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\section*{Baum-Connes conjecture}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{index_theory}{}\paragraph*{{Index theory}}\label{index_theory}

[[!include index theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Theorems}{Theorems}\dotfill \pageref*{Theorems} \linebreak
\noindent\hyperlink{greenrosenbergjulg_theorem}{Green-Rosenberg-Julg theorem}\dotfill \pageref*{greenrosenbergjulg_theorem} \linebreak
\noindent\hyperlink{related_theorems}{Related theorems}\dotfill \pageref*{related_theorems} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{introductions_and_surveys}{Introductions and surveys}\dotfill \pageref*{introductions_and_surveys} \linebreak
\noindent\hyperlink{original_articles}{Original articles}\dotfill \pageref*{original_articles} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The Baum-Connes conjecture asserts that under suitable technical conditions, the [[operator K-theory]]/[[KK-theory]] of the [[groupoid convolution algebra]] of an [[action groupoid|action]] [[topological groupoid]] $X//G$ is equivalent to the \emph{$G$-[[equivariant cohomology|equivariant]]} [[topological K-theory]]/[[equivariant operator K-theory]]/[[equivariant KK-theory]] of $X$. Moreover, it says that this equivalences is exhibited by the \emph{[[analytic assembly map]]}. (Just the [[injective map|injectivity]] of this map is related to the [[Novikov conjecture]].)

The original version of the Baum-Connes conjecture (\hyperlink{BaumConnes00}{Baum-Connes}) stated for a suitable [[topological group]] $G$ that with $X = E G \simeq \ast$ the point incarnated as the $G$-[[universal principal bundle]] with its free $G$-[[action]] the [[analytic assembly map]] (the $G$-equivariant [[index]] map)

\begin{displaymath}
K_\bullet^G(E G) \stackrel{}{\to} K_\bullet(C(\mathbf{B}G))
  \,,
\end{displaymath}
from the $G$-[[equivariant K-theory]] of $E G$ to the [[operator K-theory]] of the [[group algebra]], hence the [[groupoid convolution algebra]] of the [[delooping]] groupoid $\mathbf{B}G \simeq \ast // G$, is an [[isomorphism]].

This Baum-Connes conjecture is known to be true for

\begin{itemize}%
\item [[compact topological groups]],
\item [[abelian groups]],
\item [[Lie groups]] with finitely many [[connected components]];
\item $p$-adic groups,
\item adelic groups.

\end{itemize}
It is not known if the conjecture is true for all [[discrete groups]].

Later the statement was generalized (\hyperlink{Tu99}{Tu 99}) to more general groupoids.

In (\hyperlink{Kasparov88}{Kasparov 88}) the refinement of the [[analytic assembly map]] to [[equivariant cohomology|equivariant]] [[KK-theory]] is given, and called the \emph{descent map}. This is of the form (recalled as \hyperlink{Blackadar}{Blackadar, theorem 20.6.2})

\begin{displaymath}
KK^G(A,B) \to KK(G \ltimes A, G \ltimes B)
\end{displaymath}
where on the left we have $G$-[[equivariant KK-theory]] and on the right ordinary [[KK-theory]] of [[crossed product C\emph{-algebras]] (which by the discussion there are models for the [[groupoid convolution algebras]] of $G$-[[action groupoids]]).}

This is an [[isomorphism]] at least for $G$ a [[compact topological group]] and restricted to [[operator K-theory]] (hence to the first argument being $\mathbb{C}$) and for $G$ a [[discrete group]] and restricted to [[K-homology]] (hence ot the second argument being $\mathbb{C}$). In this form this is the \textbf{Green-Julg theorem}, see \hyperlink{GreenJulgTheorem}{below}.

\hypertarget{Theorems}{}\subsection*{{Theorems}}\label{Theorems}

\hypertarget{greenrosenbergjulg_theorem}{}\subsubsection*{{Green-Rosenberg-Julg theorem}}\label{greenrosenbergjulg_theorem}

The Green-Rosenberg-Julg theorem identifies [[equivariant K-theory]] with the [[operator K-theory]] of [[crossed product algebras]].

\begin{theorem}
\label{GreenJulgTheorem}\hypertarget{GreenJulgTheorem}{}
\textbf{(Green-Julg theorem)}

Let $G$ be a [[topological group]] acting on a [[C\emph{-algebra]] $A$.}

\begin{enumerate}%
\item If $G$ is a [[compact topological group]] then the descent map

\begin{displaymath}
KK^G(\mathbb{C}, A) \to KK(\mathbb{C},G\ltimes A)
\end{displaymath}
is an [[isomorphism]], identifying the [[equivariant operator K-theory]] of $A$ with the ordinary [[operator K-theory]] of the [[crossed product C\emph{-algebra]] $G \ltimes A$.}


\item if $G$ is a [[discrete group]] then the descent map

\begin{displaymath}
KK^G(A, \mathbb{C}) \to KK(G \ltimes A, \mathbb{C})
\end{displaymath}
is an [[isomorphism]], identifying the equivariant [[K-homology]] of $A$ with the ordinary [[K-homology]] of the [[crossed product C\emph{-algebra]] $G \ltimes A$.}



\end{enumerate}
\end{theorem}
This goes back to (\hyperlink{Green82}{Green 82}), (\hyperlink{Julg81}{Julg 81}). A [[KK-theory]]-proof is in (\hyperlink{Echterhoff}{Echterhoff, theorem 0.2}); a textbook account is in (\hyperlink{Blackadar}{Blackadar, 11.7, 20.2.7}). See also around (\hyperlink{Land13}{Land 13, prop. 41}).

\hypertarget{related_theorems}{}\subsection*{{Related theorems}}\label{related_theorems}

\begin{itemize}%
\item [[Atiyah-Segal completion theorem]], [[Green-Julg theorem]]

\end{itemize}
[[!include Segal completion -- table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{introductions_and_surveys}{}\subsubsection*{{Introductions and surveys}}\label{introductions_and_surveys}

Introductions and surveys include

\begin{itemize}%
\item [[Alain Valette]], \emph{Introduction to the Baum-Connes conjecture} (\href{http://www.univ-orleans.fr/mapmo/membres/chatterji/Valette.pdf}{pdf})


\item [[Nigel Higson]], \emph{The Baum-Connes conjecture} (\href{http://media.cit.utexas.edu/math-grasp/Higson_supplemental_1.pdf}{pdf})


\item [[Paul Baum]], \emph{The Baum-Connes conjecture, localisation of categories and quantum groups}, 2008 (\href{http://www.mimuw.edu.pl/~pwit/TOK/sem8/files/Baum_bcclcqg.pdf}{pdf})



\end{itemize}
Textbook discussion is in sections 11.7 and 20.2.7 of

\begin{itemize}%
\item [[Bruce Blackadar]], \emph{[[K-Theory for Operator Algebras]]}

\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Baum%E2%80%93Connes_conjecture}{Baum-Connes conjecture}}

\end{itemize}
\hypertarget{original_articles}{}\subsubsection*{{Original articles}}\label{original_articles}

The original article is

\begin{itemize}%
\item [[Paul Baum]], [[Alain Connes]], \emph{K-theory for Lie groups and foliations}, Enseign. Math. 46 (2000), 3--42.

\end{itemize}
Proof of the conjecture for hyperbolic groups is in

\begin{itemize}%
\item [[Vincent Lafforgue]], \emph{La conjecture de Baum-Connes \`a{} coefficients pour les groupes hyperboliques}, Journal of Noncommutative Geometry, Volume 6, Issue 1, 2012, pages 1-197 (\href{http://arxiv.org/abs/1201.4653}{arXiv:1201.4653})

\end{itemize}
Technical subtleties are discussed in

\begin{itemize}%
\item [[Nigel Higson]], [[Vincent Lafforgue]], [[Georges Skandalis]], \emph{Counterexamples to the Baum-Connes conhecture} (\href{http://www.personal.psu.edu/ndh2/math/Papers_files/Higson,%20Lafforgue,%20Skandalis%20-%202002%20-%20Counterexamples%20to%20the%20Baum-Connes%20conjecture.pdf}{pdf})

\end{itemize}
The generalization to [[Lie groupoids]] is due to

\begin{itemize}%
\item [[Jean-Louis Tu]], \emph{The Baum-Connes conjecture for groupoids}, 1999 ([[JLTBaumConnesForGroupoids.pdf:file]])

\end{itemize}
Proofs for some cases are in

\begin{itemize}%
\item [[Georges Skandalis]], [[Jean-Louis Tu]], G. Yu, \emph{The coarse Baum-Connes conjecture and groupoids} (\href{http://www.math.univ-metz.fr/~tu/publi/coarse.pdf}{pdf})

\end{itemize}
[[KK-theory]] tools and the descent map are introduced in

\begin{itemize}%
\item [[Gennady Kasparov]], \emph{Equivariant KK-theory and the Novikov conjecture}, Inventiones Mathematicae, vol. 91, p.147, 1988(\href{http://adsabs.harvard.edu/abs/1988InMat..91..147K}{web})

\end{itemize}
The ``Green-Julg theorem'' for commutative algebra and finite group is due to [[Michael Atiyah]], for commutative algebra and general group due to

\begin{itemize}%
\item P. Green, \emph{Equivariant if-theory and crossed product C\emph{- algebras\_, pp. 337-338 in Operator algebras and applications (Kingston, Ont., 1980), vol. 1, edited by R. V. Kadison, Proc. Sympos. Pure Math. 38, Amer. Math. Soc, Providence, 1982.}}

\end{itemize}
and the general case is due to an unpublished result by Green and [[Jonathan Rosenberg]] and independently due to

\begin{itemize}%
\item P. Julg, \emph{K-theorie equivariante et produits croises}, C. R. Acad. Sci. Paris Ser. I Math. 292:13 1981, 629-632.

\end{itemize}
Further discussion is in

\begin{itemize}%
\item Siegfried Echterhoff, \emph{The Green-Julg theorem} \href{http://wwwmath.uni-muenster.de/u/paravici/Focused-Semester/lecturenotes/green-julg-Echterhoff.pdf}{pdf}

\end{itemize}
\begin{itemize}%
\item Walther Paravicini, \emph{A generalised Green-Julg theorem for proper groupoids and Banach algebras}, (\href{http://arxiv.org/abs/0902.4365}{arXiv:0902.4365})


\item V. Lafforgue, \href{http://www.mmas.univ-metz.fr/~gnc/bibliographie/HarmonicAnalysis/LafforgueICM.pdf}{pdf}



\end{itemize}
A modification of the Baum-Connes conjecture with coefficient where many counterexamples (to the conjecture with coefficients) are eliminated is in

\begin{itemize}%
\item [[Paul Baum]], Erik Guentner, Rufus Willett, \emph{Expanders, exact crossed products, and the Baum-Connes conjecture}, \href{http://arxiv.org/abs/1311.2343}{arxiv/1311.2343}

\end{itemize}
Discussion in terms of [[localization]]/[[homotopy theory]] is in

\begin{itemize}%
\item [[Ralf Meyer]], [[Ryszard Nest]], \emph{The Baum-Connes conjecture via localisation of categories, Topology 45 (2006), no. 2, 209--259.}

\end{itemize}
[[!redirects Green-Julg theorem]]



\end{document}