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

## TL;DR

This paper proves the faithfulness of the Fock representation for a nuclear $C^*$-algebra generated by $q_{ij}$-commuting isometries and describes its ideal structure related to compact operators.

## Contribution

It establishes the faithfulness of the Fock representation for a class of $C^*$-algebras with $q_{ij}$-commuting isometries and characterizes an ideal isomorphic to compact operators.

## Key findings

- The $C^*$-algebra $Isom_{q_{ij}}$ is nuclear.
- The Fock representation of $Isom_{q_{ij}}$ is faithful.
- An ideal in $Isom_{q_{ij}}$ is isomorphic to the algebra of compact operators.

## 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.