1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

2.5. C*-algebras associated to discrete groups 45 Definition 2.5.6. A function <p: r ____, C is said to be positive definite ...

46 2. Nuclear and Exact C* -Algebras Proposition 2.5.8. Let A c r be a subgroup. There is a canonical inclusion C*(A) c C*(r). P ...

2.5. C* -algebras associated to discrete groups 47 Theorem 2.5.11. Let <p: r ---t CC be a function with <p(e) = 1. The fol ...

48 2. Nuclear and Exact C* -Algebras A. That is, EI(.\(s)) = XA(s).\(s) for s Er, where XA is the characteristic function of A. ...

2.6. Amenable groups 49 Note that this implies the existence of an approximate invariant mean given by normalized characteristic ...

50 2. Nuclear and Exact C*-Algebras Here is the simplest example of something nonamenable. Example 2.6.7 (Nonabelian free groups ...

2.6. Amenable groups (8) for any finite subset E c r, we have 1 11TEI 2: "-sll = 1; (9) C~(r) is nuclear;^18 (10) L(r) is semidi ...

52 2. Nuclear and Exact C* -Algebras Hence we have E f 1 IF(μ, r)jdr = E > L llsμ - μ111 = f 1 L lsF(μ, r) L F(μ, r)jdr. Jo s ...

2.6. Amenable groups 53 operator S = 111 l:::sEE As is self-adjoint. Thus, for any c: > 0, we can find a unit vector e E £^2 ...

54 2. Nuclear and Exact C* -Algebras It follows that for each fixed pair g, h E I', converges to (xo 9 , oh) = ahg-1 as k ----+ ...

Type I C* -algebras 55 2.7. Type I C*-algebras We now consider the class of C* -algebras characterized by having nice rep- ...

56 2. Nuclear and Exact C* -Algebras of projections PD = 1 ® qn, where the index set is the set of all finite subsets n c 1i par ...

Type I C* -algebras 57 Proposition 2.7.7. If A is subhomogeneous, then A is type I (hence nu- clear). Proof. For each natu ...

58 2. Nuclear and Exact C* -Algebras 2.8. References Proposition 2.3.8 is due to Effros-Lance; they are largely responsible for ...

Chapter 3 Tensor Products This chapter is devoted to tensor products in the C -context. Those with a low tolerance for operator ...

60 3. Tensor Products facts which help us rationalize the absence of formal proof in this section: All of the results are simple ...

3.1. Algebraic tensor products 61 Note that the vector space structures on X 0 Y and Xx Y are completely different. For example, ...

62 3. Tensor Products Both of these propositions are simple applications of universality. For example, if cp: X -+ C, 'l/J: Y -+ ...

3.1. Algebraic tensor products 63 (i.e., cpj(xi) = Oi,j) and let ·'lj; E Y* be arbitrary. Then for each 1 :S j :S n we have 0 = ...

64 3. Tensor Products To see that cp 8 'lj; is injective is a simple exercise using what we know about bases and linear independ ...