1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

7.2. The representation theorem 245 Proof. With the help of the previous lemma we'll show that Definition 7.2.1 implies a strong ...

246 7. Quasidiagonal C* -Algebras E > 0 are given, then one can find a finite-rank projection which almost commutes (up to E) ...

7.3. Homotopy invariance 247 Exercise 7.2.4. Use the spectral theorem to show that {N} C IIB(H) is a quasidiagonal set, for ever ...

248 7. Quasidiagonal C* -Algebras Given a EA, of norm one, we consider the inner derivation Da(b) = ba - ab, b EA. Assuming b al ...

7.3. Homotopy invariance 249 Proof. Let CC JIB(JC) be a faithful, essential representation and let O"t: B----+ C be a homotopy o ...

250 7. Quasidiagonal C* -Algebras and hence V is a partial isometry. Thus VV is a finite-rank projection in JE(ffi~ JC). A sligh ...

Homotopy invariance 251 down the commutator in matrix form, you will see that one must estimate the norms of elements of t ...

252 7. Quasidiagonal C* -Algebras From the Bott periodicity theorem it follows that the K-theory of an arbitrary A is isomorphic ...

7.4. Two more examples 253 Proof. We first recall a basic operator theory fact: If T E IIB('h'.) is a con- traction, then [ T Jl ...

254 7. Quasidiagonal C* -Algebras Remark 7.4.3. Free groups actually enjoy a stronger property: Every rep- resentation can be ap ...

7.5. External approximation 255 A crucial remark is that we can even arrange a certain amount of com- mutativity. Namely, since ...

256 7. Quasidiagonal C* -Algebras for all a E A. Letting B = C*(<I>(A)), it is clear that B is RFD. The remainder of the p ...

7.5. External approximation 257 Exercise 7.2.4). However, A is evidently isomorphic to C(S EB S EB S) c B(£^2 (N) EB £^2 (N) EB ...

258 7. Quasidiagonal C* -Algebras for all a, b E J. It follows that 1/Jn is almost multiplicative on !f?n(J). More precisely, is ...

7.5. External approximation 259 It will be convenient to introduce some notation. Let i.p: A--+ IIB(H) and 'ljJ: A--+ IIB(JC) be ...

260 7. Quasidiagonal C* -Algebras first. Finally the last line also follows from Voiculescu's Theorem (Corollary 1.7.7) since we ...

Chapter 8 AF Embeddability Deciding when a particular C* -algebra is isomorphic to a subalgebra of some AF algebra-i.e., when it ...

262 8. AF Embeddability Proposition 8.1.1 (Approximately commuting diagrams). Given a C- algebra A, assume there exist finite-di ...

8.1. Stable uniqueness 263 To get a feel for the type of thing we're after, here's a very important first step. Proposition 8.1. ...

264 8. AF Embeddability By construction we have n1 o 0"1 = 0"2. If you really understand the argument so far, the rest of the pr ...