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

TL;DR

This paper constructs a variety of irreducible representations for a $C^*$-algebra generated by isometries with $q_{ij}$-commutation relations, extending the representation theory of Cuntz algebras through deformation techniques.

Contribution

It introduces an infinite family of unitarily non-equivalent irreducible representations for $q_{ij}$-commuting isometries, generalizing Cuntz algebra representations.

Findings

Constructed unitarily non-equivalent irreducible representations

Representations are deformations of Cuntz algebra representations

Extended understanding of $C^*$-algebras with $q_{ij}$-relations

Abstract

For $C^*$-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 $\mathcal O_d$.