# 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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