BaumConnesConjecture 2009-02-19 11:16:18 2009-02-19 11:50:16 Conjecture assembly map BCC OKT the Baum-Connes conjecture (BCC) irreducible unitary representations operator K-theory (OKT) K-homology of a group C*-algebra of a group proper actions of a group % this is the default PlanetPhysics preamble. as your % 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}} \subsection{Introduction} The goal of \PMlinkexternal{the Baum-Connes conjecture}{http://www.math.vanderbilt.edu/~bisch/ncgoa07/talks/roe1_NCGOA07.pdf} is to understand irreducible, unitary representations topologically. Furthermore, the relationship between topology and representation theory is mediated by elliptic operators. ``The origins of the conjecture go back to Fredholm theory, the Atiyah-Singer index theorem and the interplay of geometry with operator K-theory as expressed in the works of Brown, Douglas and Fillmore, among many other motivating subjects.'' \subsection{The Baum--Connes Conjecture (BCC)} Thus, in operator K-theory (OKT), the Baum--Connes proposition conjectures that there is a link between the K--theory of the C*--algebra of a group and the K-homology of the corresponding classifying space of proper actions of that same group. It thus proposes that there exists a correspondence between several distinct areas of mathematics: K--homology (related to geometry), differential operator theory, and homotopy theory on the one hand, and the K-theory of the reduced C*--algebra--which is currently formulated as an analytical object--on the other hand. The BCC, if it were shown to be true, would also have some older, quite famous conjectures as consequences. For instance, the surjectivity part of BCC implies the Kadison-Kaplansky conjecture for a discrete torsion-free group, whereas the injectivity part of BCC would seem to be closely related to the earlier, Novikov conjecture. BCC may also be seen as related to Index Theory (IT), because the assembly map $ \mu $ is a type of index, that plays a major role in Alain Connes' noncommutative geometry formulation. \subsubsection{Mathematical Formulation} Let us consider $G$ to be the group $\mathbb{Z}$ (with the discrete topology; that is, not a ``topological group''). Then, every complex number $u$ with $|u| = 1$ corresponds to a $1$-dimensional irreducible representation of $G$, on which $n \in \mathbb{Z}$ acts by multiplication by $u^n$. Furthermore, these are all of the possible irreducible representations of $G$ that can be found. When $G =\, \mathbb{Z}$ every unitary representation can be uniquely decomposed into a direct sum of irreducible representations. The space of irreducible representations of $G$ carries a natural Hausdorff topology and can be studied as a commutative, standard geometric/topological space. In the general case, as for example for a non-Abelian group, ${N_G}^A$, the ``space of irreducible representations of such a group'' is no longer a commutative object, and was replaced by A. Connes by a C*-algebra which is in general a noncommutative object-- or a so-called (non-standard), noncommutative ``space''. One is especially interested in \emph{graded Hilbert spaces} $ H = H_+ \oplus H_-$. In this case an odd unbounded operator is identified with a \emph{grading--preserving functional calculus homomorphism} $$\Phi_T: f \mapsto f(T) , \mathfrak{G} \to \mathfrak{B}(H)$$ where $\mathfrak{G}$ denotes the algebra $C_0(\mathbb{R})$ graded by even and odd functions. Consider $\mathfrak{G}$ to be a second countable locally compact group (such as a countable discrete group). Then , one can define a morphism $$ \mu \colon RK^$\mathfrak{G}$_*(\underline{E$\mathfrak{G}$}) \to K_*(C^*_\lambda($\mathfrak{G}$)), $$ called the \emph{assembly map}, from the equivariant K-homology with $ \mathfrak{G}$--compact supports of the classifying space of proper actions $ \underline{E$\mathfrak{G}$} $ to the K--theory of the reduced C*-algebra of $\mathfrak{G}$. The index $*$ can be either $0$ or $1$. Alain Connes and Paul Baum proposed in 1982 the following conjecture about the morphism (assembly map) $\mu$: \begin{conjecture} The assembly map $\mu$ is an isomorphism. \end{conjecture} As the left hand side tends to be more easily accessible than the right hand side, because there are hardly any general structure theorems of the $ C^* $-algebra, one usually views the conjecture as an "explanation" of the right hand side.