Faithfulness of the Fock representation of $C^*$-algebra generated by $q_{ij}$-commuting isometries

Abstract: We consider $C^*$-algebra $Isom_{q_{ij}}$ generated by $n$ isometries $a_1, \ldots, a_n$ satisfying the relations $a_i^* a_j = q_{ij} a_j a_i^*$ with $\max |q_{ij}| < 1$. This $C^*$-algebra is shown to be nuclear. We prove that the Fock representation of $Isom_{q_{ij}}$ is faithful. Further we describe an ideal in $Isom_{q_{ij}}$ which is isomorphic to the algebra of compact operators.