1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

10.2. Exact C* -algebras 305 Proposition 10.2.8. For any discrete group I', the C*-algebra C*({Ag 0 Ag: g EI'}) c Ct(I') 0max C~ ...

306 10. Summary the proof of Theorem 5.1.10). Since (B Q9 A) ><Irr~ B Q9 (A ><Ir I') in this case (Exercise 4.1.3), ...

10.3. Quasidiagonal C* -algebras 307 Proof. Let p: A -+ M(I) be the natural extension of the inclusion I ~ M(I) ([198, 2.2.14]). ...

308 10. Summary Proof. Let i c A/ I be a finite set and let E > 0. Let X C A/ I be a finite- dimensional operator system whic ...

10.4: Open problems 309 Tensor products. It isn't known if the maximal tensor product of QD C* -algebras is again QD. For spatia ...

310 10. Summary (1) (Connes's embedding problem) Every separable II1-factor embeds into the ultraproduct Rw of the hyperfinite I ...

Part 2 Special Topics ...

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Chapter 11 Simple C* -Algebras This chapter contains a variety of results related to simple C* -algebras. Our first goal is to a ...

314 11. Simple C*-Algebras To a generalized inductive system, we can associate a limit C* -algebra just as for usual inductive s ...

11.1. Generalized inductive limits 315 on polynomials provide better and better approximations, to get represent- ing sequences ...

316 11. Simple C* -Algebras for all n. Let dn be the dimension of Sn. With a little work, one can find natural numbers PI < q ...

11.2. NF and strong NF algebras 317 Showing that A is isomorphic to the induced limit is reasonably straight- forward - one veri ...

318 11. Simple C*-Algebras Remark 11.2.2. If a u.c.p. map r.p: A -+ E is a complete order embedding, then r.p-^1 : r.p(A)-+ A is ...

11.2. NF and strong NF algebras 319 commutative diagram Ai -------> Ai EB A2 -------> Ai EB A2 EB A3 ------->... ------ ...

320 11. Simple C* -Algebras and c.c.p. maps am: A-+ Ms(m)(C), f3m: Ms(m)(C) -+A with all of the following properties: (1) llam(a ...

11.2. NF and strong NF algebras 321 Proof. We will choose 6 > O later. Let A = EBk= 1 JE(e~(k)) c JE(ea) with EB e~(k) = ea, ...

322 11. Simple C*-Algebras that 114? -1/illcb =max lllfz -1/izllcb < c:. Moreover, 1,0 is completely isometric since the map ...

11.3. Inner quasidiagonality 323 and use the t.pn,k, n > k, for connecting maps. Since we have defined Ck+i+l to be the algeb ...

324 11. Simple C* -Algebras At the other end of the spectrum, if A is simple and QD, then it is necessarily inner QD as well. In ...