On $q$-tensor products of Cuntz algebras

Abstract

We consider the $C^{\ast }$-algebra ${\mathcal{E}}_{n,m}^q$, which is a $q$-twist of two Cuntz-Toeplitz algebras. For the case $|q|<1$, we give an explicit formula which untwists the $q$-deformation showing that the isomorphism class of ${\mathcal{E}}_{n,m}^q$ does not depend on $q$. For the case $|q|=1$, we give an explicit description of all ideals in ${\mathcal{E}}_{n,m}^q$. In particular, we show that ${\mathcal{E}}_{n,m}^q$ contains a unique largest ideal ${\mathcal{M}}_q$. We identify ${\mathcal{E}}_{n,m}^q/{\mathcal{M}}_q$ with the Rieffel deformation of ${\mathcal{O}}_n\otimes {\mathcal{O}}_m$ and use a K-theoretical argument to show that the isomorphism class does not depend on $q$. The latter result holds true in a more general setting of multiparameter deformations.

Accepted to publication in International Journal of Mathematics. arXiv admin note: substantial text overlap with arXiv:1812.08530