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K-Theory for Group C -Algebras

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K-Theory for Group C -Algebras K-theory for group C∗-algebras Paul F. Baum1 and Rub´enJ. S´anchez-Garc´ıa2 1 Mathematics Department 206 McAllister Building The Pennsylvania State University University Park, PA 16802 USA [email protected] 2 Mathematisches Institut Heinrich-Heine-Universit¨atD¨usseldorf Universit¨atsstr. 1 40225 D¨usseldorf Germany [email protected] Final draft (Last compiled on July 30, 2009) Introduction These notes are based on a lecture course given by the first author in the Sedano Winter School on K-theory held in Sedano, Spain, on January 22- 27th of 2007. They aim at introducing K-theory of C∗-algebras, equivariant K-homology and KK-theory in the context of the Baum-Connes conjecture. We start by giving the main definitions, examples and properties of C∗- algebras in Section 1. A central construction is the reduced C∗-algebra of a locally compact, Hausdorff, second countable group G. In Section 2 we define K-theory for C∗-algebras, state the Bott periodicity theorem and establish the connection with Atiyah-Hirzebruch topological K-theory. Our main motivation will be to study the K-theory of the reduced C∗- algebra of a group G as above. The Baum-Connes conjecture asserts that these K-theory groups are isomorphic to the equivariant K-homology groups of a certain G-space, by means of the index map. The G-space is the universal example for proper actions of G, written EG. Hence we procceed by discussing proper actions in Section 3 and the universal space EG in Section 4. Equivariant K-homology is explained in Section 5. This is an equivariant version of the dual of Atiyah-Hirzebruch K-theory. Explicitly, we define the G groups Kj (X) for j = 0; 1 and X a proper G-space with compact, second countable quotient GnX. These are quotients of certain equivariant K-cycles 2 Paul F. Baum and Rub´enJ. S´anchez-Garc´ıa by homotopy, although the precise definition of homotopy is postponed. We then address the problem of extending the definition to EG, whose quotient by the G-action may not be compact. In Section 6 we concentrate on the case when G is a discrete group, and in Section 7 on the case G compact. In Section 8 we introduce KK-theory for the first time. This theory, due to Kasparov, is a generalization of both ∗ j K-theory of C -algebras and K-homology. Here we define KKG(A; C) for a separable C∗-algebra A and j = 0; 1, although we again postpone the exact G definition of homotopy. The already defined Kj (X) coincides with this group when A = C0(X). At this point we introduce a generalization of the conjecture called the Baum-Connes conjecture with coefficients, which consists in adding coeffi- cients in a G-C∗-algebra (Section 9). To fully describe the generalized conjec- ture we need to introduce Hilbert modules and the reduced crossed-product (Section 10), and to define KK-theory for pairs of C∗-algebras. This is done in the non-equivariant situation in Section 11 and in the equivariant setting in Section 12. In addition we give at this point the missing definition of ho- motopy. Finally, using equivariant KK-theory, we can insert coefficients in equivariant K-homology, and then extend it again to EG. The only ingredient of the conjecture not yet accounted for is the index map. It is defined in Section 13 via the Kasparov product and descent maps in KK-theory. We finish with a brief exposition of the history of K-theory and a discussion of Karoubi's conjecture, which symbolizes the unity of K-theory, in Section 14. We thank the editor G. Corti~nasfor his colossal patience while we were preparing this manuscript, and the referee for her or his detailed scrutiny. 1 C∗-algebras We start with some definitions and basic properties of C∗-algebras. Good references for C∗-algebra theory are [1], [15], [40] or [42]. 1.1 Definitions Definition 1. A Banach algebra is an (associative, not necessarily unital) algebra A over C with a given norm k k k k : A −! [0; 1) such that A is a complete normed algebra, that is, for all a; b 2 A, λ 2 C, 1. kλak = jλjkak, 2. ka + bk ≤ kak + kbk, 3. kak = 0 , a = 0, 4. kabk ≤ kakkbk, K-theory for group C∗-algebras 3 5. every Cauchy sequence is convergent in A (with respect to the metric d(a; b) = ka − bk). A C∗-algebra is a Banach algebra with an involution satisfying the C∗- algebra identity. Definition 2. A C∗-algebra A = (A; k k; ∗) is a Banach algebra (A; k k) with a map ∗ : A ! A; a 7! a∗ such that for all a; b 2 A, λ 2 C 1. (a + b)∗ = a∗ + b∗, 2. (λa)∗ = λa∗, 3. (ab)∗ = b∗a∗, 4. (a∗)∗ = a, 5. kaa∗k = kak2 ( C∗-algebra identity). Note that in particular kak = ka∗k for all a 2 A: for a = 0 this is clear; if a 6= 0 then kak 6= 0 and kak2 = kaa∗k ≤ kakka∗k implies kak ≤ ka∗k, and similarly ka∗k ≤ kak. A C∗-algebra is unital if it has a multiplicative unit 1 2 A.A sub-C∗- algebra is a non-empty subset of A which is a C∗-algebra with the operations and norm given on A. Definition 3. A ∗-homomorphism is an algebra homomorphism ' : A ! B such that '(a∗) = ('(a))∗, for all a 2 A. Proposition 1. If ' : A ! B is a ∗-homomorphism then k'(a)k ≤ kak for all a 2 A. In particular, ' is a (uniformly) continuous map. For a proof see, for instance, [42, Thm. 1.5.7]. 1.2 Examples We give three examples of C∗-algebras. Example 1. Let X be a Hausdorff, locally compact topological space. Let + + X = X [ fp1g be its one-point compactification. (Recall that X is Haus- dorff if and only if X is Hausdorff and locally compact.) Define the C∗-algebra + C0(X) = α : X ! C j α continuous, α(p1) = 0 ; + with operations: for all α; β 2 C0(X); p 2 X ; λ 2 C (α + β)(p) = α(p) + β(p); (λα)(p) = λα(p); (αβ)(p) = α(p)β(p); α∗(p) = α(p); kαk = sup jα(p)j : p2X 4 Paul F. Baum and Rub´enJ. S´anchez-Garc´ıa Note that if X is compact Hausdorff, then C0(X) = C(X) = fα : X ! C j α continuousg : Example 2. Let H be a Hilbert space. A Hilbert space is separable if it admits a countable (or finite) orthonormal basis. (We shall deal with separable Hilbert spaces unless explicit mention is made to the contrary.) Let L(H) be the set of bounded linear operators on H, that is, linear maps T : H ! H such that kT k = sup kT uk < 1 ; kuk=1 where kuk = hu; ui1=2. It is a complex algebra with (T + S)u = T u + Su; (λT )u = λ(T u); (TS)u = T (Su); for all T;S 2 L(H), u 2 H, λ 2 C. The norm is the operator norm kT k defined above, and T ∗ is the adjoint operator of T , that is, the unique bounded operator such that hT u; vi = hu; T ∗vi for all u; v 2 H. Example 3. Let L(H) be as above. A bounded operator is compact if it is a norm limit of operators with finite-dimensional image, that is, K(H) = fT 2 L(H) j T compact operatorg = fT 2 L(H) j dimC T (H) < 1g ; where the overline denotes closure with respect to the operator norm. K(H) is a sub-C∗-algebra of L(H). Moreover, it is an ideal of L(H) and, in fact, the only norm-closed ideal except 0 and L(H). 1.3 The reduced C∗-algebra of a group Let G be a topological group which is locally compact, Hausdorff and second countable (i.e. as a topological space it has a countable basis). There is a C∗-algebra associated to G, called the reduced C∗-algebra of G, defined as follows. Remark 1. We need G to be locally compact and Hausdorff to guarantee the existence of a Haar measure. The countability assumption makes the Hilbert space L2(G) separable and also avoids some technical difficulties when later defining Kasparov's KK-theory. K-theory for group C∗-algebras 5 Fix a left-invariant Haar measure dg on G. By left-invariant we mean that if f : G ! C is continuous with compact support then Z Z f(γg)dg = f(g)dg for all γ 2 G: G G Define the Hilbert space L2G as Z 2 n 2 o L G = u : G ! C ju(g)j dg < 1 ; G with scalar product Z hu; vi = u(g) v(g)dg G for all u; v 2 L2G. Let L(L2G) be the C∗-algebra of all bounded linear operators T : L2G ! L2G. On the other hand, define CcG = ff : G ! C j f continuous with compact supportg : It is an algebra with (f + h)(g) = f(g) + h(g); (λf)(g) = λf(g); for all f; h 2 CcG, λ 2 C, g 2 G, and multiplication given by convolution Z −1 (f ∗ h)(g0) = f(g)h(g g0) dg for all g0 2 G: G R P Remark 2. When G is discrete, G f(g)dg = G f(g) is a Haar measure, CcG is the complex group algebra C[G] and f ∗ h is the usual product in C[G].
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