1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

9.2. Tensor product properties 285 Suppose now that both I and B are locally reflexive and the extension is locally split. Let a ...

286 9. Local Reflexivity IIB('H) and N = ClM ® N ~ IIB('H) are normal representations. Of course, we may replace IIB('H) with an ...

9.2. Tensor product properties 287 Proof. Let C C A be a C* -subalgebra. We only prove the case of property C as the other two a ...

288 9. Local Reflexivity It turns out that property C' also passes to quotients (remarkably, prop- erties C and C' are equivalen ...

9.2. Tensor product properties 289 B.14). Passing to a convex combination if necessary, we may assume that the net {cpi(l)} conv ...

290 9. Local Reflexivity which is surjective (by Kaplansky's density theorem [53, Theorem I.7.3]) if we start with a large enoug ...

9.2. Tensor product properties 291 <p: E --+A such that for every x E E and f E F, lf(1.p(x) - x)I < c:llJll llxll- We reg ...

292 9. Local Reflexivity and N = JC** (canonically). Then, <T?!eB**<VfC** is a bi-normal and min- continuous u.c.p. map on ...

9.3. Equivalence of exactness and property C 293 Proof. Let <p: E _,A** be a complete contraction with dim(E) < oo. By ope ...

294 9. Local Reflexivity if its comnmtant is semidiscrete (Corollary 3.8.6) and we already know that if A is nuclear and 7r: A - ...

9.3. Equivalence of exactness and property C 295 representation v of JR. on lBl(H) -i.e., v(t)av(t)* =at( a) for every t E JR. a ...

296 9. Local Reflexivity Proof. For simplicity, let M = M ><1 °' JR. ><1 & R Since ,\ and μ are unitarily equiva ...

9.4. Corollaries 297 Proof of Theorem 9.3.3. Let M be an injective von Neumann algebra and a be a modular action so that M >& ...

298 9. Local Reflexivity Corollary 9.4.5. A separable C* -algebra is type I if and only if every sub- algebra is nuclear. Proof. ...

9.5. References 299 Hence we can find pure states <p on A and 'ljJ on B such that ( <p@'lj;) Iker er = 0 and ( <p 0 'lj ...

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Chapter 10 Summary and Open Problems Since a number of important characterizations and permanence properties are scattered throu ...

302 10. Summary Extensions. Proposition 10.1.3. If 0 ---+ I ---+ A ---+ B ---+ 0 is short exact and both I and B are nuclear, th ...

10.2. Exact C* -algebras 303 Crossed products. For amenable groups everything is nice: The crossed product is nuclear if and onl ...

304 10. Summary We have seen that exactness is also equivalent to properties C and C' (Theorem 9.3.1). Finally, as mentioned in ...