1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

11.3. Inner quasidiagonality 325 Proof. First we must argue that A;* is equal to pA ** p + ( 1 - p) A** ( 1 - p). The inclusion ...

326 11. Simple C*-Algebras To finish the proof one considers the representation 1T = 1T1 EB ... EB ?Tj: A-----+ JB(H1 EB ... EB ...

11.3. Inner quasidiagonality 327 Here is the main theorem of this section. Theorem 11.3.9. The unital C*-algebra A is strong NF ...

328 11. Simple C* -Algebras Proof. Simple QD algebras are inner QD and hence the assumption of nuclearity implies that A must be ...

11.4. Excision and Papa's technique 329 where {QY)} are orthogonal projections and 1 = aii) > a~i) > · · · > aiili) > ...

330 11. Simple C* -Algebras such that llwi-vill :Sf/ (where 8' will be determined later) and vjvi = 8i,jP, for 1 .::; i, j .::; ...

11.4. Excision and Papa's technique unit of this matrix algebra, then cutting an element x E J, we have qxq = ( f Ai) x ( f fj,j ...

332 11. Simple C* -Algebras II [q, u] 112 = llquqj_ - qj_uqll^2 = max{llqJ_u*qll^2 , llqJ_uqll^2 } = max{llquqJ_u*qll, llqu*qJ_u ...

11.4. Excision and Papa's technique 333 A II1-factor M is said to be approximately finite-dimensional ( AFD) if the following ho ...

334 11. Simple C* -Algebras in the relevant inequality). Hence there is a maximal family {Ci hEI E S, with units {qi}· We show b ...

11.5. Connes's uniqueness 'theorem 335 11.5. Connes's uniqueness theorem We close this chapter with the celebrated uniqueness th ...

336 11. Simple C* -Algebras isometry with support el,1 and range nl,1; moreover, it can be shown that llu1 - el,1ll2 is small. N ...

11.6. References 337 11.6. References All the results in the first three sections come from [18] and [19]. Theorem 11.4.6 comes ...

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A pproxirnation Properties for Groups Chapter 12 This chapter contains a number of other approximation properties, as well as a ...

340 12. Approximation Properties for Groups Note that a unitary representation 7r has almost I'-invariant vectors if and only if ...

12.1. Kazhdan's property (T) 341 Proposition 12.1.6. Let r be a group, A <1 r be a normal subgroup and ( E, 11,) be a K azhda ...

342 12. Approximation Properties for Groups ( 4) ~ (5) is a tautology. (4) =?-(2): We again prove the contrapositive. Let E1 c E ...

12.1. Kazhdan's property (T) 343 that if a convex combination e = I:~=l akek of unit vectors (ek) in 'H is close to a unit vecto ...

344 12. Approximation Properties for Groups 1 = ~ L v(s)ll~ - 7r(s)~ll^2 sES ::; max 117r(s)~ - ~112 sES for every~ EH. This imp ...