Journal of Operator Theory

Volume 43, Issue 1, Winter 2000 pp. 35-42.

Affine Temperley-Lieb algebras

Authors: Sante Gnerre
Author institution: Department of Mathematics, University of California at Berkeley, Berkeley, CA 94720, USA

Summary: Given a finite index inclusion of factors

$N\stackrel{E}{\subset}M$ , it is possible to define a representation of the Affine Temperley-Lieb algebra on the relative commutant

$N'\cap M_n$ , via the left and right multiplication by the

$e_i$ 's, and the conditional expectations

$E_n$ and

$\lambda E^{-1}$ , where

$\lambda = {\rm Ind}(E)^{-1}$ . This result generalizes a theorem by Vaughan Jones (see~[10]), where he introduces the definition of the Affine Temperley-Lieb algebra, and proves that a representation of it exists on the Hilbert spaces

$N'\cap M_n$ constructed from a finite index and extremal inclusion of

${\rm II}_1$ factors

$N\subset M$ .

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