  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.
