1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

8.1. Stable uniqueness 265 Lemma 8.1.6. Let ao, a1: A ---+ Mk(C) be homotopic *-homomorphisms and 1f: A ---+ JIB('H) be any fait ...

266 8. AF Embeddability Theorem 8.1.8. Let cro, cr1: A--+ Mk((['.) be homotopic *-homomorphisms and assume that cro is (~, c:)-a ...

8.2. Cones over exact RFD algebras 267 ' 8.2. Cones over exact RFD algebras A few more fairly simple facts and we'll be able to ...

268 8. AF Embeddability So, let Br = Mk(I)(<C) and 0"1 = PI: CA -+ Br. Now consider Br® Mk( 2 )(<C) and the pair of *-homo ...

8.3. Cones over general exact algebras 269 We say cp is (J,E)-suitable if it is (J,E)-multiplicative and there is a u.c.p. map ' ...

270 8. AF Embeddability suggests the possibility of adapting Proposition 8.1.3 to deduce AF embed- dability of all such algebras ...

8.3. Cones over general exact algebras 271 there is an (.~1, 51)-suitable u.c.p. map 81 from C into a full matrix algebra D1 and ...

272 8. AF Embeddability = (v(O) o o-(p1) EB p(l)) EB μ(1) (tg ,h(8o)) ~ no(l)Oo(l) EB μ(1). Note that we've taken a step to the ...

8.3. Cones over general exact algebras 273 Go back and forth until the N - 1 step and you have v(O)EBm(l)v(l)EB· · ·EBm(N-l)v(N- ...

274 8. AF Embeddability hence we can apply Lemma 8.3.4 to C(q) and get a (7r(q)(Jf^1 ), 81)-suitable u.c.p. map B1(q): C(q)-+ Dl ...

8.4. Homotopy invariance 275 Proposition 8.4.2. Let I be a nonunital C* -algebra. Then there exists a unital C*-algebra M(I), ca ...

276 8. AF Embeddability Proof. Let I c B be an inclusion such that B is AF and let {en} c I be an approximate unit of I. Let J c ...

8.4. Homotopy invariance 277 for all b EB, and OC(JC) 0 I= UM2(0C(JC) 0 I)U* where we identify Mz(OC(JC) 0 I) C IIB((JC 0 H) EB ...

278 8. AF Embeddability Proposition 8.4.9. Assume 0--* I--* D --* B--* 0 is exact and there is a *-homomorphic splitting (}: B - ...

8.5. A survey 279 Proof. This follows immediately from Remark 8.4.12 and Proposition 8.4.9, since Theorem 8.3.5 ensures that CA ...

280 8. AF Embeddability Since 000 contains a copy of Co(O, 1], so does B. Thus, if A is exact, we can invoke Kirchberg's embeddi ...

A survey 281 (4) the induced map a*: Ko(A)------+ Ko(A) "compresses no element." In other words, if x E Ko(A) and a*(x) :: ...

282 8. AF Embeddability (4) Is every exact QD C*-algebra AF embeddable? This, of course, is the ultimate goal as it would give a ...

Local Reflexivity and Other Tensor Product Conditions Chapter 9 This chapter introduces yet another finite-dimensional approxima ...

284 9. Local Reflexivity the existence of a modular automorphism and semifiniteness of the corre- sponding crossed product and t ...