1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

Part 3 Applications ...

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Chapter 15 Classification of Group von Neumann Algebras In recent years, C* -tensor product theory and amenable actions of group ...

408 15. Group von Neumann Algebras Definition 15.1.2. We say the group r is bi-exact relative to g if it is exact and there exis ...

15.1. Noninjective relative commutants 409 (.\ ® .\)(s)(l ® p)(t) (cf. Fell's absorption principle). Letμ: r --t Prob(I') be a m ...

410 15. Group von Neumann Algebras We primarily consider p in the case where P = B' n pMp for a pro- jection p E JJ1 and a diffu ...

15.2. On bi-exactness 411 15.2. On bi-exactness Definition 15.2.1. Let r be a group and g be a family of subgroups of I'. For f ...

412 15. Group von Neumann Algebras 'Pl: X ---+ b.gr which is r x r-equivariant, where the right action of r on b.gr is trivial. ...

15.2. On bi-exactness 413 Lemma 15.2.6. Let r be an exact group, Y c r be an amenable subgroup and g be a family of subgroups of ...

414 15. Group von Neumann Algebras Exercises Exercise 15.2.1. Let X be a compact r-space and Prob(X) be the state space of C(X) ...

15.3. Examples 415 Theorem 15.3.4. Let ri, ... , r n be hyperbolic groups and Ni, ... , Nm be noninjective II1 -factors. If ther ...

416 15. Group von Neumann Algebras Lemma 15.3.7. For g ={A}, one has lim I supp(y)I = 0 yt->oo/Q ll((yt)ll. Proof. We first c ...

15.3. Examples 417 This yields the second assertion. For the third assertion, we observe that ((ytx) (p )-((yt) (p) is nonzero o ...

418 15. Group van Neumann Algebras Amalgamated free products. Proposition 15.3.12. Let I' = I'i *A I'2 be an amalgamated free pr ...

15.3. Examples 419 of A, we can apply Lemma 5.2.8 and deduce that snrio -+ z. Hence we obtain the contradiction c: ~ lim n lf(sn ...

420 15. Group von Neumann Algebras 15.4. References This chapter grew out of [133, 137, 135], which contain some additional resu ...

Herrera's Approximation Problem Chapter 16 Paul Halmos introduced the notion of a quasidiagonal operator (on a separa- ble Hilbe ...

422 16. Herrera's Approximation Problem Remark 16.1.3. Note that Proposition 7.2.3 shows T E QV(H) if and only if {T} is a quasi ...

16.2. C'-preliminaries 423 16.2. C*-preliminaries To solve Herrero's problem, we will need a technical generalization of Dadar- ...

424 16. Herrera's Approximation Problem To ease notation, let B = 7r(A). We have a short exact sequence 0 ---+ lK ( 1i) ---+ A + ...