nLab Baum-Connes conjecture

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 topological groupoid $X//G$ is equivalent to the $G$-equivariant topological K-theory/equivariant operator K-theory/equivariant KK-theory of $X$. Moreover, it says that this equivalence is exhibited by the analytic assembly map. (Just the injectivity of this map is related to the Novikov conjecture.)

The original version of the Baum-Connes conjecture (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)

$$K_\bullet^G(E G) \stackrel{}{\to} K_\bullet(C(\mathbf{B}G)) \,,$$

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

• compact topological groups,
• abelian groups,
• Lie groups with finitely many connected components;
• $p$-adic groups,
• adelic groups.

It is not known if the conjecture is true for all discrete groups.

Later the statement was generalized (Tu 99) to more general groupoids.

In (Kasparov 88) the refinement of the analytic assembly map to equivariant KK-theory is given, and called the descent map. This is of the form (recalled as Blackadar, theorem 20.6.2)

$$KK^G(A,B) \to KK(G \ltimes A, G \ltimes B)$$

where on the left we have $G$-equivariant KK-theory and on the right ordinary KK-theory of crossed product C*-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 Green-Julg theorem, see below.

Theorems

Green-Rosenberg-Julg theorem

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

Theorem

(Green-Julg theorem)

Let $G$ be a topological group acting on a C*-algebra $A$.

1. If $G$ is a compact topological group then the descent map

   $$KK^G(\mathbb{C}, A) \to KK(\mathbb{C},G\ltimes A)$$

   is an isomorphism, identifying the equivariant operator K-theory of $A$ with the ordinary operator K-theory of the crossed product C*-algebra $G \ltimes A$.

2. if $G$ is a discrete group then the descent map

   $$KK^G(A, \mathbb{C}) \to KK(G \ltimes A, \mathbb{C})$$

   is an isomorphism, identifying the equivariant K-homology of $A$ with the ordinary K-homology of the crossed product C*-algebra $G \ltimes A$.

This goes back to (Green 82), (Julg 81). A KK-theory-proof is in (Echterhoff, theorem 0.2); a textbook account is in (Blackadar, 11.7, 20.2.7). See also around (Land 13, prop. 41).

Related theorems

• Atiyah-Segal completion theorem, Green-Julg theorem

• delocalized equivariant cohomology

(equivariant) cohomology	representing spectrum	equivariant cohomology of the point $\ast$	cohomology of classifying space $B G$
(equivariant) ordinary cohomology	HZ	Borel equivariance $H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})$
(equivariant) complex K-theory	KU	representation ring $KU_G(\ast) \simeq R_{\mathbb{C}}(G)$	Atiyah-Segal completion theorem $R(G) \simeq KU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {KU_G(\ast)} \simeq KU(B G)$
(equivariant) complex cobordism cohomology	MU	$MU_G(\ast)$	completion theorem for complex cobordism cohomology $MU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {MU_G(\ast)} \simeq MU(B G)$
(equivariant) algebraic K-theory	$K \mathbb{F}_p$	representation ring $(K \mathbb{F}_p)_G(\ast) \simeq R_p(G)$	Rector completion theorem $R_{\mathbb{F}_p}(G) \simeq K (\mathbb{F}_p)_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {(K \mathbb{F}_p)_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Rector+completion+theorem">Rector 73</a>}}{\simeq} \!\!\!\!\!\! K \mathbb{F}_p(B G)$
(equivariant) stable cohomotopy	$K \mathbb{F}_1 \overset{\text{<a href="stable cohomotopy#StableCohomotopyIsAlgebraicKTheoryOverFieldWithOneElement">Segal 74</a>}}{\simeq}$ S	Burnside ring $\mathbb{S}_G(\ast) \simeq A(G)$	Segal-Carlsson completion theorem $A(G) \overset{\text{<a href="https://ncatlab.org/nlab/show/Burnside+ring+is+equivariant+stable+cohomotopy+of+the+point">Segal 71</a>}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Segal-Carlsson+completion+theorem">Carlsson 84</a>}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G)$

References

Introductions and surveys

Introductions and surveys include

• Alain Valette, Introduction to the Baum-Connes conjecture (pdf)

• Nigel Higson, The Baum-Connes conjecture (pdf)

• Paul Baum, The Baum-Connes conjecture, localisation of categories and quantum groups, 2008 (pdf)

Textbook discussion:

• Bruce Blackadar, Section 11.7 and 20.2.7 of: K-Theory for Operator Algebras, Cambridge University Press 1986, second ed. 1999 (doi:10.1007/978-1-4613-9572-0, pdf)

• Guido Mislin, Alain Valette, Proper Group Actions and the Baum-Connes Conjecture, Advanced Courses in Mathematics CRM Barcelona, Springer 2003 (doi:10.1007/978-3-0348-8089-3)

See also

• Wikipedia, Baum-Connes conjecture

Original articles

The original article is

• Paul Baum, Alain Connes, K-theory for Lie groups and foliations, Enseign. Math. 46 (2000), 3–42.

Proof of the conjecture for hyperbolic groups is in

• Vincent Lafforgue, La conjecture de Baum-Connes à coefficients pour les groupes hyperboliques, Journal of Noncommutative Geometry, Volume 6, Issue 1, 2012, pages 1-197 (arXiv:1201.4653)

Technical subtleties are discussed in

• Nigel Higson, Vincent Lafforgue, Georges Skandalis, Counterexamples to the Baum-Connes conhecture (pdf)

The generalization to Lie groupoids is due to

• Jean-Louis Tu, The Baum-Connes conjecture for groupoids, 1999 (pdf)

Proofs for some cases are in

• Georges Skandalis, Jean-Louis Tu, G. Yu, The coarse Baum-Connes conjecture and groupoids (pdf)

KK-theory tools and the descent map are introduced in

• Gennady Kasparov, Equivariant KK-theory and the Novikov conjecture, Inventiones Mathematicae, vol. 91, p.147, 1988(web)

The “Green-Julg theorem” for commutative algebra and finite group is due to Michael Atiyah, for commutative algebra and general group due to

• P. Green, Equivariant if-theory and crossed product C- 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.

and the general case is due to an unpublished result by Green and Jonathan Rosenberg and independently due to

• P. Julg, K-theorie equivariante et produits croises, C. R. Acad. Sci. Paris Ser. I Math. 292:13 1981, 629-632.

Further discussion is in

• Siegfried Echterhoff, The Green-Julg theorem pdf

• Walther Paravicini, A generalised Green-Julg theorem for proper groupoids and Banach algebras, (arXiv:0902.4365)

• V. Lafforgue, pdf

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

• Paul Baum, Erik Guentner, Rufus Willett, Expanders, exact crossed products, and the Baum-Connes conjecture, arxiv/1311.2343

Discussion in terms of localization/homotopy theory is in

• Ralf Meyer, Ryszard Nest, The Baum-Connes conjecture via localisation of categories, Topology 45 (2006), no. 2, 209–259.