By Gert Kjaergard Pedersen

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**Extra resources for C*-algebras and their automorphism groups**

**Example text**

21] V. Paulsen, Completely bounded maps and dilations. Pitman Research Notes in Mathematics Series, 146 Longman, New York, 1986. [22] G. J. Lie Theory 8 (1998), no. 1, 163–172. [23] S. Wassermann, Exact C ∗ -algebras and related topics. Lecture Notes Series, 19. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1994. [24] Daniel T. Wise, Cubulating Small Cancellation Groups. Geometric and Functional Analysis 14, no. 1, 150–214. [25] G. Yu, The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space.

Let p˜ : p∗ T M → T M be the natural vector bundle homomorphism over p. Using the last equation, Stokes’ theorem and d(X∗ p˜∗ Ψ(g)) = p∗ E(g) we get: R(X2 , g, ω) − R(X1 , g, ω) = d p∗ ω ∧ X∗ p˜∗ Ψ(g) + R(X1 , X2 , ω) = I×(M\V ) = p∗ (ω ∧ E(g)) + R(X1 , X2 , ω) − I×M = R(X1 , X2 , ω) For the last equality note that ω ∧ E(g) = 0 for dimensional reasons. Still assuming that X1 and X2 have non-degenerate zeros we next treat the case of a general non-degenerate homotopy X, whose zero set is not necessarily contained in a simply connected subset.

8] U. Haagerup, An example of a non-nuclear C ∗ -algebra which has the metric approximation property. Inventiones Math. 50 (1979), 279–293. Approximation Properties 35 [9] U. Haagerup, J. Kraus, Approximation properties for group C ∗ -algebras and group von Neumann algebras. Trans. Amer. Math. Soc. 344 (1994), no. 2, 667–699. [10] P. de la Harpe, Groupes hyperboliques, alg`ebres d’op´erateurs et un th´eor`eme de Jolissaint. C. R. Acad. Sci. Paris Ser. I 307 (1988), 771–774 [11] P. de la Harpe, A.