By Eduard Casas-Alvero, Gerald E. Welters, Sebastian Xambo-Descamps

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5. Exercises. 13. (Mishchenko-Fomenko) Let A be a C ∗ -algebra. 3. , does not depend on the choice of decomposition. On the other hand, show by example that it is not necessarily true that D has closed range, and hence it is not necessarily true that we can deﬁne Ind D as [ker D] − [coker D]. 14. Let G be a ﬁnite group of order n. Show that Cr∗ (G) = CG ∼ = Mdim(σ) (C), and K0 (Cr∗ (G)) ∼ = Zc , K1 (Cr∗ (G)) = 0, b σ∈G where G is the set of irreducible representations of G and c = #(G) is the number of conjugacy classes in G.

24. Construct complete Riemannian metrics g on R2 for which ∗ K∗ (Corb (X)), X = (R2 , g) is not isomorphic to K∗ (pt), and give an example of an application to index theory on X. (Hint: The Coarse Baum-Connes Conjecture is valid for the open cone on a compact metrizable space Y . ) References 1. Michael Atiyah, Elliptic operators, discrete groups and von Neumann algebras, Colloque “Analyse et Topologie” en l’Honneur de Henri Cartan (Orsay, 1974), Soc. Math. France, Paris, 1976, pp. 43–72. Ast´ erisque, No.

1 (Green-Julg [44]). If G is compact, there is a natural isomorphism −∗ (X). K∗ (C ∗ (G, X)) ∼ = KG 6(Strong) Morita equivalence (see [71]) is one of the most useful equivalence relations on the class of C ∗ -algebras. When A and B are separable C ∗ -algebras, it has a simple characterization [8]: A and B are strongly Morita equivalent if and only if A ⊗ K ∼ = B ⊗ K, where K is the C ∗ -algebra of compact operators on a separable, inﬁnite-dimensional Hilbert space. APPLICATIONS OF NON-COMMUTATIVE GEOMETRY TO TOPOLOGY 25 There are many other results relating the structure of C ∗ (G, X) to the topology of the transformation group (G, X); the reader interested in this topic can see the surveys [68], [69], and [66] for an introduction and references.