1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

Nuclear and Exact C* -Algebras: Definitions, Basic Facts and Examples Chapter 2 Nuclearity and exactness have dominated the C*-s ...

26 2. Nuclear and Exact C* -Algebras Note that nuclear maps are automatically c.c.p. If A, B and e are unital, we'll soon see th ...

2.1. Nuclear maps 27 Proof. Let { PihEI be a net of finite-rank projections which increases to the identity (i.e., i ~ j :::}-Pi ...

28 2. Nuclear and Exact C* -Algebras Exercise 2.1.7 (Dependence on range). If B: A-+ Bis nuclear and CC B is a C*-subalgebra wit ...

2.2. Nonunital technicalities 29 of B, then we are done. Though that won't be the case in general, we do have 0 S cp(lA) S llc,o ...

30 2. Nuclear and Exact C* -Algebras maps ,(/;n : Mk( n) ( <C) EB <C --+ B and one checks that { ,(/;n o <Pn} converges ...

2.2. Nonunital technicalities 31 The right maps to replace the 1/Jn's are slightly technical to describe but there is nothing de ...

32 2. Nuclear and Exact C*-Algebras Since the sum of positive operators is positive, it is easy to check that the 7/Jn's are u.c ...

2.3. Nuclear and exact C*-algebras 33 Diagrammatically, a C -algebra is exact if there exist a faithful repre- sentation 7r: A__ ...

34 2. Nuclear and Exact C*-Algebras We must show that for. each finite set J = { ai, ... , ak} C A and E > 0 there exists SEC ...

2.3. Nuclear and exact C*-algebras 35 Remark 2.3. 7. The proof shows that the set of factorable maps which have contractive fact ...

36 2. Nuclear and Exact C* -Algebras Exercises Exercise 2.3.1 (Finite sets and c:'s). Give the proper "local" formulations of nu ...

2.3. Nuclear and exact C*-algebras 37 Exercise 2.3.8 (Separable versus nonseparable). Show that a C-algebra is exact if and only ...

38 2. Nuclear and Exact C*-Algebras Exercise 2.3.13. The proof of Proposition 2.3.8 goes through verbatim if one only assumes th ...

2.4. First examples 39 we have lf(x) - f(y)I < E for any pair of points x, y E Ui. Let Yi E Ui be arbitrarily chosen and {0-1 ...

40 2. Nuclear and Exact C*-Algebras and where P E Mn(Co(X)) is a projection. Hence various exercises imply this corollary.^10 D ...

First examples 41 Actually, it suffices to. show that we can always find two orthogonal projections P1,P2 which are Murray ...

42 2. Nuclear and Exact C*-Algebras 2.5. C*-algebras associated to discrete groups This section contains a bare-bones introducti ...

C* -algebras associated to discrete groups 43 Example 2.5.1. If r = Z, then C~(r) = C('JI'), the continuous func- tions on ...

44 2. Nuclear and Exact C*-Algebras right regular representation p: r , IBS(£^2 (I')) - i.e., L(I') = p(C[r])' and L(I')' = p(C[ ...