Completely Positive Maps on Coxeter Groups, Deformed Commutation Relations, and Operator Spaces

TL;DR

This paper proves that certain mappings on Coxeter groups, defined by self-adjoint contractions satisfying braid relations, are completely positive and explores their connection to deformed commutation relations and operator spaces.

Contribution

It establishes the complete positivity of quasi-multiplicative maps on Coxeter groups linked to braid relations and connects these to deformed commutation relations and operator space theory.

Findings

Quasi-multiplicative maps on Coxeter groups are completely positive.

Deformed commutation relations relate to operator spaces.

Von Neumann algebras generated are typically non-injective.

Abstract

In this article we prove that quasi-multiplicative (with respect to the usual length function) mappings on the permutation group $\SSn$ (or, more generally, on arbitrary amenable Coxeter groups), determined by self-adjoint contractions fulfilling the braid or Yang-Baxter 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 self-adjoint 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.