Completely Positive Maps on Coxeter Groups, Deformed Commutation Relations, and Operator Spaces
Abstract
In this article we prove that quasimultiplicative (with respect to the usual length function) mappings on the permutation group $\SSn$ (or, more generally, on arbitrary amenable Coxeter groups), determined by selfadjoint contractions fulfilling the braid or YangBaxter relations, are completely positive. We point out the connection of this result with the construction of a Fock representation of the deformed commutation relations $d_id_j^*\sum_{r,s} t_{js}^{ir} d_r^*d_s=\delta_{ij}\id$, where the matrix $t_{js}^{ir}$ is given by a selfadjoint contraction fulfilling the braid relation. Such deformed commutation relations give examples for operator spaces as considered by Effros, Ruan and Pisier. The corresponding von Neumann algebras, generated by $G_i=d_i+d_i^*$, are typically not injective.
 Publication:

arXiv eprints
 Pub Date:
 August 1994
 arXiv:
 arXiv:functan/9408002
 Bibcode:
 1994funct.an..8002B
 Keywords:

 Mathematics  Functional Analysis;
 Mathematics  Operator Algebras
 EPrint:
 26 pages, amstex 3.0