CstarAlgebra 2009-01-10 19:27:37 2009-01-10 19:35:27 Definition C*-algebra % this is the default PlanetPhysics preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym, enumerate} \usepackage{xypic, xspace} \usepackage[mathscr]{eucal} \usepackage[dvips]{graphicx} \usepackage[curve]{xy} % define commands here \theoremstyle{plain} \newtheorem{lemma}{Lemma}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[section] \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem{example}{Example}[section] %\theoremstyle{remark} \newtheorem{remark}{Remark}[section] \newtheorem*{notation}{Notation} \newtheorem*{claim}{Claim} \renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote}}} \numberwithin{equation}{section} \newcommand{\Ad}{{\rm Ad}} \newcommand{\Aut}{{\rm Aut}} \newcommand{\Cl}{{\rm Cl}} \newcommand{\Co}{{\rm Co}} \newcommand{\DES}{{\rm DES}} \newcommand{\Diff}{{\rm Diff}} \newcommand{\Dom}{{\rm Dom}} \newcommand{\Hol}{{\rm Hol}} \newcommand{\Mon}{{\rm Mon}} \newcommand{\Hom}{{\rm Hom}} \newcommand{\Ker}{{\rm Ker}} \newcommand{\Ind}{{\rm Ind}} \newcommand{\IM}{{\rm Im}} \newcommand{\Is}{{\rm Is}} \newcommand{\ID}{{\rm id}} \newcommand{\grpL}{{\rm GL}} \newcommand{\Iso}{{\rm Iso}} \newcommand{\rO}{{\rm O}} \newcommand{\Sem}{{\rm Sem}} \newcommand{\SL}{{\rm Sl}} \newcommand{\St}{{\rm St}} \newcommand{\Sym}{{\rm Sym}} \newcommand{\Symb}{{\rm Symb}} \newcommand{\SU}{{\rm SU}} \newcommand{\Tor}{{\rm Tor}} \newcommand{\U}{{\rm U}} \newcommand{\A}{\mathcal A} \newcommand{\Ce}{\mathcal C} \newcommand{\D}{\mathcal D} \newcommand{\E}{\mathcal E} \newcommand{\F}{\mathcal F} %\newcommand{\grp}{\mathcal G} \renewcommand{\H}{\mathcal H} \renewcommand{\cL}{\mathcal L} \newcommand{\Q}{\mathcal Q} \newcommand{\R}{\mathcal R} \newcommand{\cS}{\mathcal S} \newcommand{\cU}{\mathcal U} \newcommand{\W}{\mathcal W} \newcommand{\bA}{\mathbb{A}} \newcommand{\bB}{\mathbb{B}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bD}{\mathbb{D}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bG}{\mathbb{G}} \newcommand{\bK}{\mathbb{K}} \newcommand{\bM}{\mathbb{M}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bO}{\mathbb{O}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bV}{\mathbb{V}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bfE}{\mathbf{E}} \newcommand{\bfX}{\mathbf{X}} \newcommand{\bfY}{\mathbf{Y}} \newcommand{\bfZ}{\mathbf{Z}} \renewcommand{\O}{\Omega} \renewcommand{\o}{\omega} \newcommand{\vp}{\varphi} \newcommand{\vep}{\varepsilon} \newcommand{\diag}{{\rm diag}} \newcommand{\grp}{{\mathsf{G}}} \newcommand{\dgrp}{{\mathsf{D}}} \newcommand{\desp}{{\mathsf{D}^{\rm{es}}}} \newcommand{\grpeod}{{\rm Geod}} %\newcommand{\grpeod}{{\rm geod}} \newcommand{\hgr}{{\mathsf{H}}} \newcommand{\mgr}{{\mathsf{M}}} \newcommand{\ob}{{\rm Ob}} \newcommand{\obg}{{\rm Ob(\mathsf{G)}}} \newcommand{\obgp}{{\rm Ob(\mathsf{G}')}} \newcommand{\obh}{{\rm Ob(\mathsf{H})}} \newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}} \newcommand{\grphomotop}{{\rho_2^{\square}}} \newcommand{\grpcalp}{{\mathsf{G}(\mathcal P)}} \newcommand{\rf}{{R_{\mathcal F}}} \newcommand{\grplob}{{\rm glob}} \newcommand{\loc}{{\rm loc}} \newcommand{\TOP}{{\rm TOP}} \newcommand{\wti}{\widetilde} \newcommand{\what}{\widehat} \renewcommand{\a}{\alpha} \newcommand{\be}{\beta} \newcommand{\grpa}{\grpamma} %\newcommand{\grpa}{\grpamma} \newcommand{\de}{\delta} \newcommand{\del}{\partial} \newcommand{\ka}{\kappa} \newcommand{\si}{\sigma} \newcommand{\ta}{\tau} \newcommand{\lra}{{\longrightarrow}} \newcommand{\ra}{{\rightarrow}} \newcommand{\rat}{{\rightarrowtail}} \newcommand{\ovset}[1]{\overset {#1}{\ra}} \newcommand{\ovsetl}[1]{\overset {#1}{\lra}} \newcommand{\hr}{{\hookrightarrow}} \newcommand{\<}{{\langle}} \def\baselinestretch{1.1} \hyphenation{prod-ucts} \newcommand{\sqdiagram}[9]{$$ \diagram #1 \rto^{#2} \dto_{#4}& #3 \dto^{#5} \\ #6 \rto_{#7} & #8 \enddiagram \eqno{\mbox{#9}}$$ } \def\C{C^{\ast}} \newcommand{\labto}[1]{\stackrel{#1}{\longrightarrow}} \newcommand{\quadr}[4] {\begin{pmatrix} & #1& \\[-1.1ex] #2 & & #3\\[-1.1ex]& #4& \end{pmatrix}} \def\D{\mathsf{D}} \section{Von Neumann and C*-algebras: Quantum Operator Algebra and Quantum Theories} C*-algebra has evolved as a key concept in Quantum Operator Algebra after the introduction of the von Neumann algebra for the mathematical foundation of Quantum Mechanics. The von Neumann algebra classification is simpler and studied in greater depth than that of general C*-algebra classification theory. The importance of C*-algebras for understanding the geometry of quantum state spaces (Alfsen and Schultz, 2003 \cite{AS}) cannot be overestimated. Moreover, the introduction of non-commutative C*-algebras in Noncommutative Geometry has already played important roles in expanding the Hilbert space perspective of Quantum Mechanics developed by von Neumann. Furthermore, extended quantum symmetries are currently being approached in terms of groupoid C*- convolution algebra and their representations; the latter also enter into the construction of compact quantum groupoids as developed in the Bibliography cited, and also briefly outlined here in the second section. The fundamental connections that exist between categories of $C^*$-algebras and those of von Neumann and other quantum operator algebras, such as JB- or JBL- algebras are yet to be completed and are the subject of in depth studies \cite{AS}. A \textbf{C*-algebra} is simultaneously a $*$--algebra and a Banach space -with additional conditions- as defined next. Let us consider first the definition of an \emph{involution} on a complex algebra $\mathfrak A$. \begin{definition} An \emph{involution} on a complex algebra $\mathfrak A$ is a \emph{real--linear map} $T \mapsto T^*$ such that for all $S, T \in \mathfrak A$ and $\lambda \in \bC$, we have $ T^{**} = T~,~ (ST)^* = T^* S^*~,~ (\lambda T)^* = \bar{\lambda} T^*~. $ \end{definition} A \emph{*-algebra} is said to be a complex associative algebra together with an involution $*$~. \begin{definition} A \emph{C*-algebra} is simultaneously a *-algebra and a Banach space $\mathfrak A$, satisfying for all $S, T \in \mathfrak A$~ the following conditions: $ \begin{aligned} \Vert S \circ T \Vert &\leq \Vert S \Vert ~ \Vert T \Vert~, \\ \Vert T^* T \Vert^2 & = \Vert T\Vert^2 ~. \end{aligned}$ \end{definition} One can easily verify that $\Vert A^* \Vert = \Vert A \Vert$~. By the above axioms a C*--algebra is a special case of a Banach algebra where the latter requires the above C*-norm property, but not the involution (*) property. Given Banach spaces $E, F$ the space $\mathcal L(E, F)$ of (bounded) linear operators from $E$ to $F$ forms a Banach space, where for $E=F$, the space $\mathcal L(E) = \mathcal L(E, E)$ is a Banach algebra with respect to the norm \bigbreak $\Vert T \Vert := \sup\{ \Vert Tu \Vert : u \in E~,~ \Vert u \Vert= 1 \}~. $ \bigbreak In quantum field theory one may start with a Hilbert space $H$, and consider the Banach algebra of bounded linear operators $\mathcal L(H)$ which given to be closed under the usual algebraic operations and taking adjoints, forms a $*$--algebra of bounded operators, where the adjoint operation functions as the involution, and for $T \in \mathcal L(H)$ we have~: $ \Vert T \Vert := \sup\{ ( Tu , Tu): u \in H~,~ (u,u) = 1 \}~,$ and $ \Vert Tu \Vert^2 = (Tu, Tu) = (u, T^*Tu) \leq \Vert T^* T \Vert~ \Vert u \Vert^2~.$ By a \emph{morphism between C*-algebras} $\mathfrak A,\mathfrak B$ we mean a linear map $\phi : \mathfrak A \lra \mathfrak B$, such that for all $S, T \in \mathfrak A$, the following hold~: \bigbreak $\phi(ST) = \phi(S) \phi(T)~,~ \phi(T^*) = \phi(T)^*~, $ \bigbreak where a bijective morphism is said to be an isomorphism (in which case it is then an isometry). A fundamental relation is that any norm-closed $*$-algebra $\mathcal A$ in $\mathcal L(H)$ is a \PMlinkname{C*-algebra}{CAlgebra3}, and conversely, any \PMlinkname{C*-algebra}{CAlgebra3} is isomorphic to a norm--closed $*$-algebra in $\mathcal L(H)$ for some Hilbert space $H$~. One can thus also define \emph{the category $\mathcal{C}^*$ of C*-algebras and morphisms between C*-algebras}. For a \PMlinkname{C*-algebra}{CAlgebra3} $\mathfrak A$, we say that $T \in \mathfrak A$ is \emph{self--adjoint} if $T = T^*$~. Accordingly, the self--adjoint part $\mathfrak A^{sa}$ of $\mathfrak A$ is a real vector space since we can decompose $T \in \mathfrak A^{sa}$ as ~: $ T = T' + T^{''} := \frac{1}{2} (T + T^*) + \iota (\frac{-\iota}{2})(T - T^*)~.$ A \emph{commutative} C*--algebra is one for which the associative multiplication is commutative. Given a commutative C*--algebra $\mathfrak A$, we have $\mathfrak A \cong C(Y)$, the algebra of continuous functions on a compact Hausdorff space $Y~$. The classification of {$C^*$-algebras} is far more complex than that of von Neumann algebras that provide the fundamental algebraic content of quantum state and operator spaces in quantum theories.