1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

6.3. Some motivation and examples 225 Proof. Thanks to the previous result, we only have to observe (1) ::::;:.. (3). However, i ...

226 6. Amenable Traces Statement ( 4) in the previous proposition reveals a subtlety which one should be aware of: the notion of ...

6.4. Factorization and property (T) 227 Exercise 6.3.3. Prove that r is amenable if and only if C!(r) has a finite- dimensional ...

228 6. Amenable Traces With Theorem 6.2. 7 at our disposal, the following fact is immediate. (See Exercise 6.1.4 for the identif ...

6.4. Factorization and property (T) 229 representations 7f(i): r -----+ JIB(Hi) such that each 7f(i) has no fixed vectors but th ...

230 6. Amenable Traces Proof. We can find a representation u: C*(I') --+ Iffi(H) and a finite-rank projection P E IIB(H) such th ...

6.4. Factorization and property (T) 231 Remark 6.4.9. It is possible to avoid Lemma 6.2.5 in the proof above, as follows. Write ...

232 6. Amenable Traces Proof. Let T1, ... ,Tn E GL(n,K) be given and let r be the group they generate. Let R c K be the ring gen ...

6.4. Factorization and property (T) 233 is continuous with respect to the minimal norm for all hyperbolic groups r. However, if ...

234 6. Amenable Traces Given z E F, we can write z as an integer polynomial in the k/s. Hence, substituting in our decomposition ...

6.5. References 235 Exercise 6.4.3. Let r be a discrete group with property (T). Show that r is residually finite if and only if ...

...

Chapter 7 Quasi diagonal C* -Alge bras In this chapter we present the basics of a large and intriguing class of C - algebras. Qu ...

238 7. Quasidiagonal C* -Algebras Remark 7.1.2. Completely positive maps respect linear, involutive and order structures; the de ...

7.1. The deE.nition, easy examples and obstructions 239 Another thing one should check is whether quasidiagonality passes to uni ...

240 7. Quasidiagonal C* -Algebras Proposition 7.1.9. Quasidiagonality passes to inductive limits, so long as the connecting maps ...

7.1. The defi.nition, easy examples and obstructions 241 is also injective. Since injective representations of C* -algebras are ...

242 7. Quasidiagonal C* -Algebras we will see counterexamples after describing the other obstruction to qua- sidiagonality. In s ...

7.2. The representation theorem 243 Exercise 7.1.3. Show that A is QD if and only if there exists an injective *-homomorphism A- ...

244 7. Quasidiagonal C* -Algebras for all T E ~ and llPv-vll<c: for all v EX· As usual, this local formulation is handy when ...