Journal of Operator Theory

Volume 80, Issue 1, Summer 2018 pp. 77-93.

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

Authors: Alexey Kuzmin (1) and Nikolay Pochekai (2)
Author institution: (1) Department of Mathematical Sciences, University of Gothenburg, Gothenburg, 41261, Sweden
(2) Faculty of Mathematics, National Research University Higher School of Economics, Moscow, 119048, Russia

Summary: We consider the $C^*$-algebra $\mathrm{Isom}_Q$, where $Q = (q_{ij})_{i,j=1}^n$ is a matrix of complex numbers. This algebra is generated by $n$ isometries $a_1, \ldots, a_n$ satisfying the relations $a_i^* a_j = q_{ij} a_j a_i^*$, $i \neq j$ with $\max |q_{ij}|$ less than $1$. This $C^*$-algebra is shown to be nuclear. We prove that the Fock representation of $\mathrm{Isom}_{Q}$ is faithful. Further we describe an ideal in $\mathrm{Isom}_{Q}$ which is isomorphic to the algebra of compact operators.

DOI: http://dx.doi.org/10.7900/jot.2017jun01.2172
Keywords: $C^∗$-algebra, Cuntz algebra, nuclear, $q$-deformation, Fock representation, operator algebras

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