1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

4.9. References 165 where T(<C) is the Toeplitz algebra and w is the vacuum state. (Really long hint, which some consider a p ...

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Exact Groups and Related Topics Chapter 5 We begin this chapter by defining exact groups and proving that this concept is equiva ...

168 5. Exact Groups and Related Topics width Eis the subset Tube(E) in r x r given by Tube(E) = {(s, t) EI' x I': sC^1 E E}.^2 B ...

5.1. Exact groups 169 Remark 5.1.5. Note that if k is a positive definite kernel supported in a tube (i.e., k(s, t) = 0 for all ...

170 5. Exact Groups and Related Topics 2: 0 since e is c.p. and [.A(sisj^1 )]i,j 2: 0 in Mn(C~(r)) (Example 1.5.13). Since <p ...

5.1. Exact groups 171 (2) the left translation action of r on the Stone-Cech compactification ,er is amenable (i.e., for any fin ...

172 5. Exact Groups and Related Topics Fix NE N and define m: lF2-+ Prob(lF2) by l N-1 mt= NL Ot(k)· k=O Clearly, all mt are sup ...

5.1. Exact groups e Figure 1. The Oayley graph of 1F2 and the boundary olF 2 s.x f. Figure 2. Amenability of lF 2 acting on olF ...

174 5. Exact Groups and Related Topics Proof. Assume first that r is KW-exact and let 0 -----+ I -----+ A -----+ A/ I -----+ 0 b ...

5.1. Exact groups 175 4.1.4). Thus we have an embedding (A><lrr)0C~(r) c JE(7i'.0£^2 (r)0£^2 (r)). Letting V: £^2 (r) --+ ...

176 5. Exact Groups and Related Topics map 'T/: Y --->Prob( A) c Prob(I') such that maxmaxsup ll(cr(p)-^1 s-^1 cr(sp)).'TJY - ...

5.2. Groups acting on trees 177 Proof. We first claim that for every E > 0 and finite subset E c r, there exists a continuous ...

178 5. Exact Groups and Related Topics This implies r !ri(x) dm(x) < r I: 11s.c - (^3 ·x11 dm(x) + 8 ; < s Jx JxsEE and we ...

5.2. Groups acting on trees 179 = μs.x,s.y for every s E E and ( x, y) E X x Y. 0 Remark 5.2.2. This result really does generali ...

180 5. Exact Groups and Related Topics The lemma above is really a special case. Indeed, essentially the same proof yields the f ...

5.2. Groups acting on trees 181 We observe that for each n there exists at most one x(n) such that {a : xa(n) = x(n)} EU and suc ...

182 5. Exact Groups and Related Topics Lemma 5.2.8. Let r be a group and T be a tree on which r acts. Let (sn) be a net in r suc ...

5.3. Hyperbolic groups 183 The set Nr(A) is called the r-tubular neighborhood of A in K. For subsets A, B c K, the Hausdorff dis ...

184 5. Exact Groups and Related Topics Proposition 5.3.4. Let K be a hyperbolic graph. Then there exists 6 > 0 such that ever ...