A class of representations of $C^{\ast }$-algebra generated by $q_{ij}$-commuting isometries

Abstract

For $C^{\ast }$-algebra generated by a finite family of isometries $s_j$, $j=1,\dots ,d$ satisfying $q_{ij}$-commutation relations \[ s_j^* s_j = I, \quad s_j^* s_k = q_{ij}s_ks_j^*, \qquad q_{ij} = \bar q_{ji}, |q_{ij}|<1, \ 1\le i \ne j \le d, \] we construct an infinite family of unitarily non-equivalent irreducible representations. These representations are deformations of the corresponding class of representations of the Cuntz algebra ${\mathcal{O}}_d$.