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**Example text**

Similarly, the map φ is completely positive iﬀ φn is positive for all n; it is completely isometric iﬀ φn is isometric for all n ≥ 1 and φ is completely contractive iﬀ the maps φn are contractions ( φn ≤ 1) for all n. 5]. When A is unital and φ is a complete contraction, then φ is completely positive if and only if φ(1) = φ . Approximation Properties 27 With these notions at hand we can introduce one of the main approximation properties. 1. A C ∗ -algebra A is nuclear iﬀ it has the following completely positive approximation property (CPAP): The identity map id : A → A can be approximated in the point-norm topology by ﬁnite rank completely positive contractions.

2, 169–203. [15] V. Laﬀorgue, A proof of property (RD) for cocompact lattices of SL(3, R) and SL(3, C). J. Lie Theory 10 (2000), no. C. Lance, On nuclear C ∗ -algebras. J. Functional Analysis 12 (1973), 157–176 [17] H. Leptin, Sur l’alg`ebre de Fourier d’un groupe localement compact. R. Acad. Sci. Paris S´er. A-B 266 (1968) A1180–A1182. [18] V. Mathai, Heat kernels and the range of the trace on completions of twisted group algebras. Contemporary Mathematics 398 (2006), 321–346. D. Reeves, The geometry of cube complexes and the complexity of their fundamental groups.

The zero set of such a section then gives rise to σ satisfying (9). So for two vector ﬁelds with non-degenerate zeros this construction yields a one chain cs(X1 , X2 ), well deﬁned up to a boundary, satisfying ∂ cs(X1 , X2 ) = eX2 − eX1 . Let us extend this to vector ﬁelds with isolated singularities. Suppose X is a vector ﬁeld with isolated singularities. For every zero x ∈ X we choose an embedded ball Bx centered at x, assuming all Bx are disjoint. Set B := x∈X Bx . Choose a vector ﬁeld with non-degenerate zeros X that coincides with X on M \B.