0 A ug 1 99 4 COMPLETELY POSITIVE MAPS ON COXETER GROUPS , DEFORMED COMMUTATION RELATIONS , AND OPERATOR SPACES

In this article we prove that quasi-multiplicative (with respect to the usual length function) mappings on the permutation group Sn (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 did ∗ j − ∑ r,s t ir jsd ∗ rds = δij1, where the matrix t js 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 Gi = di + d ∗ i , are typically not injective. Typeset by AMS-TEX 2MAREK BOŻEJKO AND ROLAND SPEICHER 1 INSTYTUT MATEMATYCZNY UNIWERSYTET WROC