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

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We have to find some X' ~So). E Let Matfin(SO) with IIX'II = IIYII; we may assume that Y is irreducible. Now there exists a decomposition Y=Y 1+"'+Y k with Yj symmetric matrices in Matl(SO) for j=l,,···,k. Set Let e E IR~ be a Perron-Frobenius vector for Y and define f = Then X' f = IIYllf, so that f IIX'II = IIXII. l. is a Perron-Frobenius vector for X'. 1 of [Hofj. Now we state again the main result of this section, due to G. Skandalis. 4. For any S cIRone has ~S) = ~~S». 5. Norms of graphs and integral matrices Proof.

For any S cIRone has ~S) = ~~S». 5. Norms of graphs and integral matrices Proof. It is obvious from considering I-by-l matrices that A'(8) C A'(A'(8)). To show the theorem, set T = A'(8) and let X E Mat m,n(T)j one has to find some Z E Mat fin (8) with IIZII = IIXII· For any pair (i,j) with 1 s i ~ m and 1 ~ j s n, we choose an integer Pi,j ~ 1 and a symmetric matrix Y.. of size p.. over 8 with IIY··II = X... Let p be the product of IJ IJ the p. ' s. Write Z.. for 1 ® l,j l,j IJ ••• ® y .. ® I,j ••• ® the (i,j) th place.

We end this section with the finite dimensional case of the coupling constant theorem of Murray and von Neumann (see Theorem X in [MvN Ij, and also Theorem X in [MvN IV]). This is a digression motivated by the importance of the theorem for II1-factors (see Chapter 3). Consider a factor M= En~(V), where V is of dimension p, over 1<, and a representation 'If of M in a vector space W. Assume that 'If is of multiplicity d, so that diIl1«W) = dp" and view 'If as an inclusion M c F with F = Endl«W).

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