International Congress of Mathematicians

Banach KK-theory and the Baum-Connes Conjecture 797CA(EI,E 0 ) satisfy vu — Id# 0 £ ICA(EQ) and uv — Id^ £ KA(EI).If (E, T) is a Fredholm module over A and 9 : A —t B a unital morphism then(E (E)A B,T ® 1) is a Fredholm module over B (here E (E)A B is the completion ofE ®^s B for the maximal Banach norm such that ||x ® b\\ < ||X||E:||&||B for x £ Eand 6 £ B).Yet A[0,1] be the Banach algebra of continuous functions from [0,1] to A withthe norm ||/|| = sup tG r 01 n ||/(i)|U and 9n,9i : A[0,1] —t A the evaluations at 0 and1. Two Fredholm modules on A are said to be homotopic if they are the images by9o and 9i of a Fredholm module over A[0,1].Theorem 1.1.2 There is a functorial bijection between Kn(A) and the set of homotopyclasses of Fredholm modules over A, for any unital Banach algebra A.Let (En,Ei,u,v) be a Fredholm module over A. Its index, i.e. the correspondingelement in K 0 (A), is constructed as follows. It is possible to find n £ N andw £ K,A(A U , EI) such that (u, w) £ CA(EQ ® A n , E{) is surjective. Its kernel is thenfinitely generated projective and the index is the formal difference of Ker((u,wj)and A n .An ungraded Fredholm module over A is the data of a (ungraded) right Banachmodule E over A, and T £ CA(E) such that T 2 — Id# £ KA(E). There is afunctorialbijection between KfiA) and the set of homotopy classes of ungraded Fredholmmodules.For a non-unital algebra A, K 0 (A) = Ker(K 0 (Ä) -+ K 0 (C) = Z) and KfiA) =Ki(A) where A = A © CI. In particular every idempotent in Mk(A) gives a classin Kn(A) but in general not all classes in Kn(A) are obtained in this way. Thedefinition of a Fredholm module should be slightly modified for non-unital Banachalgebras, but the theorem 1.1.2 remains true.1.2. Statement of the Baum-Connes conjectureLet G be a second countable, locally compact group. We fix a left-invariantHaar measure dg on G. Denote by C c (G) the convolution algebra of complex-valuedcontinuous compactly supported functions on G. The convolution of /,/' £ C C (G)is given by / * f'(g) = J G f(h)f'(hr 1 g)dh for any g £ G.When G is discrete and dg is the counting measure, C c (G) is also denoted byCC? and if e g denotes the delta function at g £ G (equal to 1 at g and 0 elsewhere),(e g ) ge G is a basis of CG and the convolution product is given by e g e g i = e gg i.The completion of C C (G) for the norm ||/||LI = J G \f(g)\dg is a Banach algebraand is denoted by L 1 (G).For any / £ C C (G) let A(/) be the operator /' H> / * /' on L 2 (G). Thecompletion of C C (G) by the operator norm ||/|| re d = \\Hf)\\c(L 2 (G)) ' 1S ealled thereduced G*-algebra of G and denoted by C* ed (G). If G is discrete (e g ') g i e G is anorthonormal basis of L 2 (G) and X(e g ) : e g i >-¥ e gg i.For any / £ C C (G), ||/||j,i > ||/|| re d and L 1 (G) is a dense subalgebra ofC* ed (G). We denote by i : L 1 (G) —¥ C* ed (G) the inclusion.