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

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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 .

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