1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

5.6. Groupoids 205 Cc(G(o)). Let U c G be an open subset on which both s and r are homeomorphisms. Then, for every f E Cc( G) wi ...

206 5. Exact Groups and Related Topics We note that sup1'EG lh(1)I = supxEG(o) h(x) for a positive-type function h. Also, h E Cc ...

5.6. Groupoids 207 is algebraically positive, where gp("!) = (1 - lf(r('Y))l^2 )^112 gp("f). Take an element e E Cc(G(o)) such t ...

208 5. Exact Groups and Related Topics Thus, writing z = L, >..(fj)<g>aj, we have z*z E Co(G(0))0A C C)'.(G)®maxA. Sinc ...

References 209 a E Mn(<C). For simplicity, let az = 'lj;(>..(Jz)). Then every az is a contraction such that ((az@ 1@ ...

210 5. Exact Groups and Related Topics [6] for more. The Gromov boundary of a hyperbolic space was defined by- care to guess? -G ...

Amenable Traces and Kirchberg's Factorization Property Chapter 6 In this chapter we explore another duality between finite-dimen ...

212 6. Amenable Traces A 2 Let T be a tracial state on A, L^2 (A, T) be the GNS space, AC L (A, T) be the canonical image of A a ...

6.1. Traces and the right regular representation 213 (JS J)'. With these simple observations and the technical calculations of t ...

214 6. Amenable Traces Exercise 6.1.6. Let tr be the unique tracial state on Mn(C). Show that the product map arising from the l ...

6.2. Amenable traces 215 It follows that if b is a linear combination of unitaries in A, then <p(Tb) = <p(bT) and, since u ...

216 6. Amenable Traces for all finite-rank h, k. Letting hk = ulhkl be the polar decomposition, one checks that which completes ...

6.2. Amenable traces 217 Lemma 6.2.5. Let h E JB(H) be a positive, finite-rank operator with ra- tional eigenvalues and such tha ...

218 6. Amenable Traces and, by the Powers-St0rmer inequality, 1 1 1 1/2 1 2 1 l -Tr(h2uh2u*) = 2JJh -uh2u*Jb :S 2JJh-uhu*J[1. We ...

6.2. Amenable traces 219 1 1/2 :::::; 2llh - uhu*ll1 + llh-uhu*ll1 , so the proof is complete. D If w is a state on B, then we d ...

220 6. Amenable Traces (3) the positive linear functional μ 7 : A 0 A^0 P --+ <C is continuous with respect to the minimal te ...

6.2. Amenable traces 221 and they are still u.c.p. (Exercise 6.1.7). For notational convenience, we let 'P~P : A^0 P --+ Mk(n) ( ...

222 6. Amenable Traces Tu(h.>.. ·)are the corresponding states, then the net ('1.b.>..)IM converges to Tin the weak topolo ...

6.3. Some motivation and examples 223 In [41] another important characterization of amenable traces was given; it also illustrat ...

224 6. Amenable Traces As usual, group C* -algebras provide some insightful examples. Recall that the left translation action of ...