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Group C*-Algebras and K-Theory

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Group C*-Algebras and K-Theory Group C*-Algebras and K-theory ¡ Nigel Higson and Erik Guentner ¢ Department of Mathematics, Pennsylvania State University University Park, PA 16802 [email protected] £ Department of Mathematical Sciences University of Hawaii, Manoa 2565 McCarthy Mall, Keller 401A Honolulu, HI 96822 [email protected] Preface These notes are about the formulation of the Baum-Connes conjecture in operator algebra theory and the proofs of some cases of it. They are aimed at readers who ¤ have some prior familiarity with -theory for ¥§¦ -algebras (up to and including the Bott Periodicity theorem). I hope the notes will be suitable for a second course in operator ¤ -theory. The lectures begin by reviewing ¤ -theory and the Bott periodicity theorem. Much of the Baum-Connes theory has to do with broadening the periodicity theo- rem in one way or another, and for this reason quite some time is spent formulating and proving the theorem in a way which is suited to later extensions. Following that, the lectures turn to the machinery of bivariant ¤ -theory and the formulation of the Baum-Connes conjecture. The main objective of the notes is reached in Lec- ture 4, where the conjecture is proved for groups which act properly and isometrically on affine Euclidean spaces. The remaining lectures deal with partial results which are important in applications and with counterxamples to various overly optimistic strengthenings of the conjecture. Despite their length the notes are not complete in every detail, and the reader will have to turn to the references, or his own inner resources, to fill some gaps. In ad- dition the lectures contain no discussion of applications or connections to geometry, topology and harmonic analysis, nor do they cover the remarkable work of Vincent Lafforgue. For the former see [7]; for the latter see [62, 44]. The notes are based on joint work carried out over a period of many years now with many people: Paul Baum, Alain Connes, Erik Guentner, Gennadi Kasparov, Vincent Lafforgue, John Roe, Georges Skandalis and Jody Trout. It is a pleasure to thank them all. I am especially grateful to Erik Guentner for writing the first draft of these notes and for his valuable assistance throughout their creation. Both authors were partially supported by NSF grants during the preparation of this paper. Nigel Higson 138 Nigel Higson and Erik Guentner 1 K-Theory In the first three lectures we shall be developing machinery needed to formulate the Baum-Connes conjecture and prove some cases of it. We shall presume some prior ¤ familiarity with ¥¨¦ -algebra -theory, but we shall also develop a `spectral' picture of ¤ -theory from scratch. In Lecture 1 we shall prove the Bott periodicity theorem in ¤ ¥¨¦ -algebra -theory in a way which will be suited to generalization in subsequent lectures. 1.1 Review of K-theory ¤ We begin by briefly reviewing the rudiments of ¥§¦ -algebra -theory, up to and ¤ including the Bott periodicity theorem. As the reader knows, ¥©¦ -algebra -theory is a development of the topological ¤ -theory of Atiyah and Hirzebruch [4]. But the basic definition is completely algebraic in nature: ¤ Definition 1.1. Let be a ring with a multiplicative unit. The group is the abelian group generated by the set of isomorphism classes of finitely generated and projective (unital, right) -modules, subject to the relations . Remark 1.1. Functional analysts usually prefer to formulate the basic definition in terms of equivalence classes idempotents in the matrix rings "! . This is be- cause in several contexts idempotents arise more naturally than modules. We shall use both definitions below, bearing in mind that they are related by associating to an ! %'#( idempotent #%$& ! the projective module . ¤ The group is functorial in since associated to a ring homomorphism ,)+-(. * )+* there is an induction operation on modules, . ¤ Most of the elementary algebraic theory of the functor is a consequence of a structure theorem involving pull-back diagrams like this one: /10 / 87 7 87 87 4 :9 ¡ ¡>=@? ?A¡ ¡ ;$< > AB 65 . 0 /32 ¡ 4 / * 2 Theorem 1.1. Assume that in the above diagram at least one of the two homomor- ¡ phisms into * is surjective. If and are finitely generated and projective mod- ¡ ?E ?A¡ ¡ DC ) * ules over and , and if is an isomorphism of modules, ¦ ¦ then the -module 8F F JF F 9 = ¡ I ¡ ¡ 665 G$H -KLM -'KNB is finitely generated and projective. Moreover, up to isomorphism, every finitely gen- OP erated and projective module over has this form. Group C*-Algebras and K-theory 139 This is proved in the first few pages of Milnor's algebraic ¤ -theory book [49]. The theorem describes projective modules over in terms of projective modules ¡ over , projective modules over , and invertible maps between projective mod- ¤ *© ules over * . It leads very naturally to the definition of a group in terms of invertible matrices, but at this point the purely algebraic and the ¥©¦ -algebraic theo- ries diverge, as a result of an important homotopy invariance principle. 9 ¥Q¦ Q R KS ¥Q¦ Definition 1.2. Let be a -algebra. Denote by the -algebra of contin- 9 R KT uous functions from the unit interval into . JU ¥§¦ We shall similarly denote by the -algebra of continuous functions from U a compact space into a ¥Q¦ -algebra . ¥Q¦ Theorem 1.2. Let be a -algebra with unit. If is a finitely generated and pro- 9 R KT jective module over § then the induced modules over obtained by evaluation 9 9 R KT K($X R KS OP at RV$W and are isomorphic to one another. As a result, ¤ -theory is a homotopy functor in the sense of the following defini- tion: ¥§¦ Definition 1.3. A homotopy of Y -homomorphisms between -algebras is a family 87 9 `<$a R KT `b,)_Zc[ of homomorphisms Z\[]C ^)_* ( ), for which the maps are 7 $d ¥§¦ continuous, for all . A functor on the category of -algebras is a homotopy [ functor if all the homomorphisms Z in any homotopy induce one and the same map [ Z ]Ce >)+ *§ . ¤ ¤ We shall now define the -theory group . ¦ ¦ ¥ f! ¥ Definition 1.4. Let be a -algebra with unit. Denote by the -algebra I g h(i;! of g matrices with entries in and denote by the group of invert- h(ij! h(ij!lknm ible elements in ! . View as a subgroup of each via the embeddings p oR o,) Rrqsut ¤ w v h(i x Denote by the direct limit of the component groups ! : ¤ >y{z}| v h(i>! x ~ ) t h(i ! Remark 1.2. This is a group, thanks to the group structure in , and in fact an wn abelian group since is homotopic to , and hence to . Returning to our pullback diagram and Theorem 1.1, it is now straightforward to derive all but the dotted part of the following six-term `Mayer-Vietoris' exact sequence of ¤ -theory groups: ¤ w ¤ w ¤ ¤ ¡ *§ / / (1) O ¤ ¤ ¤ ¤ ¡ *© o o t 140 Nigel Higson and Erik Guentner The diagram is completed (along the dotted arrow) as follows. Consider first the pullback diagram 0 / / 4 / 0 2 ¡ * 4 / 2 involving algebras of functions on the circle . The Mayer-Vietoris sequence asso- ciated to it, ¤ w ¤ 8 ¤ 8 ¤ 8 ¡ x x x * x / / (2) O 8 8 ¤ 8 ¤ ¤ ¤ 8 ¡ 9 * x x x x o o K$ maps to the Mayer-Vietoris sequence (1) via the operation of evaluation at , and in fact this map is the projection onto a direct summand since has a one-sided inverse consisting of the inclusion of the constant functions into the various algebras of functions on . The complementary summands are computed using the following two results: ¥Q¦ Theorem 1.3. Let be a -algebra. The kernel of the evaluation homomorphism ¤ ¤ w \C M) ¤ P is naturally isomorphic to . O This is a simple application of the partial Mayer-Vietoris sequence (think of 8 9 Q R KS as assembled by a pullback operation from two copies of ). ¥Q¦ Theorem 1.4. Let be a -algebra. The kernel of the evaluation homomorphism ¤ ¤ \C M) ¤ is naturally isomorphic to . This is much harder; it is one formulation of the Bott periodicity theorem. But granting ourselves the result for a moment, we can complete the diagram ¤ (1) by the simple device of viewing its horizontal reflection (with the -groups on the top) as a direct summand of the diagram (2). The required connecting ¤ ¤ *©) map C appears as a direct summand of the connecting map ¤ 8 ¤ 8 C x>) x * . The full Mayer-Vietoris sequence is a powerful computational tool, especially U ¤ U ) x for commutative algebras. For example it implies that the functors , constitute a cohomology theory on compact spaces (as in algebraic topology). A simple consequence is the formula Group C*-Algebras and K-theory 141 ¡ ¤ w 8 ¤ ¤ w xM which is a perhaps more familiar formulation of Bott periodicity. Let us conclude our review of ¤ -theory with a quick look at the proof of Theo- rem 1.4. The launching point is the definition of a map ¤ w ¤ C j) by associating to the class of an idempotent #$& f! the element J E ~ # K #¨ M (3) 8 in h(i>! . The following argument (due to Atiyah and Bott [6]) then shows that this Bott homomorphism is an isomorphism onto the kernel of the evaluation ¤ ¤ cC map x ) . The key step is to show is surjective; the proof in injectivity is a minor elaboration of the surjectivity argument3 and we shall not comment on it further. By an approximation argument involving trigonometric polynomials the proof of surjectivity quickly reduces to showing that a polynomial loop of invertible matrices J 9 9 \ 'SL $< u! which defines a element of the kernel of the evaluation map must lie in the image of 8 .
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