1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

D. Positive Definite Functions 465 Proof. First suppose that llmkllcb ::S 1. By Wittstock's factorization theo- rem for complete ...

466 D. Positive Defi.nite Functions Let r be a discrete group and cp: r ---+ CC be a function. Abusing notation, we denote by m' ...

D. Positive De:fi.nite Functions 467 Proof. We first prove the assertion for a E CHr) 0lIB(£^2 ) and f E (C~(r) 0 JIB(.€^2 ))*. ...

468 D. Positive Definite Functions Cocycles of unitary representations. Let (n, 11) be a unitary represen- tation of r. Then, a ...

D. Positive Deflnite Functions 469 definite. In particular, for any 1-cocycle b on a group r and I > 0, the function <p~ o ...

470 D. Positive Definite Functions Prove that there exists a unique vector(, called the circumcenter of V, such that V is contai ...

Appendix E Groups and Graphs Definition of graphs. Definition E.1. A graph X consists of a vertex set V = V(X) and an edge set E ...

472 E. Groups and Graphs Proof. Form, n with m ~ n, we define h(m, n) = d(y, xm) + d(xm, xn) - d(y, xn)· Note that h decreases ( ...

E. Groups and Graphs 473 We note that srxs-^1 = rs.x. The stabilizer of an edge (x, y) EE is rCx,y) = rx nrY. Every group acts b ...

474 E. Groups and Graphs :::; 21 ~ 1 2: llf(x) - f(v)ll^2 , (x,y)EE where m = lvl-^1 l:xEV v(x)f(x) EH is the mean off. Proof. T ...

E. Groups and Graphs 475 is the combinatorial Laplacian of Xn (modulo identification of £^2 (r n) and L^2 (rn)). Let £^2 (rn)^0 ...

476 E. Groups and Graphs (1) r* contains r as a subgroup and r* is generated by rand a distin- guished element z of infinite ord ...

E. Groups and Graphs 477 i.e., w = td^1 std^2 s .. · stdn for some di, ... , dn E {1, 2}. It follows that w(O, oo) c (-oo, 0) a ...

478 E. Groups and Graphs conclude that kl!_.1! 111~1 L A(s)llJIB(£2(rp,q)) = 1. sErpk(S) That is, the Baumslag-Solitar group r p ...

Bimodules over von Neumann Algebras Appendix F In this appendix we study bimodules over von Neumann algebras, restrict- ing our ...

480 F. Bimodules over von Neumann Algebras Example F.2. We recall that the formula llxll2 = T(x*x)^112 defines a Hilbertian norm ...

F. Bimodules over von Neumann Algebras 481 Theorem F. 7. Let M be a finite van Neumann algebra. Then, there exists a unique cond ...

482 F. Bimodules over van Neumann Algebras the right M--action, we have PE Iffi(R^2 ) ® M, where M acts on L^2 (M) from the left ...

F. Bimodules over von Neumann Algebras 483 the identity bimodule L^2 (M) as a space of square integrable closed operators affili ...

484 F. Bimodules over von Neumann Algebras We briefly review Jones's basic construction .. Let A c M be a von Neumann subalgebra ...