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

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

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**Example text**

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. 5. Let M be a multi-matrix subalgebra of a factor F with 1 E M c F and let q EMU CF(M) be a nonzero idempotent. Then (a) qMq is a multi-matrix algebra. (b) CqFq(qMq) = qCF(M)q. Proof. C F (qMq)q. CF(M)q. M)q.. 4.

Introduction 31 MO =N( Ml =M( ... ( Mk ( Mk+1 ... are immediately determined by that of N (M. These towers have a rich' structure, the further study of which requires the introduction of traces. A trace on M is a linear map tr: M -+ I< such that tr(xy) = tr(yx) for x,y E M. It is faithful if the bilinear form (x,y)'" tr(xy) is non-degenerate. If I< is given as an extension of the real field IR, a trace tr is positive if tr(e) ~ 0 for any idempotent e in M. A trace tr on M = m ED i=l Mat (I<) is completely described by the row vector Pi S= (tr(fi))l<_i/m where f.

Generated by a multi-matrix pair N C M. We assume that there exists a Markov trace tr of some modulus f3 E 1<* on N c M. 1. 6. Let MO C M1 be a multi-matrix pair on which there exists a Markov trace tr of modulus {3. For each k ~ 1, let Mk and Ek be as above. Then (i) Mk is generated by M1 and El" ",Ek- 1. 3EiEjEi = Ei if EiEj = EjEi if IHI = 1, IH I ~ 2. 7. In the "generic case" (see below), it is remarkable that (ii) is a complete set of relations for the Ei' s. ). For any {3E 1<* and for any integer k ~ 1, the I<-algebra (with unit) A{3,k is defined by the presentation with generators (1'" "(k-1 and with relations: 2 (i = fi {3fi f/ i fj fj = fj = fjfj if Ii-j I = 1 if li-J'I ~ 2.