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39 pages; latex2e
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Duality, compact groups, nontrivial center,
tensor C*-categories, Hilbert C*-modules
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\title{\bf Duality of compact groups and
Hilbert C*-systems for C*-algebras with a nontrivial center}
\author{
{\sc Hellmut Baumg\"artel}\\[2mm]
{\footnotesize Mathematical Institute, University of Potsdam,}\\
{\footnotesize Am Neuen Palais 10, PF 601 553,} \\
{\footnotesize D-14415 Potsdam, Germany.} \\[1mm]
{\footnotesize baumg@rz.uni-potsdam.de} \\
\and
{\sc Fernando Lled\'o} \\[2mm]
{\footnotesize Institute for Pure and Applied Mathematics,} \\
{\footnotesize RWTH-Aachen, Templergraben 55,} \\
{\footnotesize D-52062 Aachen, Germany.} \\[1mm]
{\footnotesize lledo@iram.rwth-aachen.de}}
\date{\today{}}
\begin{document}
\maketitle
\begin{abstract}
The new duality theory for compact groups established
in \cite{Doplicher89b} by Doplicher and Roberts (DR)
characterizes the dual object
$\wh {\al G.}$ of a compact group $\al G.$ by means of
a particular type of tensor C*-category $\al T.$ which we
call a DR-category. $\al T.$ may be realized as a full subcategory
of the category of endomorphisms
of a suitable C*-algebra $\al A.$ with trivial center
$\al Z.(\al A.)=\al Z.=\C\1$. In this context the DR-duality
theory gives a bijective correspondence between the pair
$\{\al A.,\al T.\}$ and a special kind of C*-dynamical system
$\{\al F.,\alpha_\al G.\}$ that, in addition, contains
explicitly the representation category of the compact group
$\al G.$ and which we call a Hilbert C*-system.
The C*-algebra $\al A.$ is the fixed point algebra of
the group action and $\al A.'\cap\al F.=\C\1$.
In the present paper we prove a duality theory for compact
groups in the case when the C*-algebra $\al A.$ has a nontrivial
center $\al Z.\supset\C\1$ and the relative commutant of the
corresponding Hilbert C*-system satisfies the minimality condition
\[
\al A.'\cap\al F.=\al Z.\,,
\]
as well as a technical condition called regularity.
The abstract characterization of the
mentioned Hilbert C*-system is expressed by means
of an inclusion of
C*-categories $\al T._\c < \al T.$, where $\al T._\c$ is a suitable
DR-category and $\al T.$ a full subcategory of the category of
endomorphisms of $\al A.$. Both categories have the same objects
and the arrows of $\al T.$ can be generated from the arrows of
$\al T._\c$ and the center $\al Z.$.
A crucial new element that appears in the present analysis is an
abelian group $\ot C.(\al G.)$, which we call the chain group
of $\al G.$, and that can be constructed from certain equivalence
relation defined on $\wh{\al G.}$.
The chain group can be related to the character group of the center
of $\al G.$ and
determines the action of irreducible endomorphisms
of $\al A.$ when restricted to $\al Z.$.
Moreover, $\ot C.(\al G.)$ encodes the possibility of defining
a symmetry $\epsilon$ also for the larger category $\al T.$ of the previous
inclusion.
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
The superselection theory in algebraic quantum field theory,
as stated by the
Doplicher-Haag-Roberts~(DHR) selection criterion
\cite{bHaag92,DHR69a,DHR69b},
led to a profound body of work, culminating in
the general Doplicher-Roberts~(DR) duality theory for compact groups
\cite{Doplicher89b}.
The DHR criterion selects a distinguished class of ``admissible''
representations of a quasilocal algebra $\al A.$ of observables,
which has trivial center
$\al Z.:=\al Z.(\al A.)=\C\1$.
This corresponds to the selection of a
so-called DR-category $\al T.$,
which is a full subcategory of the category of
endomorphisms of the C*-algebra $\al A.$ (see Definition~\ref{DRCat} below).
Furthermore, from this endomorphism category
$\al T.$
the
DR-analysis constructs a C*-algebra $\al F.\supset\al A.$
together with a compact group action
$\alpha:\al G.\ni g\to\alpha_{g}\in\aut\al F.$
such that:
\begin{itemize}
\item
$\al A.$ is the fixed point algebra of this action
\item
$\al T.$ coincides with the category of all ``canonical
endomorphisms" of $\al A.$, associated with the pair
$\{\al F.,\alpha_{\al G.}\}$ (cf.~Subsection~\ref{SubCanEn}).
\end{itemize}
$\al F.$
is called a Hilbert extension of $\al A.$ in \cite{bBaumgaertel92}.
Physically, $\al F.$ is identified as a field algebra and
$\al G.$ with a global gauge group of the system.
The pair $\{\al F.,\alpha_{\al G.}\}$, which we call
{\em Hilbert C*-system} (cf.~Definition~\ref{defs2-1};
the name {\em crossed product} is also used),
is uniquely determined by
$\al T.$
up to
$\al A.$-module isomorphisms. Conversely,
$\{\al F.,\alpha_{\al G.}\}$
determines uniquely its category of all canonical endomorphisms.
Therefore $\{\al T.,\al A.\}$ can be seen as the abstract side
of the representation category of a compact group, while
$\{\al F.,\alpha_{\al G.}\}$ corresponds
to the concrete side of the representation category of
$\al G.$, and, roughly, any irreducible representations
of $\al G.$ is explicitly realized within
the Hilbert C*-system.
One can state the equivalence of the ``selection
principle", given by
$\al T.$
and the ``symmetry principle", given by the compact group
$\al G.$. This is one of the crucial theorems of the
Doplicher-Roberts theory.
In the DR-theory the center $\al Z.$ of the C*-algebra $\al A.$
plays a peculiar role:
As stated above, if $\al A.$ corresponds
to the inductive limit of a net of local C*-algebras indexed by
open and bounded regions of Minkowski space, then the triviality
of the center of $\al A.$ is a consequence of standard assumptions
on the net of local C*-algebras. But, in general, the C*-algebra appearing
in the DR-theorem does not need to be a quasilocal algebra and, in fact,
one has to assume explicitly that $\al Z.=\C\1$
in this context (see \cite[Theorem~6.1]{Doplicher89b}).
Finally, we quote from the introduction of the article
\cite{Doplicher89b}: ``There is, however, no
known analogue of Theorem~4.1 of \cite{Doplicher89a} for a C$^*$-algebra
with a non-trivial center and hence nothing resembling a ``duality''
in this more general setting.''
The aim of the present paper is to show that
{\em there is} a duality theory for compact groups in the
case of a nontrivial center, if the relative commutant of the
corresponding Hilbert C*-system satisfies the following minimality
condition:
\begin{equation}\label{1afz}
\al A.'\cap\al F.=\al Z.
\end{equation}
(cf.~Theorem~\ref{Teo2}). The essence of the
previous result is that now the abstract characterization of the
mentioned Hilbert C*-system is expressed by means
of an inclusion of
C*-categories $\al T._\c < \al T.$, where $\al T._\c$ is a suitable
DR-category and $\al T.$ a full subcategory of the category of
endomorphisms of $\al A.$. Both categories have the same objects
and the arrows of $\al T.$ can be generated from the arrows of
$\al T._\c$ and the center $\al Z.$.
Several new elements appear in the generalization of the DR-theory
studied here. The crucial one is an
abelian group $\ot C.(\al G.)$, which we call the {\em chain group}
of $\al G.$, and that can be constructed from certain equivalence
relation defined on $\wh{\al G.}$, the dual object of the compact
group $\al G.$. The chain group, which is interesting
in itself and can be related to the character group of the center
of $\al G.$, determines the action of irreducible endomorphisms
of $\al A.$ when restricted to the center $\al Z.(\al A.)$.
Moreover, $\ot C.(\al G.)$ appears explicitly in the construction
of a family of examples realizing the inclusion of categories
$\al T._\c<\al T.$ mentioned above (cf.~Section~\ref{TrivialChainHom}).
Finally, the chain group encodes also the possibility of defining
a symmetry $\epsilon$ also for the larger category $\al T.$ of the previous
inclusion.
There are several reasons that motivate the generalization of
the DR-theory for systems satisfying the minimality condition
(\ref{1afz}) for the relative commutant:
\begin{itemize}
\item[(i)]
In this context there is a nice intrinsic characterization
of the Hilbert C*-systems satisfying (\ref{1afz}) and a further
technical condition called regularity (cf.~Theorems~\ref{Teo1}
and \ref{Teo2}). One can also prove
several results in the spirit of the DR-theory: for example,
the category $\al T.$ is isomorphic to
the category $\al M._\al G.$ of all free right Hilbert
$\al Z.$-modules generated by the algebraic Hilbert spaces in
$\al T._\al G.$ (cf.~Proposition~\ref{prop0}).
\item[(ii)]
In the context of compact groups, the equation (\ref{1afz})
is also convenient for technical reasons. The minimality of
the relative commutant implies that irreducible endomorphisms are
mutually disjoint (cf.~Proposition~\ref{disj}) and this fact is
crucial to have a nice decomposition of objects in terms of irreducible
ones (cf.~Proposition~\ref{DecompEnd}).
\item[(iii)]
The nontriviality of the center gives also the possibility to a more
geometrical interpretation of the DR-theory. Indeed, from
Gelfand's theorem we have $\al Z.\cong C(\Gamma)$, $\Gamma$ a compact
Hausdorff space, and in certain situations the
Hilbert C*-system $\{\al F.,\alpha_{\al G.}\}$ is a bundle over $\Gamma$,
where the Hilbert C*-system corresponding to the base point
$\lambda\in\Gamma$ is of a DR-type with the same group $\al G.$.
Here the chain group
plays again an important role.
This more geometrical line of research has lead to recent developments
in the context of vector bundles
(cf.~\cite{pVasselli03a,pVasselli03b,Vasselli03}).
\item[(iv)]
There are physically relevant examples that satisfy the
condition (\ref{1afz}).
For example, this equation is presented in \cite{Mack90}
as a ``new principle". Moreover, the elements of the center $\al Z.$
of $\al A.$ may be interpreted as classical observables contained
in the quasilocal algebra.
\item[(v)] The present generalization of the DR-theory in the context
minimal and regular Hilbert C*-systems has also found application
in the context of superselection theory for systems carrying quantum
constraints (see \cite{pBaumgaertel03} as well as
\cite{Grundling85,Lledo00} for a C*-algebraic formulation of
the theory of quantum constraints).
\end{itemize}
The paper is structured in 9 sections: in Section~\ref{BasicHCS}
we introduce the notion of a Hilbert C*-system
(cf.~Definition~\ref{defs2-1})
and give a detailed account of its properties.
Hilbert C*-systems are special
types of C*-dynamical systems $\{\al F.,\alpha_{\al G.}\}$ that,
in addition, contain the information of the representation category
of $\al G.$. They also satisfy important properties,
which are interesting in themselves, as for example:
$\al F.$ is simple iff the fixed point algebra $\al A.$ is simple
(cf.~Proposition~\ref{SimpleAF}); one can naturally introduce spectral
subspaces of $\al F.$ and prove Parseval-type equations for a suitable
$\al A.$-valued scalar product on $\al F.$ (cf.~Proposition~\ref{prop1}).
Finally, Hilbert C*-systems provide a natural and
concrete frame to describe the
DR-theory as well as the generalization
to the nontrivial center situation that we study here.
In Section~\ref{TwoEx} we study the
important relation between two C*-categories
$\al T._\al G.$ and $\al T.$
that are naturally associated with a Hilbert C*-system.
In general, $\al T._\al G.$ is a subcategory of $\al T.$
and this inclusion turns out to be characteristic for the inverse
result stated in Theorem~\ref{Teo2}. In
Section~\ref{MinRegSect}
the main duality theorems are stated in the context of minimal
and regular Hilbert C*-systems. The next section defines
the notion of an irreducible object and introduces the
chain group of $\al G.$, denoted by $\ot C.(\al G.)$.
We give examples of chain groups for several finite and
compact Lie groups and state the conjecture that
the chain group is isomorphic to the character group of the
center of $\al G.$. There is a close relation between
the chain group and the set of irreducible canonical
endomorphisms: an irreducible canonical endomorphism of $\al A.$
restricted to the center $\al Z.$ turns out to be an
automorphism of $\al Z.$. We show that
there is a group homomorphism between the chain group and
the subgroup of $\mr aut.\al Z.$ generated by irreducible
objects (cf.~Theorem~\ref{EndoChain}). One of the typical
difficulties in the context of a nontrivial center is that
$\al Z.$ is not stable under the action of a general canonical
endomorphism $\sigma$, i.e.
\[
\sigma(\al Z.)\not\subset\al Z.\,.
\]
In this section we also give
an explicit formula in terms of isotypical projections
that describes the action of reducible endomorphisms
restricted to the center (cf.~Theorem~\ref{GeneralZMap}).
In Section~\ref{RealInclu} we construct a family of examples
that satisfy the requirements of the pair of
categories $\al T._\c<\al T.$ considered in Theorem~\ref{Teo2}.
In Section~\ref{TrivialChainHom} we analyze the
situation where the homomorphism between the chain group and
the subgroup of $\mr aut.\al Z.$ generated by irreducible
objects is trivial. In this case $\al Z.$ becomes the common
center of $\al A.$ {\it and} $\al F.$. We can therefore
decompose these algebras, which in this section are assumed
to be separable, w.r.t.~$\al Z.$. Then the
Hilbert C*-system $\{\al F.,\alpha_{\al G.}\}$ becomes
a bundle over $\Gamma:=\mr spec.\al Z.$ and
the fibre Hilbert C*-system corresponding to the base point
$\lambda\in\Gamma$ is of a DR-type with the same group $\al G.$.
That means, in particular, that
the fixed point algebra associated with $\lambda$ has a trivial
center. Another simplifying condition of the present situation
is the fact that any canonical endomorphism acts trivially
on the center, i.e.~$\rho\rest\al Z.=\mr id.\rest\al Z.$. Moreover,
we show that in this case the minimality condition already implies
the regularity of the corresponding Hilbert C*-system
(cf.~Corollary~\ref{Mintoreg}). The special situation studied
in this section is also related to the notion of extention of C*-categories
by abelian C*-algebras (cf.~\cite{Vasselli03}).
Some conclusions connecting the present analysis to related
lines of research are stated in Section~\ref{Conclu}. Finally,
the paper contains
an appendix recalling the decomposition
of a C*-algebra w.r.t.~its center.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Basic properties of Hilbert C*-systems}\label{BasicHCS}
In this section we summarize the structures from superselection
theory which we need. For proofs, we refer to the literature
if possible, otherwise proofs are included in this paper.
Below $\al F.$ will always denote a unital C*-algebra.
A Hilbert space $\al H.\subset
\al F.$ is called {\it algebraic} if the scalar product $\langle\cdot,\cdot
\rangle$ of $\al H.$ is given by
$\langle A,B\rangle\un := A^{\ast}B$ for $A,\; B\in\al H.\,.$ Henceforth,
we consider only finite-dimensional algebraic Hilbert spaces. The
support
$\hbox{supp}\,\al H.$
of $\al H.$ is defined by
$\hbox{supp}\,\al H.:=\sum_{j=1}^{d}\Phi_j\Phi_{j}^{\ast}$,
where $\{\Phi_j\,\big|\,
j=1,\ldots,\,d\}$ is any orthonormal basis of $\al H..$
Unless otherwise specified, we assume below that
each considered algebraic Hilbert space $\al H.$
satisfies ${\rm supp}\,\al H.
=\un.$
We also fix a compact
C*-dynamical system
$\{\al F.,\al G.,\alpha\}$,
i.e.
$\al G.$
is a compact group and
$\alpha:\al G.\ni g\to\alpha_{g}\in\aut\al F.$
is a pointwise norm-continuous morphism.
For $D\in\wh{\al G.}$
(the dual of $\al G.$) its {\it spectral projection}
$\Pi_{\mt D.}\in\al L.(\al F.)$
is defined by
\begin{eqnarray*}
\Pi_{{\mt D.}} (F)&:=&\int_{\al G.}\ol\chi_{\mt D.} (g).\,\alpha_{g}(F)\,dg
\quad\hbox{for all}\quad F\in\al F., \\[1mm]
\hbox{where:}\quad\qquad
\chi_{\mt D.} (g)&:=&\dim{D}\cdot\tr\pi(g),\quad\pi\in D\,.
\end{eqnarray*}
The spectrum of
$\alpha_{\al G.}$
can then be defined by
\[
\spec\alpha_{\al G.}:=\set D\in\wh{\al G.}, \Pi_{\mt D.}\not=0.\,.
\]
Note that
$\spec\alpha_{\al G.}$
coincides with the so-called
Arveson spectrum of
$\alpha_{\al G.}$
(see e.g.~\cite{Baumgaertel95}).
Our central object of study is:
\begin{defi}\label{defs2-1}
The compact C*-dynamical system
$\{\al F.,\al G.,\alpha\}$
is called a {\bf Hilbert C*-system} if
for each $D\in\wh{\al G.}$
there is an algebraic Hilbert space
$\al H._{\mt D.}\subset\Pi_{\mt D.}\al F.,$
such that
$\alpha_{\al G.}$
acts invariantly on
$\al H._{\mt D.},$
and the unitary representation
$\alpha_\al G.\rest\al H._{\mt D.}$
is in the equivalence class
$D\in\wh{\al G.}.$
\end{defi}
We are mainly interested in Hilbert C*-systems whose fixed point
algebras coincide such that they appear as extensions of it.
\begin{defi}\label{defs2-2}
A Hilbert C*-system
$\{\al F.,\al G.,\alpha\}$
is called a {\bf Hilbert
extension} of a C*-algebra $\al A.\subset\al F.$ if $\al A.$ is the
fixed point algebra of ${\al G.}.$
Two Hilbert extensions
$\{\al F._i,\,\al G.\,,\alpha^{i}\},\;i=1,\,2$
of $\al A.$
(w.r.t.~the same group $\al G.$)
are called
$\al A.\hbox{\bf-module isomorphic}$
if there is an isomorphism
$\tau:\al F._1\to\al F._2$
such that
$\tau(A)=A$
for
$A\in\al A.,$
and
$\tau$
intertwines the group actions, i.e.
$\tau\circ\alpha^{1}_g=\alpha^{2}_g\circ\tau$, $g\in\al G.$.
\end{defi}
\begin{rem}
\begin{itemize}
\item[(i)]
For a Hilbert C*-system
$\{\al F.,\al G.,\alpha\}$
one has
$\hbox{spec}\,\alpha_{\al G.}=\wh{\al G.}$
and the morphism
$\alpha:\al G.\to \hbox{Aut}\,\al F.$
is necessarily faithful. So, since
$\al G.$
is compact and
$\hbox{Aut}\,\al F.$
is Hausdorff w.r.t.~the topology of pointwise norm-convergence,
$\alpha$
is a homeomorphism of
$\al G.$
onto its image. Thus
$\al G.$
and
$\alpha_{\al G.}$
are isomorphic as topological groups.
\item[(ii)]
Group automorphisms of
$\al G.$
lead to $\al A.$-module isomorphic Hilbert extensions of
$\al A.$,
i.e.~if
$\{\al F.,\al G.,\alpha\}$
is a Hilbert extension of
$\al A.$
and
$\xi$
an automorphism of
$\al G.$,
then the Hilbert extensions
$\{\al F.,\al G.,\alpha\}$
and
$\{\al F.,\al G.,\alpha\circ\xi\}$
are $\al A.$-module isomorphic.
Therefore, the Hilbert C*-system
$\{\al F.,\al G.,\alpha\}$
depends, up to $\al A.$-module isomorphisms, only on
$\alpha_{\al G.}$,
which is isomorphic to
$\al G.$.
In other words, up to $\al A.$-module isomorphy we may identify
$\al G.$
and
$\alpha_{\al G.}\subset\aut\al F.$
neglecting the action
$\alpha$
which has no relevance from this point of view. Therefore in the
following, unless it is otherwise specified, we use the notation
$\{\al F.,\al G.\}$
for a Hilbert extension of
$\al A.$,
where
$\al G.\subset\aut\al F.$.
\item[(iii)]
As mentioned above,
Hilbert C*-systems arise in DHR-superselection theory
(cf.~\cite{bBaumgaertel92,bBaumgaertel95}). Mathematically,
there are constructions by means of
tensor products
$\al B.$
of Cuntz algebras
$\al O._{\al H._{u}},\;\al B.=\otimes_{u\in\ob\,\al R.}\al O._{\al H._{u}},$
where
$\al R.$
is a category whose objects $u$ are finite-dimensional continuous
unitary representations
of a compact group
$\al G.$
on Hilbert spaces
$\al H._{u}$
with
$\dim\,\al H._{u}>1$
and whose arrows are the corresponding intertwining operators
(cf.~\cite[Section~7]{Doplicher88}). In these examples the center
$\al Z.$ of the fixed point algebra
$\al A.$
is trivial.
Further examples
in the context of the CAR-algebra
with an abelian group
$\al G.=\T$
and nontrivial
center $\al Z.$ are given
in \cite{Baumgaertel01}. In Section~\ref{RealInclu} we construct
a family of examples of minimal and regular Hilbert C*-systems
for nonabelian groups and with nontrivial $\al Z.$.
\end{itemize}
\end{rem}
\begin{rem}
\label{remark1}
A Hilbert C*-system is a very highly structured object;-
below we list some important properties (for details, consult
~\cite{bBaumgaertel95,bBaumgaertel92}):
\begin{itemize}
\item[(i)]
Given two $\al G.$-invariant
algebraic Hilbert spaces
$\al H.,\al K.\subset\al F.,$
then
$\spa(\al H.\cdot\al K.)$
is also a $\al G.$-invariant
algebraic Hilbert space
which we will
briefly denote by $\al H.\cdot\al K..$
It carries the tensor product of the representations of $\al G.$
carried by $\al H.$ and $\al K..$
\item[(ii)]
Let
$\al H.,\al K.$
as before but not necessarily of support $\1$:
There is a natural isometric embedding of
$\al L.(\al H.,\al K.)$
into
$\al F.$
given by
\[
\al L.(\al H.,\al K.)\ni T\to\al J.(T):=\sum_{j,k}
t_{jk}\Psi_{j}\Phi^{\ast}_{k},\quad t_{jk}\in \C,
\]
where
$\{\Phi_{k}\}_{k}$ resp.~$\{\Psi_{j}\}_{j}$
are orthonormal basis of
$\al H.$ resp.~$\al K.$ and where
\[
T(\Phi_{k})=\sum_{j}t_{jk}\Psi_{j},
\]
i.e.
$(t_{j,k})$
is the matrix of $T$ w.r.t.~these orthonormal basis. One has
\[
T(\Phi)=\al J.(T)\cdot\Phi,\quad \Phi\in\al H..
\]
For simplicity of notation we will often put $\wwh T.:=\al J.(T)$.
Moreover, we have
$\wwh T.\in\al A.$
iff
$T\in\al L._{\al G.}(\al H.,\al K.)$,
where
$\al L._{\al G.}(\al H.,\al K.)$
denotes the linear subspace of
$\al L.(\al H.,\al K.)$
consisting of all intertwining operators of the representations
of
$\al G.$
on
$\al H.$ and $\al K.$ (cf.~\cite[p.~222]{bBaumgaertel92}).
\item[(iii)]
Generally for a Hilbert C*-system, the assignment $D\to\al H._{\mt D.}$
is not unique.
If
$U\in \al A.$
is unitary then also
$U\al H._{\mt D.}\subset\Pi_{{\mt D.}}\al F.$
is an $\al G.$-invariant algebraic Hilbert space carrying
a representation in
$D$. Note that each
$\al G.$-invariant algebraic
Hilbert space
$\al K.$
which carries a representation of
$D$
is of this form, i.e.~there is a unitary
$V\in\al A.$
such that
$\al K.=V\al H._{{\mt D.}}.$
\item[(iv)]
There is a useful partial order on the $\al G.$-invariant
algebraic Hilbert spaces. We define
$\al H.<\al K.$
to mean that there is an orthoprojection
$E$
on
$\al K.$
such that
$E\al K.$
is invariant w.r.t.
$\al G.$
and the representation
$\al G.\rest\al H.$
is unitarily equivalent to
$\al G.\rest E\al K.$.
Note that
$\al H.<\al K.$
iff there is an isometry
$V\in\al A.$
such that
$VV^{\ast}=:E$
is a projection of
$\al K.$,
i.e.
$V\al H.=E\al K.$
(use (ii)).
\item[(v)]
Given a Hilbert C*-system
$\HS$ a useful
*-subalgebra of
$\al F.$ is
\[
\al F._{\rm fin}:= \set F\in\al F.,\Pi_{\mt D.} F\not=0
\quad\hbox{for only finitely many $D\in\wh{\al G.}$}.
\]
which is dense in $\al F.$ w.r.t.~the C*-norm (cf.~\cite{Shiga55}).
\item[(vi)]
The spectral projections
satisfy:
\begin{eqnarray*}
\Pi_{{\mt D.}_1}\Pi_{{\mt D.}_2}
&=& \Pi_{{\mt D.}_2}\Pi_{{\mt D.}_1}=
\delta\s{\mt D.}_1{\mt D.}_2.\Pi_{{\mt D.}_1}\\[1mm]
\|\Pi_{\mt D.}\| &\leq& d({D})^{3/2}\;,
\qquad d({D}):=\dim(\al H._{\mt D.})\;, \\[1mm]
\Pi_{\mt D.}\al F. &=& \spa(\al AH._{\mt D.})\;, \\[1mm]
\Pi_{{\mt D.}}(AFB) &=& A\cdot\Pi_{{\mt D.}}(F)\cdot B,\quad
A,B\in\al A.,\,F\in\al F.\;, \\[1mm]
\al A. &=& \Pi_{\iota}\al F.\;,
\end{eqnarray*}
where
$\iota\in\wh{\al G.}$
denotes the trivial representation of $\al G..$
\item[(vii)]
In $\al F.$ there is an $\al A.$-scalar product given by
${\langle F,\, G\rangle_{\al A.}:=\Pi_\iota FG^*},$ w.r.t.~which
the spectral projections are symmetric, i.e.
$\bra\Pi_{\mt D.} F,G.=\bra F,\Pi_{\mt D.} G.$
for all $F,\; G\in\al F.,$ $D\in\wh{\al G.}$.
Using the $\al A.$-scalar product one can define a norm on
$\al F.$,
called the $\al A.$-norm
\[
\vert F\vert_{\al A.}:=\Vert\langle F,F\rangle\Vert^{1/2},\quad
f\in \al F..
\]
Note that
$\vert F\vert_{\al A.}\leq \Vert F\Vert$
and that
$\al F.$
in general is not closed w.r.t.~the $\al A.$-norm.
\end{itemize}
\end{rem}
The following result confirms the importance and naturalness of the
previously defined norm $|\cdot|_\al A.$ in the context of Hilbert
C*-systems. This norm plays also a fundamental role in the
so-called inverse
superselection theory which reconstructs the Hilbert C*-system from
the data $\al A.$ and a suitable family of endomorphisms of $\al A.$
(cf.~\cite{Baumgaertel97,bBaumgaertel95,Lledo01a}).
\begin{pro}
\label{prop1}
Let
$\HS$
be a Hilbert C*-system, then for
each $F\in\al F.$ we have
\begin{equation}\label{FG}
F=\sum_{{\mt D.}\in\wh{\al G.}}\Pi_{\mt D.} F
\end{equation}
where the sum on the right hand side is convergent w.r.t.~the
$\al A.$-norm
and we have Parseval's equation:
\begin{equation}\label{PEq}
\langle F,F\rangle_{\al A.}
=\sum_{{\mt D.}\in\wh{\al G.}}\langle\Pi_{\mt D.} F,\Pi_{\mt D.}
F\rangle_{\al A.}
\;.
\end{equation}
\end{pro}
\begin{beweis}
Let
$\Gamma\subset\wh{\al G.},\;\hbox{card}\,\Gamma<\infty$.
The set
$\{\Gamma\}$
of all such subsets of
$\wh{\al G.}$
is a directed net. The assertion (\ref{FG}) means
\[
\sum_{{\mt D.}\in\wh{\al G.}}\Pi_{{\mt D.}}F:=
\lim_{\Gamma\to\wh{\al G.}}F_{\Gamma},
\]
where
\[
F_{\Gamma}:=\sum_{{\mt D.}\in\Gamma}\Pi_{{\mt D.}}F,
\]
and ``lim'' means convergence w.r.t.~the $\al A.$-norm. On the
other hand, if
$\Gamma$
is fixed, we put
\[
G_{\Gamma}=G_{\Gamma}(C_{{\mt D.}},\;D\in\Gamma):=
\sum_{{\mt D.}\in\Gamma}C_{{\mt D.}},\quad
C_{{\mt D.}}\in\Pi_{{\mt D.}}\al F..
\]
Then
$G_{\Gamma}\in\al F._{\rm fin}$.
By a simple calculation one obtains
\[
\langle F-G_{\Gamma},F-G_{\Gamma}\rangle_{\al A.}=
\langle F,\,F\rangle_{\al A.}-\sum_{{\mt D.}\in\Gamma}\langle\Pi_{{\mt D.}}F,
\Pi_{{\mt D.}}F\rangle_{\al A.} +
\sum_{{\mt D.}\in\Gamma}\langle \Pi_{{\mt D.}}F-C_{{\mt D.}},
\Pi_{{\mt D.}}F-C_{{\mt D.}}\rangle_{\al A.}.
\]
Since
\[
\sum_{{\mt D.}\in\Gamma}\langle\Pi_{{\mt D.}}F-C_{{\mt D.}},
\Pi_{{\mt D.}}F-C_{{\mt D.}}\rangle_{\al A.}\geq 0
\]
we obtain
\begin{eqnarray}
\langle F-G_{\Gamma},F-G_{\Gamma}\rangle_{\al A.} &\geq&
\langle F,\,F\rangle_{\al A.}-\sum_{{\mt D.}\in\Gamma}
\langle\Pi_{{\mt D.}}F,\,
\Pi_{{\mt D.}}F\rangle_{\al A.}\label{lasteq} \\[1mm]
&=& \Big\langle F-\sum_{{\mt D.}\in\Gamma}\Pi_{{\mt D.}}F,\,
F-\sum_{{\mt D.}\in\Gamma}\Pi_{{\mt D.}}F\Big\rangle_{\al A.}\geq 0.
\nonumber
\end{eqnarray}
Therefore
\[
\langle F,\,
F\rangle_{\al A.}\geq \sum_{{\mt D.}\in\Gamma}
\langle\Pi_{{\mt D.}}F,\Pi_{{\mt D.}}F\rangle_{\al A.}
\]
Since
$\Vert X\Vert\geq\vert X\vert_{\al A.}$
for all
$X\in\al F.$
we have
\[
\Vert F-G_{\Gamma}\Vert\geq\vert F-G_{\Gamma}\vert_{\al A.}\geq
\vert F-F_{\Gamma}\vert_{\al A.}.
\]
According to Shiga's theorem (see ~\cite{Shiga55}) the left hand side
can be chosen arbitrary small for suitable
$\Gamma$
and suitable coefficients
$C_{{\mt D.}}$. Hence
$\vert F-F_{\Gamma}\vert_{\al A.}\to 0$
for
$\Gamma\to\wh{\al G.}$
follows. This is (\ref{FG}) and this implies
\[
\lim_{\Gamma\to\wh{\al G.}}\Vert\langle F,F\rangle_{\al A.}-
\sum_{{\mt D.}\in\Gamma}\langle\Pi_{{\mt D.}}F,\Pi_{{\mt D.}}F\rangle_{\al A.}
\Vert=0,
\]
which proves (\ref{PEq}).
\end{beweis}
Note that (\ref{FG}) does not
in general converge
w.r.t.~the C*-norm $\Vert\cdot\Vert$.
\begin{cor}
(i) Each $F\in\al F.$ is uniquely determined by its projections
$\Pi_{\mt D.} F,$ $D\in\wh{\al G.},$ i.e.~$F=0$
iff $\Pi_{\mt D.} F =0$ for all $ D\in\wh{\al G.}.$
\chop
(ii) We have that $\big|\Pi_{\mt D.}\big|_{\al A.}=1$
for all $D\in\wh{\al G.}$,
where $\vert\cdot\vert_{\al A.}$
denotes the operator norm of $\Pi_{\mt D.}$ w.r.t.~the norm
$\vert\cdot\vert_{\al A.}$ in $\al F.$.
\end{cor}
\begin{pro}\label{SimpleAF}
$\al F.$
is simple iff
$\al A.$
is simple.
\end{pro}
\begin{beweis}
Let $\al D.\lhd\al A.$ be a nontrivial closed 2-sided ideal and consider
\[
\al E._r:=\mr clo._{\|\cdot\|}\;
\mr span.\{ \al D.\,\al H._\mt D.\mid D\in\wh{\al G.}\}\,.
\]
$\al E._r$ is a closed right ideal in $\al F.$: Indeed, recall that
\[
\al F.:=\mr clo._{\|\cdot\|}\;
\mr span.\{ \al A.\,\al H._\mt D.\mid D\in\wh{\al G.}\}\,
\]
and take $A\in\al A.$. Then
\[
\al D.\,\al H._\mt D.\;A\al H._{\mt D.'}
=\underbrace{\al D.\rho_\mt D.(A)}_{^{\in\al D.}}\;
\al H._\mt D.\,\al H._{\mt D.'} \subset\al E._r\,,
\quad D,D'\in\wg\,,
\]
where the latter inclusion follows from the fact that the
tensor product $\al H._\mt D.\,\al H._{\mt D.'}$ can be decomposed in
terms of irreducible algebraic Hilbert spaces.
This shows that
\[
\mr span.\{ \al D.\,\al H._\mt D.\mid D\in\wh{\al G.}\}
\cdot \mr span.\{ \al A.\,\al H._{\mt D.'}\mid D'\in\wh{\al G.}\}
\subset\al E._r\,.
\]
Take now $\{F_n\}_n\subset
\mr span.\{ \al A.\,\al H._{\mt D.'}\mid D'\in\wh{\al G.}\}$
such that $F_n\to F\in\al F.$. Then for any
$E_0\in\mr span.\{ \al D.\,\al H._\mt D.\mid D\in\wh{\al G.}\}$
we have $E_0F\in\al E._r$, because $\al E._r\ni E_0F_n\to E_0F$.
Similarly one can show that $EF\in\al E._r$ for all
$E\in\al E._r$, $F\in\al F.$, hence $\al E._r$ is a closed
right ideal in $\al F.$.
This implies that $\al E.:=\al E._r\cap\al E._r^*\lhd\al F.$ is a nonzero
closed 2-sided ideal in $\al F.$, which is proper since
$\1\not\in\al E.$.
To show the reverse implication, let $\al E.\lhd\al F.$
be a nontrivial closed 2-sided ideal and put
$\al D.:=\al E.\cap\al A.$, which is a closed 2-sided ideal
in $\al A.$. We show next that $\al D.=\{0\}$ implies
$\al E.=\{0\}$ contradicting the nontriviality assumption
of the ideal $\al E.$: take
$U_0\Psi_0\in\al E.\cap(\al A.\al H._{D_0})$ with $U_0$ a
unitary in $\al A.$. Then $U_0\Psi_0\Psi_0^*\in\al E.\cap\al A.
=\al D.=\{0\}$, hence $\Psi_0\Psi_0^*=0$ and $\Psi_0=0$.
This show that $\al D.$ is nonzero. Finally, $\al D.$ is
proper because $\1\not\in\al D.$.
\end{beweis}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Two natural examples of C*-categories associated with a Hilbert
C*-system}\label{TwoEx}
In the following we introduce two important examples of
C*-categories that naturally appear in the context of Hilbert
C*-systems. For convenience of the reader we
recall next the definition of this type of categories
(see \cite{Doplicher89b,bMaclane98}).
A category $\ot T.$ with objects $\alpha,\beta$ etc.~is called a
{\em C*-category}
if each arrow space $(\alpha,\beta)$ is a complex Banach space where
the bilinear composition of arrows $R,S$ satisfies
$\|R\circ S\|\leq \|R\|\,\|S\|$
and there is an antilinear involutive contravariant functor
$^*\colon\ot T.\to\ot T.$ such that the analogue of the C*-property is
satisfied, i.e.~if $R\in(\alpha,\beta)$, $R^*\in (\beta,\alpha)$, then
$\|R^*\circ R\|= \|R\|^2$. The category $\ot T.$ is said to be
a {\em tensor} C*-category if there is an associative functor
$\circ\colon\ \ot T.\times\ot T.\to\ot T.$ with unit $\iota$ and
commuting with *. Further given two arrows
$R\in(\alpha,\beta)$, $R'\in(\alpha',\beta')$ there is a tensor product
of arrows denoted by $R\times R'\in (\alpha\alpha',\beta\beta')$. The
mapping $R,R'\to R\times R'$ is associative and bilinear and we have
(denoting by $1_\iota$ the identity of $\iota$)
\begin{equation}\label{Neutral}
1_\iota\times R=R\times 1_\iota=R\;,\quad (R\times R')^*=R^*\times R'^*\,,
\end{equation}
as well as the interchange law
\[
(S\circ R)\times (S'\circ R')=S\times S'\circ R\times R'\,,
\]
whenever the left hand side is defined. (We adopt as in
\cite{Doplicher89b} the convention of evaluating $\times$ before $\circ$.)
It is immediate to check from the previous properties that
$(\iota,\iota)$ is an abelian unital C*-algebra.
We may now generalize the notion of irreducible object introduced in
\cite[Section~5]{Lledo97b} (see also \cite{Lledo01a}) to the
present general situation.
\begin{defi}\label{AllgIrr}
Let $\ot T.$ be a strict tensor C*-category. Then $\rho\in\mr Ob.\ot T.$
is called {\bf irreducible} if
\[
(\rho,\rho)=1_\rho\times (\iota,\iota) \,.
\]
We denote the set of all irreducible objects in $\ot T.$ by
$\mb Irr.{\ot T.}$.
\end{defi}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The category $\al T._\al G.$ of all $\al G.$-invariant algebraic
Hilbert spaces}\label{GCanEnd}
The $\al G.$-invariant algebraic Hilbert spaces
$\al H.$
of
$\{\al F.,\al G.\}$, satisfying $\mr supp.\al H.=\1$,
form the objects of a C*-category
$\al T._{\al G.}$
whose arrows are given by the
$\al J.(\al L._{\al G.}(\al H.,\al K.))\subset\al A.$.
In this context the tensor product of objects is given by the
product in $\al F.$, $\iota=\C\1$ and $(\iota,\iota)=\C\1$.
$\al H.$ is irreducible iff $\al J.(\al L._{\al G.}(\al H.,\al H.))=\C\1$
(Schur's lemma).
We will focus next on the additional structure of $\al T._\al G.$.
For this recall the partial order in
$\hbox{Ob}\,\al T._{\al G.}$
given in Remark~\ref{remark1}~(iv).
If
$\al K.\in\hbox{Ob}\,\al T._{\al G.}$
is given, an object
$\al H.<\al K.$
is called a {\it subobject} of
$\al K.$.
If
$E\in\al J.(\al L._{\al G.}(\al K.))$
is an orthoprojection
$0