In this paper we continue the study of a new class of states on
quantum spin chains, the so-called \cfc\ states, for which an
explicit formula for all finite volume expectations can be given
(see Section~1).
A special subclass of the \cfc\ states was first introduced as a
possible quantum generalization of Markov chains \cite\AFri. On the
other hand, examples of such states appeared in the physics
literature \cite\AKLT\ in the study of the ground states of certain
one-dimensional antiferromagnets. They were obtained in that context
by a different construction, as so-called valence bond solid (VBS)
states. In \cite\FCS\ and papers cited therein, we showed that
suitable generalizations of these two constructions are, in fact,
equivalent, and we began a systematic study
%\cite{\FCS,\FCSE,\GAP,\TREE,\FCD}
of this class of states.
We isolated a special subclass, the ``\pg'' states
such that each \cfc\ state can be written as a local function of a
\pg\ one. Each such state has a unique convex \dec\ into
extremal periodic \pg\ states. No further \dec\ is possible: we
showed that every extremal periodic \pg\ state is the unique ground
state of a suitable translation invariant finite range interaction,
and, as a corollary, that these basic states are pure states of the
infinite tensor product algebra.
Thus in contrast to the classical case, where translation invariant
pure states are necessarily homogeneous product states, our
construction yields a large set (in fact, a w*-dense set \cite\FCD)
of translation invariant pure states in the quantum case.
In this paper we show that the purity criterion used in \cite\FCS\
is also necessary, and establish several other necessary and
sufficient conditions for the purity of a \cfc\ state. At the same
time we address the important question to what extent the objects
from which we construct a \cfc\ state are uniquely determined by the
state.
The paper is organized as follows. In Section~1 we introduce the basic
construction of \cfc\ states in terms of a finite dimensional
auxiliary algebra $\B$ and a completely positive map $\E$ and we
formulate our main results. Section~2 is devoted to an analysis of the
algebraic structure of the auxiliary algebra $\B$ and the map $\E$.
In Section~3 we investigate the implications of vanishing
mean entropy for a \cfc\ state, and prove the results stated in
Section~1.
\beginsection 1. Main Results
We now introduce some basic notation and terminology. For each
$i\in\Ir$ we consider a copy $\A_i$ of the algebra $\A$ of $d\times
d$-matrices with complex entries. For a finite subset
$\Lambda\subset\Ir$, we put $\A_\Lambda= \bigotimes_{i\in\Lambda}
\A_i$. The chain algebra $\chain\A$ is then the C*-inductive limit of
the local algebras $\A_\Lambda$, \ie $\chain\A$ is the C*-norm closure
of $\bigcup_\Lambda \A_\Lambda$. A state $\om$ on $\chain\A$ is
defined by its collection of correlation functions, \ie its
restrictions to the local subalgebras $\A_\Lambda$. It suffices to
specify $\om(A_m\otimes\cdots A_n)$ for any choice of $m\leq n\in
\Ir$, and $A_i\in\A_i$, $i=m,\ldots n$. There is an evident action of
the group $(\Ir,+)$ as translation automorphisms on $\chain\A$. We
will denote the elementary translation automorphism by $\gamma$. A
state $\om$ is called translation invariant if it is invariant under
this action. Obviously, a translation invariant state is completely
determined by specifying the $\om(A_1\otimes\cdots A_n)$ for all
$n=1,2,\ldots$, and $A_i\in\A$.
\cfc\ states are now defined by giving an explicit formula for the
correlation functions in terms of certain auxiliary objects. These
are a finite dimensional C*-algebra $\B$ (\ie a direct sum of matrix
algebras), a positive element $e\in\B$, a positive functional $\rho$
on $\B$, and a \cp\ map $\E:\A\otimes\B\to\B$. We always assume that
$\E$, $e$ and $\rho$ satisfy the {\it compatibility conditions}
$\E(\idty\otimes e)= e$ and $\rho\bigl(\E(\idty\otimes B)\bigr)=
\rho(B)$ for all $B\in \B$. Furthermore we assume that $e$ and $\rho$
satisfy the {\it normalization condition\/} $\rho(e)=1$. We use the
abbreviation $\E_A(B)=\E(A\otimes B)$ for $A\in\A$, $B\in\B$, \ie for
fixed $A\in\A$ we consider $\E_A$ as an operator on $\B$. The formula
defining the ``{\bf \cfc\ state generated by} $(\B,\E,e,\rho)$'' is
$$\om(A_1\otimes\cdots A_n)=
\rho\left(\E_{A_1}\circ\cdots\E_{A_n}(e)\right) \quad. \eqno(1.1)$$
The compatibility and normalization conditions ensure that this state
is well-defined and translation invariant. The positivity of $\om$ is
ensured by the positivity of $\rho$ and $e$ and by the complete
positivity of $\E$.
The term {\it finitely correlated\/} refers to the following property:
$\chain\A$ can be written as $\A^L\otimes\A^R$, where $\A^L$ is
generated by the $\A_i, i\leq 0$ and $\A^R$ by the $\A_i, i> 0$.
For any $X\in\A^L$ we define a linear functional $\om_X: \A^R\to
\Cx:\om_X(Y)= \om(X\otimes Y)$. The state $\om$ is called finitely
correlated if the functionals $\set{\om_X\stt X\in\A^L}$ form a
finite dimensional linear space. In \cite\FCS\ it was shown that any
finitely correlated state has a representation similar to that of
formula (1.1). In general, however, $\B$ need not be a C*-algebra, but
has to be taken a finite dimensional matrix ordered space \cite\CE.
The term ``\cfc'' emphasizes that $\B$ is assumed to be a finite
dimensional C*-algebra equipped with its natural matrix order.
In \cite\FCS\ it was shown that the objects $(\B,\E,e,\rho)$
generating a \cfc\ state $\om$ can always be chosen such that $e=
\idty_\B$ and $\rho$ is faithful. In this situation we will say that
\triple{#1} is a {\bf standard triple\/} for $\om$. In the sequel we
are always considering standard triples.
The simplest kind of \cp\ maps are those implemented by operators on
the underlying Hilbert spaces. Specifically, we consider the case
where $\B= \M_k$ is the algebra of $k\times k$-matrices, and $\E$ is
of the form
$$\E(A\otimes B)= V^*(A\otimes B)V \quad, \eqno(1.2) $$
where $V: \Cx^k\to \Cx^d\otimes\Cx^k$ is an isometry.
(Recall that $\A$ has been chosen as the algebra $\M_d$ of $d\times d$
matrices.) This special form of $\E$ is equivalent to $\E$ being a
``pure'' map, which means that it has no non-trivial \dec s $\E=
\E^{(1)}+ \E^{(2)}$ into other \cp\ maps $\E^{(i)}$ (see section 4
of \cite\FCS).
\iproclaim Definition~1.1.
A \cfc\ state $\om$ is called {\bf \pg} if there exists a standard
triple \triple{#1} generating $\om$ such that $\E$ is of the form
$X\in\A\otimes\B\mapsto \E(X)= V^*XV\in\B$, where $\B$ is a
*-subalgebra of $\M_k$ containing the identity $\idty_k$ of $\M_k$,
and $V:\Cx^k\to \Cx^d\otimes\Cx^k$ is an isometry.
\eproclaim
We emphasize that a state is called \pg\ if it is generated by {\it
some} triple of the required form; there will always be many other
triples generating the same state which are not of that form.
The notion of \pg\ states plays a central r\^ole in this paper, in
which we address the following main questions:\nl
{\bf Question~1:}\nl
Is there a canonical choice for the triple \triple{#1}
generating a given \cfc\ state $\om$?\nl
{\bf Question~2:}\nl
Which triples \triple{#1} generate pure states $\om$?\nl
\noindent
{\bf About Question~1 (uniqueness)}:
As it is clear that one can always enlarge the algebra $\B$ by adding
direct summands which are irrelevant for generating $\om$, one should
avoid this trivial sort of non-uniqueness by imposing a minimality
condition on \triple{#1}. In the following definition we describe
the minimality condition we will use, and also the precise meaning
of ``isomorphism'' up to which we will understand ``uniqueness''.
\iproclaim Definition~1.2.
A standard triple \triple{#1} generating a \cfc\ state $\om$ is called
{\bf minimal} if there is no proper subalgebra of $\B$ containing
$\idty$ which is invariant under all operators $\E_A,\, A\in\A$. \nl
Two standard triples \triple{#1\up1} and \triple{#1\up2} are called
{\bf isomorphic} if there is a C*-isomorphism $\tau: \B\up1\to \B\up2$
such that
$$\E\up2\circ\bigl(\id_\A\otimes\tau\bigr)
=\tau\circ\E\up1 \quad,$$
and $\rho\up2\circ\tau=\rho\up1$.
\eproclaim
It is easy to check that every \cfc\ state is generated by a minimal
triple (see Lemma~2.5 of \cite\FCS\ ), and also that isomorphic
triples generate the same state. Unfortunately, minimal triples are
not unique in general. The following example shows that even the
stronger condition of ``minimal dimension of $\B$'' is not enough to
force a unique generating triple. Consider any minimal triple
\triple{#1}, and let $\F: \B\to \B$ denote the operator $\F(B)=
\lambda B+ (1- \lambda)\rho(B)\idty$ for some fixed $\lambda\in(0,1)$.
Since $\F(\idty)= \idty$ and $\rho\circ\F= \rho$ the inverse of this
operator is of the same form with $\lambda$ replaced by $1/\lambda$.
Of course, $\F^{-1}$ is not positive. We define $\E\up1(A\otimes B)=
\E(A\otimes\F(B))$ and $\E\up2(A\otimes B)= \F(\E(A\otimes B))$.
Then both triples $(\B,\E\up i,\rho)$ generate the same state.
However, in spite of the fact that $\E\up1_A$ and $\E\up2_A$ are
similar as linear operators via the invertible transformation $\F$
there is in general no transformation with a positive inverse (\ie
no C*-isomorphism) with this property. Hence the two triples
generate the same state, but are not isomorphic in the sense of
Definition~1.2. This lack of uniqueness is due to the constraints on
the positivity structure of $\B$ imposed by taking this space as a
C*-algebra. In particular, a properly ordered subspace of a
C*-algebra cannot in general be given an algebraic structure, which
induces the positive cone inherited from the algebra. Thus we cannot
pass from an algebra to the minimal $\E_\A$-invariant linear
subspace without leaving the category of C*-algebras. In the
category of general matrix ordered spaces and their natural
isomorphisms, we can obtain uniqueness of minimal triples, precisely
by demanding the minimality of $\B$ as a (matrix-~) ordered linear
space. The problem with this approach, however, is that far less is
known about the structure of matrix ordered spaces than is known
about C*-algebras.
On the other hand, in the case of \pg\ states we do have uniqueness in
the sense of Definition~1.2:
\iproclaim Theorem~1.3.
Let $\om$ be a \pg\ \cfc\ state. Then up to isomorphism there is only
one minimal triple \triple{#1} which generates $\om$.
\eproclaim
The proof of this result is given at the end of Section 3. It plays
an important r\^ole in the analysis of symmetries \cite\FSY\ of
\cfc\ states: by uniqueness every automorphism of $\A$ under which a
\pg\ state $\om$ is invariant can be lifted to a symmetry of the
generating triple.
\noindent
{\bf About Question~2 (purity)}:
We will establish several different ways of characterizing the \cfc\
pure states of the chain algebra $\chain\A$. An essential ingredient
in the proof of the conditions we will obtain, is the notion of
entropy density of a translation invariant state $\om$ on
$\chain\A$, which we now review briefly. For each finite segment
$\Lambda= \bracks{m,n}\subset\Ir$ the restriction
$\om\rstr\A_\Lambda$ is a state on a finite matrix algebra, which is
of the form $\om(A)= \tr(D_\om^\Lambda A)$ for a unique density
matrix $D_\om^\Lambda\in\A_\Lambda$. We can thus define the usual
von Neumann entropy
$S(\om\rstr\A_\Lambda)= -\tr(D_\om^\Lambda\ln D_\om^\Lambda)$. Being
a concave function, $S$ is a measure of the mixedness of a state.
$S$ is always non-negative and it vanishes precisely for pure
states. The entropy density, or ``mean entropy'' of $\om$ is then
defined as
$$ \Sm(\om)= \lim_{\Lambda\nearrow\Ir} {1\over\abs\Lambda}
S(\om\rstr\A_\Lambda) \quad,\eqno(1.4)$$
where the limit is over all segments $\Lambda= \bracks{m,n}$ such that
$m\to -\infty$ and $n\to +\infty$. Here the division by the number of
sites in $\Lambda$ makes $\Sm$ a measure of the ``mixedness per
site''. It turns out \cite{\BraRo} that $\Sm$ is an affine function
of $\om$. Thus the set of states $\om$ with $\Sm(\om)= 0$ is a convex
set, and even a face of the simplex of translation invariant states
of $\chain\A$, \ie all translation invariant convex components of a
state with vanishing entropy density have again this property.
If $\E$ has a decomposition into two completely positive
summands, we may substitute this decomposition into formula (1.1) at
any given site, thus obtaining a convex decomposition of $\om$. We
therefore expect $\om$ to have positive ``mixedness per site'',
i.e.\ positive entropy density. This intuition is borne out by the
following Theorem, which is one of our main technical tools in this
paper.
\iproclaim Theorem~1.4.
The entropy density of a \cfc\ state vanishes if and only if it is
\pg.
\eproclaim
This Theorem is also proven at the end of section 3, together with
the following one, which summarizes our results regarding the purity
of \cfc\ states.
\iproclaim Theorem~1.5.
Let $\om$ be a \cfc\ state. Then the following are equivalent:
\item{(i)}
$\om$ is pure
\item{(ii)}
$\Sm(\om)= 0$ and $\om$ is spatially clustering, i.e.\ for
$A,A'\in\A_\Lambda$:
$$ \lim_{n\to\infty}
\om\bigl(A\otimes\underbrace{\idty\otimes\cdots\idty}
_{n\ {\rm times}} \otimes A'\bigr)
=\om(A)\om(A')
\quad.$$
\item{(iii)}
$\om$ is \pg\ and the operator $\E_\idty$ has trivial peripheral
spectrum, \ie the only eigenvector of $\E_\idty$ with eigenvalue of
modulus 1, is $\idty$
\item{(iv)}
$\om$ is generated by a minimal triple \triple{#1}, with $\B= \M_k$,
and $\E(X)= V^*XV$ with an isometry $V: \Cx^k\to
\Cx^d\otimes\Cx^k$.
\eproclaim
Using this Theorem it is straightforward to construct an abundance
of translation invariant pure states on $\chain\A$. We briefly
address the question how the set of states constructed in this way
is positioned inside the space of translation invariant $\chain\A$.
We showed in \cite\FCS\ that the \cfc\ states form a dense face in
that space. Here we show (in Proposition~2.7) that the \pg\ states,
too, form a dense face. In a forthcoming paper \cite\FCD, we show
that the minimality assumption in Theorem~1.5.(4) is satisfied for
generic $V$. Therefore the pure translations invariant states are
w*-dense in the translation invariant states of the chain.
%%%%%%%%%%%%%%%%%
% file fcp2.tex %
%%%%%%%%%%%%%%%%%
\beginsection 2. The central flow of a \cfc\ state
In this section we study the basic \dec s of \cfc\ states, and their
generating triples. Every unity preserving \cp\ map satisfies the
``2-positivity''-inequality $\E(X^*X)\geq \E(X)^*\E(X)$. The
following Lemma \cite\CHOI\ gives a very useful characterization of
the case of equality in this operator inequality. We will usually
apply this Lemma to the map $\E: \A\otimes\B\to \B$.
\iproclaim Lemma~2.1.
Let $\E: \A\to \B$ be a unity preserving \cp\ map between unital
C*-algebras, and let $X\in\A$ be such that $\E(X^*X)= \E(X)^*\E(X)$.
Then $\E(YX)= \E(Y)\E(X)$ for all $Y\in\A$.
\eproclaim
\iproclaim Proposition~2.2.
Let \triple{#1} be a minimal triple in the sense of Definition~1.2
and let $\ZoE=\set{B\in\B\mid \Eh(B)=B}$ denote the eigenspace of
$\Eh$ for eigenvalue $1$.
Then $\E\bigl(Y(\idty\otimes B)\bigr)= \E(Y)B$ and
$\E\bigl((\idty\otimes B)Y\bigr)= B\E(Y)$ for all $B\in\ZoE$ and
$Y\in\A\otimes\B$. Moreover, $\ZoE$ is a subalgebra contained in the
center of $\B$.
\eproclaim
\proof:
Let $B\in\ZoE$, \ie $\Eh(B)= B$. We then find by the complete
positivity of $\E$:
$$X= \E\bigl((\idty\otimes B^*)(\idty\otimes B)\bigr)-
\E(\idty\otimes B^*)\E(\idty\otimes B)= \E(\idty\otimes
B^*B)-B^*B \geq 0 \quad.$$
Since $\rho$ is invariant under $\Eh$, $\rho(X)= \rho(B^*B- B^*B)=
0$, and since $\rho$ is faithful this implies $X= 0$. Hence by Lemma~1
we have $\E\bigl(Y(\idty\otimes B)\bigr)= \E(Y)\E(\idty\otimes B)=
\E(Y)B$. This relation implies immediately that $\ZoE$ is a
subalgebra. Because $\E$ preserves adjoints $\ZoE$ is a *-subalgebra,
and the second equation follows from the first by taking adjoints.
In order to prove the centrality of $\ZoE$, define $\D_{k+1}$ as the
C*-algebra generated by $\D_k\cup\E(\A\otimes\D_k)$ with $\D_0=
\Cx\idty$. Clearly, every $B\in\ZoE$ commutes with $\D_0$, and for
$D\in\D_k,\ A\in\A$ we have
$$ \bracks{B,\E(A\otimes D)}
= \E(\bracks{\idty\otimes B,\ A\otimes D})
= \E(A\otimes\bracks{B,D})=0 $$
by induction on $k$. By minimality of $\B$, the sets $\D_k$ generate
$\B$, so $\ZoE$ is in the center.
\QED
As $\ZoE$ is a finite dimensional abelian algebra all the
information about its structure is contained in the set of its
minimal projections. For any
minimal projection $P$ we consider the algebra $\B_P=P\B P$. Clearly
$\B=\bigoplus \B_P$, where the direct sum runs over all minimal
projections $P\in \ZoE$.
Then by Proposition~2.2 $\E(\A\otimes\B_P)\subset\B_P$, and we can
define the restriction $\E_P=\E\rstr\A\otimes\B_P$ and $\E=\bigoplus
\E_P$. With $\rho_P=\rho(P)^{-1}\rho\rstr\B_P$ we thus have a triple
\triple{#1_P} generating a \cfc\ state $\om_P$. A direct expression
for $\om_P$ is
$$\om_P(A_1\otimes\cdots A_n)
=\rho(P)^{-1}\rho\left(\E_{A_1}\circ\cdots\E_{A_n}(P)\right)
\quad.\eqno(2.1)$$
All states $\om_P$ are translation invariant and it is obvious that
$\om=\sum \rho(P)\om_P$. Since $P$ is minimal in $\ZoE$ it is the
unique eigenvector of $(\E_P)_\idty$, i.e.\ $\Zo(\E_P)=\Cx
P=\Cx\idty_{\B_P}$. From this one readily concludes that $\om_P$ is
clustering in the mean, and hence \cite{\BraRo, 4.3.10, 4.3.11}
{\it ergodic}, \ie extremal in the convex set of translation
invariant states. The \dec\ of a translation invariant state into
ergodic components is unique, hence the states $\om_P$ are uniquely
determined by $\om$. However, we are not assured in general that the
$\om_P$ are all different for different $P$. This has to do with the
lack of uniqueness in the representation of a state in terms of
\triple{#1}.
Lemma~2.1 can be applied in yet another situation. Rather than
looking at fixed points of $\Eh$, we can study fixed points of $\E_U$
for other elements $U\in\A$. Since each $\E_U$ is the direct sum of
the corresponding operators for the ergodic components of $\om$, and
since there is no intrinsic connection between the ergodic
components, it is best to study this problem for one ergodic
component at a time, \ie for ergodic \cfc\ states.
\iproclaim Proposition~2.3.
Let \triple{#1} be minimal. Assume that $\ZoE=\Cx\idty$, and that
the one-site restriction of the \cfc\ state generated by
\triple{#1} is faithful. Let $U\in\A$ with $\norm U\leq1$, and
$\bU\in\B$ with $\bU\neq0$, and
$$ \E(U\otimes\bU)=\bU \quad.\eqno(2.2) $$
Then $U$ is unitary, and $\bU$ is uniquely determined by $U$ and
$(2.2)$ up to a scalar factor, which may be chosen so that $\bU$
becomes unitary as well. Moreover,
$$ \E\bigl(Y(U\otimes\bU)\bigr)=\E(Y)\bU
\qquad\hbox{for all $Y\in\A\otimes\B$.}\eqno(2.3)$$
Let $\Ge$ denote the set of unitary $U\in\A$ for which a $\bU$
satisfying $(2.2)$ exists. Then $\Ge$ is a subgroup of the unitary
group of $\A$. For every $U\in \Ge$, $\bU$ commutes with $\rho$, \ie
for all $B\in\B, \rho(\bU^* B \bU)=\rho(B)$.
\eproclaim
\proof:
Let $X=U\otimes\bU$. Then
$$\eqalign{
0&\leq\rho\bigl(\E(X^*X)-\E(X)^*\E(X)\bigr)
=\rho\bigl(\E(U^*U\otimes\bU^*\bU)-\bU^*\bU\bigr) \cr
&\leq \rho\bigl(\E(\idty\otimes\bU^*\bU)-\bU^*\bU\bigr)
=\rho\bigl(\bU^*\bU-\bU^*\bU\bigr)=0
\quad.}$$
Hence $(2.3)$ follows from the faithfulness of $\rho$ and Lemma~1.
The above string of inequalities also implies that
$\E(\idty\otimes\bU^*\bU)=\bU^*\bU$ so that $\bU^*\bU\in\ZoE$ is a
non-zero multiple of the identity. We choose $\bU$ unitary from now
on. Since
$\om(U^*U)=\rho(\E(U^*U\otimes\idty))
=\rho(\E(U^*U\otimes\bU^*\bU))=\rho(\bU^*\bU)=1$
the faithfulness of the
one-site restriction of $\om$ implies $U^*U=\idty$. By taking the
adjoint of $(2.2)$ we find that $\Ge$ is closed under taking
inverses, and equation $(2.3)$ implies that it is closed under
products.
Finally we check that $\rho(\bU^* B \bU)=\rho(B)$, for all $B\in\B$.
Using (2.3) and the invariance of $\rho$ under $\Eh$ we have:
$\rho(\bU^* B \bU)=\rho(\Eh(\bU^* B \bU))=\rho(\bU^*\Eh(B)\bU)$ and
hence $\rho(\bU^*\cdot\bU)$ is an invariant state for $\Eh$ and as
$\ZoE=\Cx\idty$ this implies $\rho(\bU^*\cdot\bU)=\rho$.
\QED
For the \dec\ theory we are especially interested in the subgroup of
elements with $U=\zeta^{-1}\idty$, \ie in the solutions of the
eigenvalue equation $\Eh(\bU)=\zeta\bU$ with $\abs\zeta=1$. Since
$\norm{\Eh}=1$ these eigenvalues are on the rim of the spectral
circle, and will be called {\it peripheral eigenvalues} of $\Eh$. In
the Perron-Frobenius theory of matrices of transition probabilities,
such eigenvalues determine the periodic \dec\ of a stationary Markov
chain. Similar results in the non-commutative case have been
obtained in \cite\Albev. However, these results only cover the case
$\A=\Cx$ (in our notation). In the following Proposition we apply
Proposition~2.3 to obtain the necessary refinements. It also yields
the periodic \dec s of $\om$, and the fine structure of the minimal
projections of $\ZoE$.
\iproclaim Proposition~2.4.
Let \triple{#1} be minimal, and
$$\Ze=\set{B\in\B\stt \exists n\in\Nl\ \Eh^n(B)=B}
\quad.$$
Then $\Ze$ is spanned by the eigenvectors of $\Eh$ of modulus one,
and is an algebra contained in the center of $\B$.
Let $\Pi$ denote the set of minimal projections in $\Ze$. Then
$\eta(P)=\Eh(P)$ defines a bijective map $\eta:\Pi\to\Pi$, and, more
generally,
$$\E\bigl(Y(\idty\otimes P)\bigr)
=\E(Y)\eta(P)
\qquad\hbox{for all $Y\in\A\otimes\B$ and $P\in\Pi$.}
$$
Moreover, if $\eta^n(P)=P$ the operator $\Eh^n\rstr P\B\to P\B$ has
no eigenvalues of modulus one except the simple eigenvalue $1$.
\eproclaim
\proof:
By the observations following the proof of Proposition~2.2, $\E$ is
the direct sum of maps $\E_P:\A\otimes\B_P\to\B_P$ with $P$ a
minimal projection in $\ZoE$. It suffices to consider one of these
summands, \ie we may assume that $\ZoE=\Cx\idty$. Let
$\zeta\idty\in\Ge$ for some $\zeta$ with modulus one and let
$\bU\in\B$ be such that $\E(\zeta\idty\otimes\bU)=\bU$. Then since
$\Ge$ is a group we have $\Eh(\bU^n)=\zeta^{-n}\bU^n$. Since $\B$ is
finite dimensional, the spectrum of $\Eh$ is a finite set, so that
we must have $\zeta^p=1$ for some $p\in\Nl$. By adjusting the phase
of $\bU$ we can also make $\bU^p=\idty$. Now let
$\bU=\sum_{m=0}^{p-1}\zeta^mP_m$ be the spectral resolution of
$\bU$, where $\zeta=\exp(2\pi i/p)$. Then for $r= 0,\ldots,p-1$:
$$\sum\zeta^{mr}\Eh(P_m)= \Eh(\bU^r)= \zeta^{-r}\bU=
\sum\zeta^{mr}P_{m+ 1} \quad,$$
where $P_p\equiv P_0$. Hence, as $\det(\zeta^{ij})\neq 0$, we can
conclude that $\Eh(P_m)=P_{m+1}$.Thus $\eta$ is the cyclic shift on
$\Pi\cong\set{0,\ldots,p-1}$. Inserting the resolution into
$\E\bigl(Y(\idty\otimes\bU)\bigr)=\zeta\E(Y)\bU$ gives the relation
stated in the Proposition.
Clearly, $P_i\in\Ze$. Conversely, let $C\in\Ze$, \ie $\Eh^n(C)=C$
for some $n$. It is clear from the construction that apart from the
$p$\th roots of unity all eigenvalues of $\Eh$ have modulus strictly
less than one. Hence $C=\Eh^{pn}(C)$ must be in the eigenspace of
$\Eh^p$ for eigenvalue $1$, hence must be a linear combination of
the $P_m$. It follows that $\Ze$ is generated by the $P_m$, and that
they are precisely the minimal projections of $\Ze$.
\QED
We will call $\eta:\Pi\to\Pi$ the {\it central flow} associated with
$\E$. From this flow the periodic components of the \cfc\ state
$\om$ can be obtained in the same way as the ergodic components were
defined via equation (2.1). The only difference is that the
resulting states are now no longer translation invariant, so one has
to distinguish the translates of a periodic state by keeping track
of the location (modulo a period) of the observables on which the
state is evaluated. Thus we set, for every $P\in\Pi$:
$$\om_P\,(A_1\otimes\cdots A_n)
=\rho(P)^{-1} \rho\left(\E_{A_1}\circ\cdots\E_{A_n}
(\eta^{-n}P)\right)
\quad.\eqno(2.4)$$
By construction, the flow $\eta$ applied to $P$ produces the
translates $\om_P\circ\gamma=\om_{\eta^{-1}(P)}$. Since
$\rho(\eta(P))=\rho(P)$, the average over the $p$ translates of
$\om_P$ (where $\eta^p(P)=P$) is the ergodic state associated to
$P'=\sum_{k=0}^{p-1}\eta^k(P)\in\ZoE$ via equation (2.1).
Each of the states $\om_P$ can be considered as a translation
invariant state of a regrouped chain algebra, where the single site
algebra is then the $p-$fold tensor product of $\A$. In \cite\FCS\
it is shown that these states are \cfc\ on the regrouped chain.
Since $\Eh^p$, restricted to $P\B$, has only one eigenvalue of
modulus $1$, the powers of $P\Eh^p$ contract exponentially to the
subspace of invariant elements, \ie to the projection onto the
multiples of $P$. The states $\om_P$ are therefore clustering on the
regrouped chain and so are extremal periodic on the original chain. We
thus obtain a \dec\ of $\om$ into a convex combination of periodic
states, which is uniquely determined by \triple{#1}. It is
determined by $\om$ only up to repetitions, as there is no guarantee
that states determined by different minimal projections in $\Ze$ are
different. In fact, one can construct examples in the same spirit of
the one given after Definition~1.2, that show that minimality of the
generating triple is in general not enough to guarantee that all
states in (2.1) or (2.4) be distinct. Non-uniqueness arises from the
occurrence of non-equivalent minimal representations of a component.
For the discussion of regrouped chains it is convenient to introduce
the following notations. For all integers $n\geq 1$ we define
inductively the ``iterates''
$\E^{(n)}:\A^{\otimes n}\otimes\B \to\B$ of $\E$ by
$$\E\up1= \E
\midbox{and}
\E\up{n+1}(A\otimes A_n\otimes B)= \E\bigl(A\otimes\E\up
n(A_n\otimes B)\bigr) \quad.$$
Note that $\E\up n(A_1\otimes\cdots A_n\otimes B)
=\E_{A_1}\circ\cdots\E_{A_n}(B)$
as needed in formula (1.1), and that $\E\up p$ is the operator
needed to generate $\om$ as a state on the regrouped chain. We will
therefore use the analogue of the notation $\E_A(B)=\E(A\otimes B)$
also for the iterates $\E\up n$. $\E\up n$ is completely positive
since composing and tensoring with identity maps preserves complete
positivity. As \cp\ unity preserving maps all iterates $\E\up n$ have
norm one. Hence the norm estimates
$$\norm{\E\up n_A(B)}\leq \norm{A\otimes B}= \norm{A}\norm{B}
\midbox{and}
\norm{\E\up n_A}\leq \norm{A} \eqno(2.5)$$
hold for all $A\in\A^{\otimes n}$ and $B\in\B$.
We will also need the linear subspace $\B_0\subset\B$ which is the
smallest subspace of $\B$ containing $\idty$ and invariant under all
operators $\E_A, A\in\A$. Obviously, this space is given by
$\B_0=\set{\E^{(n)}_A(\idty) \stt n\geq 1, A\in\A^{\otimes n} }$.
By the finite-dimensionality of $\B$, and because
$\E_\idty(\idty)=\idty$, there exists an integer $m$
such that actually
$$ \B_0=\set{\E^{(m)}_A(\idty)\stt A\in\A^{\otimes m}}
\quad.\eqno(2.6)$$
In the following Lemma we give a criterion in terms of of $\ZoE$,
$\Ze$, and $\B_0$ for the absence of multiple representations of
periodic components by projections $P\in\Pi$. We will see in
Proposition 3.7 that this condition is automatically satisfied for
\pg\ states.
\iproclaim Lemma~2.5.
Let \triple{#1} be a minimal triple generating a \cfc\ state $\om$.
\item{(1)}
$\ZoE\subset\B_0$ if and only if all $\om_Q$ from (2.1) with minimal
projections $Q\in\ZoE$ are distinct.
\item{(2)}
$\Ze\subset\B_0$ if and only if all $\om_P$ from (2.4) with minimal
projections $P\in\Ze$ are distinct.
\eproclaim
\proof:
Let $p$ denote a common multiple of all periods appearing in the
central flow $\eta$. If we perform a regrouping of the chain by
lumping together intervals of $p$ sites and generate $\om$,
considered as a \cfc\ state on the new chain by the triple
$(\B,\E^{(p)},\rho)$, we see that all $\om_{Q,q}$ now become ergodic
components, that $\Zo(\E^{(p)})=\Ze$, and that $\B_0$ is the same as
before. Therefore (2) follows from (1), and we need only consider
(1).
{``$\Leftarrow$''}:
Assume that $\ZoE\subset\B_0$. Then there exists a $n\geq 1$
such that for any minimal projection $Q\in\ZoE$ we have an element
$A_Q\in\A^{\otimes n}$ for which $\E^{(n)}_{A_Q}(\idty)=Q$. In the
decomposition $\B=\bigoplus \B_Q$, where $\B_Q=Q\B Q$, each of the
algebras $\B_Q $ is invariant under all operators $\E^{(m)}_A$. In
particular $\E^{(n)}_{A_{Q'}}(Q)\in\B_Q$ for all $Q'\in\ZoE$, and
hence $\E^{(n)}_{A_{Q'}}(Q)= \delta_{Q,Q'}Q$. Using this relation one
immediately gets that $\om_Q(A_{Q'})=\delta_{Q,Q'}$, proving that
indeed all $\om_Q$ are distinct.
{``$\Rightarrow$''}:
Suppose now that all $\om_Q$ are distinct. Then the $\om_Q$ are
mutally disjoint, and hence for any $\epsilon >0$ there exists an
integer $m_\epsilon$ and projections $P^{\epsilon,Q} \in\A^{\otimes
m_\epsilon}$ such that
$$\vert \om_Q(P^{\epsilon,Q'})-\delta_{Q,Q'}\vert\leq \epsilon
\eqno(2.7)$$
Define
$$Y^{Q,\epsilon}_N={1\over N}\sum_{k=0}^{N-1}\gamma^k(P^{\epsilon,Q})
$$
It is then obvious that $\Vert Y^{\epsilon,Q}_N\Vert\leq 1$ and
$\Vert Y^{\epsilon,Q}_N-\gamma(Y^{\epsilon,Q}_N)\Vert\leq 2/N$. \nl
Let $ X^{\epsilon,Q}_N=\E^{(m_\epsilon+N-1)}_{Y^{\epsilon,Q}_N}(\idty)$.
Using (2.5) we get:
$$\eqalign{
\Vert X^{\epsilon,Q}_N-\Eh(X^{\epsilon,Q}_N)\Vert&=\Vert
\E^{(m_\epsilon+N)}_{Y^{\epsilon,Q}_N-\gamma(Y^{\epsilon,Q}_N)}(\idty
)\Vert\cr
&\leq \Vert Y^{\epsilon,Q}_N-\gamma(Y^{\epsilon,Q}_N)\Vert\cr
&\leq {2\over N}\cr
}$$
and hence any cluster point of $(X^{\epsilon,Q}_N)_{N\in\Nl}$ must belong
to $\ZoE\cap\B_0$. In fact, using
$$X^{\epsilon,Q'}_N Q=
\E_{Y^{\epsilon,Q'}_N}\up{m_\epsilon+N-1}(\idty)Q=
\E_{Y^{\epsilon,Q'}_N}\up{m_\epsilon+N-1}(Q)=
QX^{\epsilon,Q'}_N \quad,$$
we obtain
$$\rho(X^{\epsilon,Q'}_NQ)=
\rho(\E_{Y^{\epsilon,Q'}_N}\up{m_\epsilon+N-1}(Q))=
\rho(Q)\om_Q(Y^{\epsilon,Q'}_N)=
\rho(Q)\om_Q(P^{\epsilon,Q'})$$
and by (2.7) and the faithfulness of $\rho$ this implies
$$\Vert(\delta_{Q,Q'}\idty-X^{\epsilon,Q}_N)P_{Q'}\Vert\leq
\epsilon \quad. $$
For any sequence of $\epsilon_N>0$ such that $\lim_N \epsilon_N=0$
we will therefore have:\hfill\break
$\lim_N X^{\epsilon_N,Q}_N=Q$, which proves the
assertion.
\QED
As a first application of Lemma~2.5, we prove in the following
Proposition that the situation described in 2.5.(1) indeed occurs
in some of the minimal triples generating $\om$.
\iproclaim Proposition~2.6.
For any \cfc\ state $\om$ there is a triple \triple{#1} such that
$\ZoE\subset\B_0$.
\eproclaim
\proof:
Let \triple{#1'} be any minimal triple generating $\om$ and consider
the direct sum decomposition $\B'=\bigoplus\B'_P$ as described in
the paragraph following Proposition~2.2. Denote by $\Pi'$ the set of
minimal projections in $\Zo(\E')$. We can then select a subset
$\Pi_0\subset\Pi'$, such that all $\om_P,\ P\in\Pi_0$, are distinct,
and $\{\om_P\mid P\in\Pi_0\}= \{\om_P\mid P\in\Pi'\}$. A triple with
the desired property is then obtained by putting:
$\B= \bigoplus_{P\in\Pi_0} \B'_P$,
$\E= \bigoplus_{P\in \Pi_0} \E'_P$ and
$\rho(\bigoplus_{P\in\Pi_0} B_P)
= \sum_{P\in\Pi_0} \sum_{P'\in\Pi';\ \om_P =\om_{P'}}
\rho'(P') \rho'_P(B_P)$.
\QED
In the sequel we are especially interested in \pg\ states. By
De\-finition~1.1 we may generate such states with an $\E$ of the
form $\E(X)=V^*XV$, and it is clear that in this case we may take
the the auxiliary algebra $\B$ as the full matrix algebra
$\tilde\B=\M_k$. Although the algebra $\tilde\B$ has no center, the
central flow of a \pg\ state may be non-trivial, because $\tilde\B$
is in general {\it not} minimal. It is useful to remark that the
ergodic and periodic components of a \pg\ state (considered as \cfc\
states on a regrouped algebra), are still \pg. This is proven in the
following Proposition.
\iproclaim Proposition~2.7.
The set of \pg\ states is a w*-dense face in the set of translation
invariant states on $\chain\A$.
\eproclaim
\proof:
% \def\Chain#1{\chain{\left(#1\right)}}%
The density of the \pg\ states follows from the stronger statement
that even the \cfc\ pure states are w*-dense in the set of
translation invariant ones \cite\FCD.
In order to show that the set of \pg\ states is convex let, for each
$i=1,\ldots,N$, $\K_i$ be a finite dimensional Hilbert space,
$V_i:\K_i\to\Cx^d\otimes\K_i$ an isometry and $\rho_i$ a state on
$\B(\K_i)$ generating a \pg\ state $\om_i$. Then for any choice of
coefficients $\lambda_i\geq0$ with $\sum_i\lambda_i=1$ we set
$\K=\bigoplus_i\K_i$, and $V:=\bigoplus_iV_i\,:\K\to\Cx^d\otimes\K$,
where we identify $\phi\otimes\bigoplus_i\psi_i$ with
$\bigoplus_i(\phi\otimes\psi_i)$ for $\phi\in\Cx^d$ and
$\psi_i\in\K_i$.
The state $\rho$ on $\B(\K)$ is defined as
$\bigoplus(\lambda_i\rho_i)$. Then it is elementary to check that
these objects generate the state $\sum_i\lambda_i\om_i$ on
$\chain\A$.
Finally, we show that the convex components of a \pg\ state $\om$
are again \pg. Let $V:\K\to\Cx^d\otimes\K$ be the isometry with
$\E(A\otimes B)=V^*(A\otimes B)V$. Let $\B$ be the smallest
subalgebra of $\B(\K)$ containing $\idty$ which is closed under all
operators $\E_A$. The ergodic components of $\om$ are labeled by the
minimal projections $P$ in $\Z_1(\E\rstr\A\otimes\B)$, and the
component $\om_P$ is generated by $\B_P=P\B P\subset\B(P\K)$, and
the suitable restrictions of $\E$ and $\rho$ to this algebra. We set
$V_P\psi=(\idty\otimes P)VP\psi$ for $\psi\in P\K$. Then for
$B\in\B_P$:
$\E(A\otimes B)=P\E(A\otimes PBP)P
=PV^*(A\otimes PBP)VP
=V_P^*(A\otimes B)V_P$.
Hence $\E\rstr\A\otimes\B_P$ is pure.
\QED
%%%%%%%%%%%%%%%%%
% file fcp3.tex %
%%%%%%%%%%%%%%%%%
\beginsection 3. States with vanishing entropy density
The main purpose of this section is the proof of Theorems 1.3, 1.4,
and 1.5. The stated propositions (with the exception of 3.7) are
partial results to this end, and are later superseded by the
corresponding theorems. We begin with two criteria for a \cfc\ state
to have vanishing entropy density. In both cases we will later also
establish the conditions for the converse.
\iproclaim Proposition 3.1.
Let $\om$ be a \cfc\ state, which is either pure (\ie extremal in
the state space of $\chain\A$) or \pg. Then $S_M(\om)=0$.
\eproclaim
\proof:
The idea of the proof is to show that for a \cfc\ state $\om$ which
is either pure or \pg, and generated by a triple \triple{#1} where
$\B\subset\M_k$, the entropy of the local restrictions of $\om$
satisfies: $S(\om\rstr\A_\Imn)\leq 2\log k$.
Suppose first that $\om$ is pure.
If $\B\subset\M_k$ we can extend $\E$ to $\A\otimes\M_k$ by
projecting first to the subalgebra $\B$. The extended $\E$ will then
generate the same state, although the new triple will no longer be
minimal. Anyhow, we can suppose $\B=\M_k$. Then the completely
positive map $\E$ is of the form $\E(X)=\sum_\alpha V_\alpha^* X
V_\alpha$ where $V_\alpha:\Cx^k\to\Cx^d\otimes\Cx^k$ and the index
$\alpha$ takes values in a finite set.
The maps
$\E^\alpha (X)= V_\alpha^* X V_\alpha $ are also completely positive
although not unity preserving. For any finite interval
$\Imn\subset\Ir$ and any choice of indices
$\alpha_m,\ldots,\alpha_n$ we can therefore define a non-negative
functional $\om_{\alpha_m,\ldots,\alpha_n}$ on $\chain\A$ by
$$ \om_{\alpha_m,\ldots,\alpha_n}
(A_{m^\prime}\otimes\cdots\otimes A_{n^\prime})
=\rho(\E_{A_{m^\prime}}\circ\cdots\E_{A_{m-1}}\circ
\E^{\alpha_m}_{A_m}\circ\cdots\circ\E^{\alpha_n}_{A_n}
\circ\E_{A_{n+1}}\circ\cdots\E_{A_{n^\prime}} (\idty))
$$
for all
$m^\prime\leq m$, $n^\prime\geq n$ and $A_{m^\prime},\ldots
A_{n^\prime}\in \A$. As
$\sum_{\alpha_m,\ldots,\alpha_n}\om_{\alpha_m,\ldots,\alpha_n}=\om$,
at least for one set of indices, say $\beta_m,\ldots,\beta_n$
the functional $\om_{\beta_m,\ldots,\beta_n}\neq 0$.
As we have assumed that $\om$ is pure there must exist a constant
$C>0$ such that $\om = C \om_{\beta_m,\ldots,\beta_n}$. The entropy
of the state $C \om_{\beta_m,\ldots,\beta_n}\rstr\A_\Imn$ can be
estimated by looking at its support. The density matrix
$D_\om^\Imn$ of $\om\rstr\A_\Imn$ can be computed using the defining
formula (1.1). We find
$$ D_\om^\Imn
=D_{\om_{\beta_m,\ldots,\beta_n}}^\Imn
=\tr_{\Cx^k} K\rho K^*
\quad,\eqno(3.1)$$
where $K:\Cx^k\to (\Cx^d)^{\otimes n-m+1}\otimes\Cx^k$,
$K=(\idty_d\otimes\cdots\idty_d\otimes V_{\beta_n})
\cdots(\idty_d\otimes V_{\beta_{m+1}})V_{\beta_m}$,
and where $\tr_{\Cx^k}$ denotes the partial trace with respect to
the first tensor factor. Writing out the sum defining this trace,
and also the spectral resolution of $\rho$ in terms of
one-dimensional eigenprojections we obtain a representation of
$D_\om^\Imn$ as the sum of $k^2$ rank one operators. It then
immediately follows that
$$ S(\om\rstr\A_\Imn)\leq 2\log k
\quad.$$
This concludes the proof of the Proposition for a pure \cfc\ state $\om$.
The case of a \pg\ state can be treated along the same lines. One
also has an expression like in (3.1) for the local density matrices,
except that now all the $V_\beta$ coincide with the $V$ defining the
\pg\ state (see (1.2)). The rest of the argument remains the same.
\QED
For studying the converses of this result, it turns out that it is
especially useful to study the state $\omom$ on the chain with
one-site algebra $\A\otimes\A$. We caution the reader that the
ergodicity of $\om$ does not imply the ergodicity of $\omom$, as is
readily seen for convex combinations of periodic states. Indeed, let
$\om$ be an ergodic state on $\chain\A$ and suppose that $\om$ has a
non-trivial decomposition into two extremal 2-periodic states:
$\om=\tover12 (\om_1+\om_2)$. The state $\omom$ is then no longer
ergodic: it has a decomposition
$\omom=\tover14\bigl(\om_1\otimes\om_1 + \om_2\otimes\om_2\bigr)
+\tover14\bigl(\om_1\otimes\om_2 + \om_2\otimes\om_1\bigr)$
into two translation invariant states.
As a first step we will convert the information about the entropy
into a property of the state $\omom$. By $\flA$ we denote the
{\it flip} in $\A$, \ie the unitary operator in $\A\otimes\A$
implementing the automorphism
$A_1\otimes A_2\mapsto A_2\otimes A_1$. Representing $\A$ as the set
of linear operators on $\Cx^d$, and $\A\otimes\A$ as the operators
on $\Cx^d\otimes\Cx^d$ we can define
$\flA\,\phi_1\otimes\phi_2=\phi_2\otimes\phi_1$, $\phi_i\in\Cx^d$.
\iproclaim Lemma 3.2.
Let $\om$ be a state on $\chain\A$ with vanishing
entropy density. Then
$$ \limsup_{n\to\infty} {-1\over n}\ln\omom\bigl(
\underbrace{\flA\otimes\cdots\flA}_{\hbox{$n$ \rm factors}}
\bigr)=0
\quad.\eqno(3.2)$$
\eproclaim
\proof:
For a translation invariant state $\om$ with associated local
density matrices $D_\om^{\Lambda}$ and any $q>1$ the R\'enyi-entropy
$R^q(\om\rstr\A_\Lambda)$ is defined by
$$ R^q(\om\rstr\A_\Lambda)
=-{1\over q-1}\log \tr (D_\om^{\Lambda})^q
\quad,$$
and the mean R\'enyi-entropy $R^q_M(\om)$ is then defined by
$$ R^q_M(\om)
=\limsup_{n-m\to\infty}{1\over n-m}R^q(\om\rstr\A_\Imn)
\quad.$$
In \cite\FCSE\ it is shown that for $1,\> A_1 e_{i_2}\rangle
\langle e_{i_2}\>,\> A_2 e_{i_1}\rangle\cr
&= \sum_{i_1,i_2}\langle e_{i_1}\otimes e_{i_2}\>,\>
A_1\otimes A_2\ e_{i_2}\otimes e_{i_1}\rangle \cr
&= \tr_{\H\otimes\H}
((A_1\otimes A_2)\ \flH)
\quad,&(3.3)\cr}$$
where
$\flH:\H\otimes\H\to\H\otimes\H
:\phi_1\otimes\phi_2\mapsto\phi_2 \otimes\phi_1$.
Applying (3.3) with $\H=(\Cx^d)^{\otimes n-m+1}$ and
$A_1=A_2=D_\om^\Imn$, we get:
$$\eqalign{
R^2(\om\rstr\A_\Imn)
&=\om\otimes\om (\flH)\cr
&=\omom(\underbrace{\flA\otimes\cdots\flA}_{
\hbox{$n-m+1$ \rm factors}})
\quad.}$$
Taking the $\limsup$ the desired result follows.
\QED
Note that this result does not use the fact that $\om$ is a \cfc\
state. However, if we do have a \cfc\ \triple{#1\up2} generating
$\omom$, expectations as the one in equation (3.2) are particularly
easy to compute. One can easily see that $\omom$ is \cfc\ iff $\om$
is \cfc. We then have
$$\omom\bigl(\flA\otimes\cdots\flA\bigr)
=\rho\bigl(\E\up2_{\flA}\circ\cdots\E\up2_{\flA}(\idty)\bigr)
=\rho\bigl(\F^n(\idty)\bigr)
\quad,$$
where
$$ \F(B)=\E\up2_{\flA}(B)=\E\up2(\flA\otimes B)
\quad.$$
Thus the asymptotics of the left hand side of (3.2) is governed by
the spectral properties of $\F$. Note that
$$ \F:B\mapsto \flA\otimes B\mapsto\E\up2(\flA\otimes B)=\F(B)$$
is a contraction as the product of two contractions. We may thus
apply the following Lemma to $\F$ as an operator on the Banach
space $X\equiv\B\up2$ with $x=\idty$, and $\xi=\rho$.
\iproclaim Lemma 3.3.
Let $X$ be a finite dimensional Banach space, and $\F$ a contraction
in $X$. Suppose that there are $x\in X$, $\xi\in X^*$, such that
$\bra\xi,\F^n x>$ is positive for all $n\in\Nl$, and that
$\limsup_n {-1\over n}\ln\bra\xi,\F^n x>=0$.
Then $1$ is an eigenvalue of $\F$.
\eproclaim
\proof:
Let $\F=\sum_\lambda(\lambda P_\lambda+N_\lambda)$ be the spectral
decomposition of $\F$ with
$$\eqalign{
P_\lambda P_\mu &=\delta_{\lambda,\mu}P_\lambda \cr
P_\lambda N_\mu &=N_\mu P_\lambda
= \delta_{\lambda,\mu}N_\lambda \cr
}$$
and $N_\lambda$ nilpotent.
Then, for the eigenvalues with $\abs{\lambda}=1$, the nilpotent part
$N_\lambda$ must vanish. Otherwise let $k$ be such that
$N_\lambda^k\neq0,\ N_\lambda^{k+1}=0$ and pick $x'\in P_\lambda X,
\xi'\in X^*$ such that $\bra\xi',\,N_\lambda^k x'>\neq0$.
Then
$$\bra\xi',\F^n x'>=\lambda^n\ \sum_{\kappa=1}^k {n\choose\kappa}
\lambda^{-\kappa}\bra\xi',N_\lambda^\kappa x'>$$
Apart from the factor in front of the sum, which is of constant
modulus, this is a polynomial of degree $k$ in $n$. This contradicts
the assumption that $\F$ is a contraction. Hence we have
$$ \bra\xi,\F^n x> = {\sum_\lambda}' \lambda^n \bra\xi,P_\lambda x>
+R_n
\quad,$$
where the prime indicates a sum only over the eigenvalues of
modulus $1$, and $\abs{R_n}\leq c\mu^n$ for some $0\leq\mu<1$.
If there are no eigenvalues of modulus $1$, then
$\limsup_n {-1\over n}\ln\bra\xi,\F^n x>
\geq \limsup_n ({-1\over n}(\ln c\,+n\ln\mu))
= -\ln\mu >0$,
contradicting the assumption.
Now for large $n$ the matrix element $\bra\xi,\F^n x>$ becomes
equal to an almost periodic function in $n$.
This can be positive only if the mean of this function (which is
equal to the constant term in the Fourier decomposition) is
positive.
Consequently, $\bra\xi,P_1 x>$ is non-zero, which proves the
assertion.
\QED
If $\Zo(\E\up2)$ is trivial so that $G(\E\up2)$ is defined, the
Lemma just states that $\flA\in G(\E\up2)$. If we take the two
factors of $\omom$ to be generated by a priori different triples
this flip induces a symmetry exchanging the two representations.
This is the basic idea of the following uniqueness result. However,
as we have noted above $\omom$ is not necessarily ergodic, and the
argument becomes slightly more involved. Combined with Proposition
3.1 the following Proposition proves Theorem 1.3 under the extra
assumption that $\om$ is ergodic.
\iproclaim Proposition 3.4.
Let $\om$ be an ergodic \cfc\ state with vanishing mean entropy. Then,
up to isomorphism in the sense of Definition 1.2, there is a
unique minimal triple \triple{#1} generating $\om$.
For this triple, $\ZoE=\Cx\idty$.
\eproclaim
\proof:
Every standard triple \triple{#1} generating $\om$ is the direct sum
of triples \triple{#1_P} satisfying the additional condition
$\Zo(\E_P)=\Cx\idty$. By ergodicity each of these triples generates
the same state $\om$. Suppose we have shown that all triples with
the additional property $\Zo(\E_P)=\Cx\idty$ are isomorphic. Then
applying an appropriate isomorphism in each summand we find that
\triple{#1} is isomorphic to a direct sum of {\it equal} triples. It
is easy to see that the subalgebra $\B'\subset\B$ of this direct
sum, which consists of the direct sum of equal elements, contains
$1$ and is invariant under $\E_\A$. By minimality we conclude that
$\B'=\B$, \ie there can be only one summand.
Let \triple{#1_1} and \triple{#1_2} be two minimal triples
generating $\om$ such that $\Zo(\E_1)=\Cx\idty$ and
$\Zo(\E_1)=\Cx\idty$. It remains to be shown that these triples are
isomorphic. $\omom$ is again a \cfc\ state and as generating triple
we take
$$ \hbox{\triple{#1\up2}}
\equiv\hbox{\triple{#1_1\otimes#1_2}}
\quad.$$
In the definition of $\E\up2$ we implicitly reshuffle the tensor
factors, \ie
$$ \E\up2\bigl((A_1\otimes A_2)\otimes(B_1\otimes B_2)\bigr)
=\E_1(A_1\otimes B_1)\otimes\E_2(A_2\otimes B_2)
\quad,$$
so that $\E\up2:(\A\otimes\A)\otimes\B\up2\to\B\up2$ as required.
It is elementary to check that we have indeed constructed a
standard triple generating $\omom$. The algebra $\B\up2$ in this
triple is minimal. Indeed a minimal algebra by definition
must be generated from the identity by algebraic operations and
operators $B\in\B\up2\mapsto\E\up2(A\otimes B)$ for all
$A\in\A\otimes\A$. If we use only elements $A=A_1\otimes\idty$ we
generate the algebra $\B_1\otimes\idty$ since $\B_1$ was assumed to
be minimal. By the same token $\idty\otimes\B_2$ lies in the minimal
algebra, hence $\B\up2=\B_1\otimes\B_2$ is minimal.
Combining Lemma 3.2 and Lemma 3.3 we conclude that the equation
$\E\up2(\flA\otimes S)=S$ has a nonzero solution $S\in\B\up2$. As
$\omom$ is not necessarily ergodic it will be useful to focus our
attention on a suitable ergodic component of $\omom$. So let $P$ be
some minimal projection in $\Zo(\E\up2)$ such that $PS\neq0$. Then
$PS$ again satisfies the equation for $\flB$. We shall focus on the
ergodic component $\tilde\om\equiv(\omom)_P$ associated to $P$
which is generated by $\tilde\B=P\B\up2\subset\B\up2$,
$\tilde\E=\E\up2\rstr(\A\otimes\A)\otimes\tilde\B$, and the state
$\tilde\rho=\rho(P)^{-1}\rho\rstr\tilde\B$. We take as $S$ its
component in $\tilde\B$, \ie $S\equiv PS$. We thus have
$\flA\in\Get$ with associated unitary $S\in\tilde\B$. Since
$\flA^2=\idty$ we get $S^2\in\Zo(\tilde\E)$ so that we may adjust
$S$ with a scalar factor to satisfy
$S^2=S^*S=P\idty_{\tilde\B}$. It is clear that
$$\tau(B)=S BS \qquad\hbox{for $B\in\tilde\B$}
\eqno(3.4)$$
defines a ``flip'' automorphism on $\tilde\B$. This automorphism will
be used below. It is elementary to check that
$$ \tau\circ\tilde\E_{A_1\otimes A_2}
= \tilde\E_{A_2\otimes A_1}\circ\tau
\quad.$$
We will now compare the two generating triples \triple{#1_1} and
\triple{#1_2} using the automorphism $\tau$.
Consider the homomorphism $\pi_1:\B_1\to\tilde\B$ with
$\pi_1(B_1)=P(B_1\otimes\idty)$, and an analogously defined
$\pi_2$.
Then a straightforward computation shows that
$$ \pi_1\bigl(\E_1(A_1\otimes B_1)\bigr)
= \tilde\E\bigl((A_1\otimes\idty)\otimes\pi_1(B_1)\bigr)
\quad.$$
We claim that both $\pi_1$ and $\pi_2$ are faithful.
Indeed, consider the ideal $\ker\pi_1\subset\B_1$.
Then it is obvious from the above relation between $\E_1$ and
$\tilde\E$ that $\ker\pi_1$ is invariant under
$\E_1(A_1\otimes\cdot)$ for all $A_1\in\A$. Since $\B_1$ is a direct
sum of finite matrix algebras, the only ideals in $\B_1$ are direct
summands.
The central projection $z\in\B_1$ with $z\B_1=\ker\pi_1$ is thus
invariant under $\E_1(\idty\otimes\cdot)$, and by the assumption
$\Zo(\E_1)=\Cx\idty$, and $\E_1(\idty\otimes\idty=\idty$, we must
have $z=0$.
As the last step in the proof of the Proposition we construct an
isomorphism $\tilde\tau$ between the triples \triple{#1_1} and
\triple{#1_2}. We first show that
$$\tau\pi_1(\B_1)=\pi_2(\B_2), \eqno(3.5)$$
where $\tau$ is the automorphism of $\tilde\B$ defined in (3.4).
Then, since $\pi_1$ and $\pi_2$ are faithful, the isomorphism
$\tilde\tau=\pi_2^{-1}\tau\pi_1:\B_1\to\B_2$ is well defined.
In order to prove (3.5), let $\D_k, k\geq0$ be the increasing
family of C*-algebras generating $\B_1$ as in the proof of
Proposition 2.2.
We shall show inductively that $\tau\pi_1(\D_k)\subset\pi_2(\B_2)$.
Since $\tau$ is a homomorphism it suffices to show that if
$A\in\A$, and $\tau\pi_1(D)=P(\idty\otimes D_2)\in\pi_2(\B_2)$,
then $\tau\pi_1(D')=P(\idty\otimes D'_2)$ for $D'= \E_1(A\otimes D)$.
But
$$\eqalign{
\tau\pi_1(D')&= S^3(D'\otimes\idty)S
= S\Bigl(\E_1\otimes\E_2\Bigr)\bigl((A\otimes\idty)
\otimes(D\otimes\idty)\bigr)S \cr
&=\tau\tilde\E((A\otimes\idty)\otimes\pi_1(D))
=\tilde\E((\idty\otimes A)\otimes\tau\pi_1(D))\cr
&=\tilde\E((\idty\otimes A)\otimes\pi_2(D_2))
=P(\idty\otimes\E_2(A\otimes D_2)) \cr
&=P(\idty\otimes D_2')
\quad.}$$
Hence $\tau\pi_1(\B_1)\subset\pi_2(\B_2)$ and in reversing the roles
of 1 and 2 we conclude that $\tau\pi_1(\B_1)=\pi_2(\B_2)$. The same
computation shows that
$\tau\pi_1\E_1(A\otimes B_1)=\pi_2\E_2(A\otimes\tilde\tau(B_1))$,
which implies $\tilde\tau\circ\E_1 = \E_2\circ\id\otimes\tilde\tau$.
The relation for the $\rho$'s which is needed to get the isomorphism
between triples follows from the uniqueness of the $\E$-invariant
state in a standard triple with $\ZoE=\Cx\idty$.
\QED
For a general \cfc\ state $\om$ generated by a minimal triple
\triple{#1} the algebra $\Ze$, defined in Proposition 2.4, is a
proper subalgebra of the center of $\B$. However, this possibility
is excluded if we impose the condition $S_M(\om)=0$.
\iproclaim Proposition 3.5.
Let $\om$ be an ergodic \cfc\ state with vanishing mean entropy.
Then in the minimal triple \triple{#1} generating $\om$ the center
of $\B$ coincides with $\Ze$.
\eproclaim
\proof:
By Proposition 3.4 we can assume that $\pi_2=\tau\pi_1$.
Suppose that $\B=\bigoplus\sa\B\sa$ is the
decomposition of $\B$ into irreducible summands.
Denote the corresponding central projections by $p\sa$.
Since $p\sa$ is central in $\pi_1(\B)$, and
$\bracks{\pi_1(\B),\pi_2(\B)}=0$,\ $\pi_1(p\sa)$
is central in $\tilde\B$.
Since $\tau$ is an inner automorphism in $\tilde\B$, we have
$\pi_1(p\sa)=\tau\pi_1(p\sa)=\pi_2(p\sa)$.
Recall that $\tilde\B$ was defined as $P\B\otimes\B$ for a suitable
minimal projection $P\in\Z_1(\E\otimes\E)$. Since $P$ is in the
center of $\B\otimes\B$, it is a sum of projections of the form
$p\sa\otimes p_\beta$. By definition of $\pi_i$ we have
$P(\idty\otimes p\sa)=\pi_2(p\sa)=\pi_1(p\sa)=P(p\sa\otimes\idty)$.
Hence $p\sa\otimes p_\beta$ can be contained in $P$ only if
$\alpha=\beta$. On the other hand $\pi_i$ is faithful so that each
$p\sa$ has to appear in the first factor at least once.
Consequently,
$$ P= \sum\sa p\sa\otimes p\sa \quad.$$
The operator $\E$ can be decomposed into blocks along the direct sum
decomposition of $\B$ as $\E=\sum_{\alpha\beta}\E_{\alpha\beta}$
with $\E_{\alpha\beta}(A\otimes B)=p\sa\E(A\otimes(p_\beta B))$.
Suppose that for some fixed $\beta$ and two different
$\alpha_1,\alpha_2$ we have $\E_{\alpha_i\beta}\neq0$. Then
$$\eqalign{
0\neq\quad&
(\E_{\alpha_1\beta}(\idty))\otimes(\E_{\alpha_2\beta}(\idty)) \cr
&=(p_{\alpha_1}\otimes p_{\alpha_2})\E\otimes\E(\idty_{\A\otimes\A}
\otimes(p_\beta\otimes p_\beta)) \cr
&\leq (p_{\alpha_1}\otimes p_{\alpha_2})\E\otimes\E
(\idty\otimes P) \cr
&= (p_{\alpha_1}\otimes p_{\alpha_2})\ P
=0 \qquad.\cr}$$
This is a contradiction, so that for each $\beta$ there must be a
unique $\alpha$ with $\E_{\alpha\beta}\neq0$.
In other words there is a permutation $\sigma$ such that
$\E(\idty\otimes p\sa)= p_{\sigma(\alpha)}$.
Hence each $p\sa$ is invariant under a suitable power of $\Eh$ and
$p\sa$ is a minimal projection in $\Ze$.
Comparing with Proposition 2.4 it is clear that we have reconstructed
the central flow of $\E$.
\QED
In Proposition 3.1 we have shown that a \pg\ \cfc\ state has zero
mean entropy. We now prove the converse under the extra assumption
of ergodicity of the \cfc\ state $\om$. Combining both results we
obtain therefore Theorem 1.4 for the ergodic case.
\iproclaim Proposition 3.6.
Any ergodic \cfc\ state $\om$ with vanishing mean entropy is \pg.
\eproclaim
\proof:
\def\Esa{\E_{\alpha}}
Using the same notation as in the proof of Proposition 3.5, let
$\Esa\equiv\E_{\alpha,\sigma(\alpha)}:
\A\otimes\B\sa\to\B_{\sigma(\alpha)}$ be one of the
non-zero summands of $\E$. This operator is ``pure'' in the
terminology of Davies \cite\DAV\ \ie a pure state $\phi$ on
$\B_{\sigma(\alpha)}$ is mapped by $\Esa$ in a pure state on
$\A\otimes\B\sa$. In order to prove this we will use the fact that a
state $\phi$ on a matrix algebra is pure iff $\phi\otimes\phi(\flip)=1$,
where $\flip$ denotes the flip operator. This is an immediate
consequence of the trace formula (3.3).
Let $S\sa$ be the component of $S\in\tilde\B=\bigoplus_\alpha
\B\sa\otimes\B\sa$ in $\B\sa\otimes\B\sa$. Since for $B\in\B\sa$ we
have $S\sa(B\otimes\idty)S\sa=\idty\otimes B$, $S\sa$ must be equal
to the flip $\flip\sa\in\B\sa\otimes\B\sa$ up to a phase $\epsilon\sa$.
Since $S\sa^2=p\sa$ this phase is further determined as
$\epsilon_\alpha =\pm1$. Now let $\phi$ be a pure state on
$\B_{\sigma(\alpha)}$. By Proposition 3.5
$\phi'\equiv\phi\circ\E\sa$ is supported by $\A\otimes\B\sa$. By
assumption
$$\def\phii{\phi\otimes\phi} \eqalign{
1=\phii(\flip_{\sigma (\alpha)})
&=\epsilon_{\sigma(\alpha)} \phii(S)
=\epsilon_{\sigma(\alpha)} \phii\bigl(\E\otimes
\E(\flA\otimes S)\bigr)\cr
&=\epsilon_{\sigma(\alpha)} \phi'\otimes\phi'(\flA\otimes S)
=\epsilon_{\sigma(\alpha)}\epsilon\sa \phi'\otimes\phi'
(\flA\otimes \flip\sa)
\quad.}$$
Since $\phi'\otimes\phi'(\flA\otimes \flip\sa)$ cannot be $-1$ it must
be $+1$, hence $\phi'$ is pure. In passing we have also shown that
$\epsilon_{\sigma(\alpha)}=\epsilon_\alpha$, \ie $S\sa=\pm \flip\sa$
with a sign independent of $\alpha$.
It is shown in Theorem 3.1 of \cite\DAV\ that a positive map between
matrix algebras is ``pure'' iff it is either $(i)$ implemented by a
linear map, $(ii)$ implemented by an antilinear map, or $(iii)$ of
rank one. Of these possibilities the second is ruled out because
$\Esa$ is not only positive but completely positive. The third would
map any operator into a multiple of the identity, which is
incompatible with the property
$\Esa(\flA\otimes \flip\sa)= \flip_{\sigma(\alpha)}$ except in the trivial
case where $\dim\B\sa=1$ for all $\alpha$ which is covered by $(i)$
anyway. Hence $\Esa$ is linearly implemented. More explicitly, if we
set $\B\sa=\B(\K\sa)$, and $\A=\B(\H)$ for some finite dimensional
Hilbert spaces $\K\sa,\H$ we find that there are isometries
$$ V\sa:\H\otimes\K\sa\to\K_{\sigma(\alpha)}
\quad,$$
which are unique up to a phase, such that $\Esa(X)=V\sa^*XV\sa$. We
can also introduce the space $\K=\bigoplus\sa\K\sa$, and take $\B$
to be represented on $\K$ in the obvious way. One easily checks that
$V:\H\otimes\K\to\K$ defined by
$V(\phi\otimes\bigoplus\sa\psi\sa)
= \sum\sa V\sa(\phi\otimes\psi\sa)$
is an isometry, and that for $A\in\A,B\in\B$ we have $\E(X)=V^*XV$.
This formula can be taken as the definition of an extension of $\E$
from the minimal algebra $\A\otimes\B$ to $\A\otimes\B(\K)$.
This extension is then a pure map generating $\om$.
\QED
Most of the results in this section were proven under the additional
assumption that the \cfc\ state $\om$ was ergodic. The following
Proposition shows that for (not necessarily ergodic) \pg\ states the
equivalent conditions of Lemma~2.5(2) hold automatically, thus
eliminating the last remaining source for non-uniqueness in the
representation of such states. It also extends Proposition~3.5 to
the non-ergodic case.
\iproclaim Proposition 3.7.
Let \triple{#1} be a minimal triple generating a \pg\ state $\om$.
Then $\Ze\subset\B_0$, and $\Ze$ is equal to the center of $\B$.
\eproclaim
\proof:
We show that, for distinct minimal projections $P,P'\in\ZoE$, the
states $\om_P$ and $\om_{P'}$ are distinct. By Proposition 2.5
$\om_P$ is again \pg\ as an ergodic components of $\om$, and by
Proposition 3.4 it has up to isomorphism a unique generating triple.
Hence if two summands $\om_P$ and $\om_{P'}$ coincide, the $\E$ acts
on corresponding summands like
$\E_A(B\oplus B')=(\E_A(B))\oplus\tau(\E_A(\tau^{-1}B'))$, where
$\tau$ is the map implementing the isomorphism. The minimal algebra
generated from $\idty\oplus\idty$ and the successive images of
$\E_A$'s therefore contains only elements with $\tau B=B'$. This
contradicts the assumption that $\B$ was minimal in the first place.
It follows that to each ergodic component corresponds a unique
minimal projection in $\ZoE$, and by Lemma 2.7 we conclude that
$\ZoE\subset\B_0$. Applying the same arguments to a regrouped chain,
we find also that $\Ze\subset\B_0$. The last statement follows by
decomposing the center of $\B$ into summands with respect to $\ZoE$,
and applying Proposition 3.5 to each summand.
\QED
\proof{\ of Theorems 1.3, 1.4, and 1.5:}
Suppose $\om$ is a \cfc\ state with vanishing entropy density. Then
since the entropy density is a positive affine functional on the set
of translation invariant states the same is true for the ergodic
components of $\om$. Applying Proposition 3.6 to each component and
using the convexity of the set of \pg\ states (Proposition 2.5) we
find that $\om$ is purely generated. This, together with Proposition
3.1, proves Theorem 1.4.
Now suppose that $\om$ is \pg, and let \triple{#1} be a minimal
triple generating $\om$.
Then by the previous Proposition each ergodic component corresponds
to a unique minimal projection in $\ZoE$. Furthermore, the ergodic
\dec\ is uniquely determined by the state $\om$ alone. Since by
Proposition 3.4 the minimal triple generating each ergodic component
is unique, the minimal triple for $\om$ is uniquely determined as
the direct sum of the triples representing the components, which
proves Theorem 1.3.
Finally it remains to prove Theorem 1.5. If $\om$ is pure, its
mean entropy vanishes by Proposition 3.1. By Theorem 1.4 this
implies that $\om$ is \pg. Thus if any one of the conditions in
Theorem 1.5 is satisfied $\om$ is \pg, and hence has a unique
minimal generating triple by Theorem 1.3.
We will argue now that any one of the conditions also implies that
$\om$ has no non-trivial \dec\ into periodic components. Since
condition (i) excludes {\it any} convex \dec\ this is trivial for
(i). From (iv) we find that $\Ze=\Cx\idty$ for a minimal triple,
which implies by Proposition 2.4 that $\Eh$ has trivial peripheral
spectrum. This implies
exponential decay of correlation functions, \ie spatial clustering
of $\om$. By standard theorems \cite\BraRo\ this implies that $\om$
is ergodic, and furthermore has no periodic \dec.
We have thus shown (iv)$\iff$(iii)$\iff$(ii)$\Leftarrow$(i). The
remaining direction (iv)$\Rightarrow$(i) was shown in Proposition 5.8
of \cite\FCS\ by demonstrating that $\om$ is the unique ground state
of a finite range Hamiltonian.
\QED
%%%%%%%%%%%%%%%%%
% file fcpr.tex %
%%%%%%%%%%%%%%%%%
%
\ACKNOW
B.N. acknowledges partial financial support from NSF Grant \# PHY-8912067.
R.F.W. was supported by the Deutsche Forschungsgemeinschaft in Bonn
with a fellowship and a travel grant.
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