By Gert Kjaergard Pedersen
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Extra resources for C*-algebras and their automorphism groups
21] V. Paulsen, Completely bounded maps and dilations. Pitman Research Notes in Mathematics Series, 146 Longman, New York, 1986.  G. J. Lie Theory 8 (1998), no. 1, 163–172.  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.  Daniel T. Wise, Cubulating Small Cancellation Groups. Geometric and Functional Analysis 14, no. 1, 150–214.  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  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.  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  P. de la Harpe, A.