quantum groups and von Neumann algebras QuantumGroupsAndVonNeumannAlgebras 2008-09-21 00:28:20 2008-10-17 11:32:14 Topic JBW-algebras $JBW$ algebras $JB$--algebra $JBWA$ $JL$ Jordan-Banach-von Neumann algebras $CQG_{lc}$ quantum groupoids Hopf and weak Hopf algebras % this is the default PlanetMath 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{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym} \usepackage{xypic} \usepackage[mathscr]{eucal} \setlength{\textwidth}{6.5in} %\setlength{\textwidth}{16cm} \setlength{\textheight}{9.0in} %\setlength{\textheight}{24cm} \hoffset=-.75in %%ps format %\hoffset=-1.0in %%hp format \voffset=-.4in \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{\GL}{{\rm GL}} \newcommand{\Iso}{{\rm Iso}} \newcommand{\Sem}{{\rm Sem}} \newcommand{\St}{{\rm St}} \newcommand{\Sym}{{\rm Sym}} \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{\G}{\mathcal G} \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}{{\mathbb G}} \newcommand{\dgrp}{{\mathbb D}} \newcommand{\desp}{{\mathbb D^{\rm{es}}}} \newcommand{\Geod}{{\rm Geod}} \newcommand{\geod}{{\rm geod}} \newcommand{\hgr}{{\mathbb H}} \newcommand{\mgr}{{\mathbb M}} \newcommand{\ob}{{\rm Ob}} \newcommand{\obg}{{\rm Ob(\mathbb G)}} \newcommand{\obgp}{{\rm Ob(\mathbb G')}} \newcommand{\obh}{{\rm Ob(\mathbb H)}} \newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}} \newcommand{\ghomotop}{{\rho_2^{\square}}} \newcommand{\gcalp}{{\mathbb G(\mathcal P)}} \newcommand{\rf}{{R_{\mathcal F}}} \newcommand{\glob}{{\rm glob}} \newcommand{\loc}{{\rm loc}} \newcommand{\TOP}{{\rm TOP}} \newcommand{\wti}{\widetilde} \newcommand{\what}{\widehat} \renewcommand{\a}{\alpha} \newcommand{\be}{\beta} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\de}{\delta} \newcommand{\del}{\partial} \newcommand{\ka}{\kappa} \newcommand{\si}{\sigma} \newcommand{\ta}{\tau} \newcommand{\med}{\medbreak} \newcommand{\medn}{\medbreak \noindent} \newcommand{\bign}{\bigbreak \noindent} \newcommand{\lra}{{\longrightarrow}} \newcommand{\ra}{{\rightarrow}} \newcommand{\rat}{{\rightarrowtail}} \newcommand{\oset}[1]{\overset {#1}{\ra}} \newcommand{\osetl}[1]{\overset {#1}{\lra}} \newcommand{\hr}{{\hookrightarrow}} \subsection{Hilbert spaces, Von Neumann algebras and Quantum Groups} John von Neumann introduced a mathematical foundation for Quantum Mechanics in the form of \PMlinkname{$W^*$-algebras}{WeakHopfCAlgebra2} of (quantum) bounded operators in a (quantum:= presumed \emph{separable}, i.e. with a countable basis) Hilbert space $H_S$. Recently, such \PMlinkname{von Neumann algebras, $W^*$}{WeakHopfCAlgebra2} and/or (more generally) C*-algebras are, for example, employed to define \PMlinkname{locally compact quantum groups $CQG_{lc}$}{LocallyCompactQuantumGroup} by equipping such \PMlinkname{algebras with a co-associative multiplication}{WeakHopfCAlgebra2} and also with associated, both left-- and right-- Haar measures, defined by two semi-finite normal weights \cite{Vainerman2003}. \subsubsection{Remark on Jordan-Banach-von Neumann (JBW) algebras, $JBWA$} A \emph{Jordan--Banach algebra} (a JB--algebra for short) is both a real Jordan algebra and a Banach space, where for all $S, T \in \mathfrak A_{\bR}$, we have \bigbreak \bigbreak \bigbreak $$ \begin{aligned} \Vert S \circ T \Vert &\leq \Vert S \Vert ~ \Vert T \Vert ~, \\ \Vert T \Vert^2 &\leq \Vert S^2 + T^2 \Vert ~. \end{aligned}$$ \bigbreak \bigbreak \bigbreak A \emph{JLB--algebra} is a $JB$--algebra $\mathfrak A_{\bR}$ together with a Poisson bracket for which it becomes a Jordan--Lie algebra $JL$ for some $\hslash^2 \geq 0$~. Such JLB--algebras often constitute the real part of several widely studied complex associative algebras. For the purpose of quantization, there are fundamental relations between \PMlinkname{$\mathfrak A^{sa}$, JLB and Poisson algebras}{JordanBanachAndJordanLieAlgebras}. \bigbreak \begin{definition} A JB--algebra which is monotone complete and admits a separating set of normal sets is called a \emph{JBW-algebra}. \end{definition} These appeared in the work of von Neumann who developed an \emph{orthomodular lattice theory of projections on $\mathcal L(H)$} on which to study \emph{quantum logic}. BW-algebras have the following property: whereas $\mathfrak A^{sa}$ is a J(L)B--algebra, the self-adjoint part of a von Neumann algebra is a JBW--algebra. \begin{thebibliography}{9} \bibitem{Vainerman2003} Leonid Vainerman. 2003. \PMlinkexternal{``Locally Compact Quantum Groups and Groupoids'': \\ Proceedings of the Meeting of Theoretical Physicists and Mathematicians}{http://planetmath.org/?op=getobj&from=books&id=160}, Strasbourg, February 21-23, 2002., Walter de Gruyter Gmbh \& Co: Berlin. \bibitem{QTF} Von Neumann and the \PMlinkexternal{Foundations of Quantum Theory.}{http://plato.stanford.edu/entries/qt-nvd/} \bibitem{Bohm66} Böhm, A., 1966, Rigged Hilbert Space and Mathematical Description of Physical Systems, {\em Physica A}, 236: 485-549. \bibitem{Bohm89} Böhm, A. and Gadella, M., 1989, \emph{Dirac Kets, Gamow Vectors and Gel'fand Triplets}, New York: Springer-Verlag. \bibitem{DJ81} Dixmier, J., 1981, \emph{Von Neumann Algebras}, Amsterdam: North-Holland Publishing Company. [First published in French in 1957: \emph{Les Algèbres d'Opérateurs dans l'Espace Hilbertien}, Paris: Gauthier-Villars.] \bibitem{GINM43} Gelfand, I. and Neumark, M., 1943, On the Imbedding of Normed Rings into the Ring of Operators in Hilbert Space, {\em Recueil Mathématique} [Matematicheskii Sbornik] Nouvelle Série, 12 [54]: 197-213. [Reprinted in C*-algebras: 1943-1993, in the series Contemporary Mathematics, 167, Providence, R.I. : American Mathematical Society, 1994.] \bibitem{Alex55} Grothendieck, A., 1955, Produits Tensoriels Topologiques et Espaces Nucléaires, \emph{Memoirs of the American Mathematical Society}, 16: 1-140. \bibitem{HS90} Horuzhy, S. S., 1990, {\em Introduction to Algebraic Quantum Field Theory}, Dordrecht: Kluwer Academic Publishers. \bibitem{JV55} J. von Neumann.,1955, {\em Mathematical Foundations of Quantum Mechanics.}, Princeton, NJ: Princeton University Press. [First published in German in 1932: {\em Mathematische Grundlagen der Quantenmechanik}, Berlin: Springer.] \bibitem{JV37} J. von Neumann, 1937, {\em Quantum Mechanics of Infinite Systems}, first published in (Rédei and Stöltzner 2001, 249-268). [A mimeographed version of a lecture given at Pauli's seminar held at the Institute for Advanced Study in 1937, John von Neumann Archive, Library of Congress, Washington, D.C.] \end{thebibliography}