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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