Content-Type: multipart/mixed; boundary="-------------0311111143887" This is a multi-part message in MIME format. ---------------0311111143887 Content-Type: text/plain; name="03-496.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-496.comments" 22D25;22D35;47L65;46L08 39 pages; latex2e lledo@iram.rwth-aachen.de baumg@rz.uni-potsdam.de ---------------0311111143887 Content-Type: text/plain; name="03-496.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-496.keywords" Duality, compact groups, nontrivial center, tensor C*-categories, Hilbert C*-modules ---------------0311111143887 Content-Type: application/x-tex; name="HCSbglle.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="HCSbglle.tex" % LATEX - File `On Hilbert C*-systems' % % % %\documentstyle[11pt]{article} \documentclass[11pt]{article} \usepackage{latexsym} \usepackage{amssymb} \usepackage{bbm} 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\renewcommand{\baselinestretch}{1.5}} \def\XB{\marginpar{ {\footnotesize\bf Change~starts----}\lower 11pt\hbox{\mathsurround=0pt$ \!\!\displaystyle{ \Bigg\downarrow}$\mathsurround=3pt}}} \def\XE{\marginpar{{\footnotesize\bf Change~ends-----}\raise 10pt\hbox{\mathsurround=0pt$ \!\!\displaystyle{ \Bigg\downarrow}$\mathsurround=3pt}}} \def\HS{{\{\al F.,\,\al G.\}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % End of the special commands % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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