# Download Coxeter Graphs and Towers of Algebras by F. M. Goodman, P. de la Harpe, V. F. R. Jones PDF

By F. M. Goodman, P. de la Harpe, V. F. R. Jones

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Additional resources for Coxeter Graphs and Towers of Algebras

Sample text

Choose Z E CF(M) with y = qzq and compute xy = q(qxq)zq = qzqxq = yx. It follows that qCF(M)q c CqFq(qMq). Consider then s t E CF(M). As tq = qt = qsq E CqFq(qCF(M)q) and one has st = sqqt = sqtq = qtqs = tqqs = ts. 2. 2 in qFq, one has Assume now that q E CF(M). (a) The linear map tp from M to qMq which sends x to qxq is a morphism of algebras and its image contains q = qlq. As M is simple, tp is an isomorphism. 2. Commutant and bicommutant 41 using CF(CF(M» = M. 2 in qFq to obtain the conclusion.

A~ Then OOy and AM are N (x,y) ... [H HI N = K (9 Mat2(K) and by (x,y) ... OOOy [~ ~ ~]. included and in pseudo-equiVal:~:O (2 1) = Mat 5(K) by but M and M are not isomorphic. quivalent to the second inclusion matrix (1 2). The next proposition is a special case of a statement which appears in exercise 17. 5. Consider two multi-matrix subalgebras M,N of a factor F with 1 ENe M c F. The inclusion matrix for CF(M) C CF(N) is the transpose of the inclusion matrix for N C M. Proof. 2). In general, write M and denote by X..

E.. , and -'-' I,J I,J -'-' I,J I,J -'-' I,J I,J = ~ x.. a ® e.. , it follows that ~ x.. ® e. lies in the commutant of A ® 1 if and only -'-' I,J I,J -'-' 1,J I,J if x· . E A' for all (i,j). I,J (b) If x = ~ x· . · commutes with A ®.. Matd(I<), then in particular it -'-' I,J l,j -I'. i ,j commutes with the matrix units 1 ® e for all (/t,v), Proof. (a) Since /t,V ~ x . ® e . = (1 ® e )x -'-' v,J /t,J /t,v j = x(l ®e/t,v) = ~ x. ® e· . -'-' 1,/t I,V i It follows that x . 1,/1 = 0 for i t- It, and Xv)v = x/t,/t .