arXiv subjects

Danylo Proskurin

Publications and source records attributed to Danylo Proskurin.

3 recordsLinked to original sources

arXiv2021

On $q$-tensor products of Cuntz algebras

We consider the $C^*$-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.

math.OA↗

arXiv2021

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

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 $\mathcal O_d$.

math.OA↗

arXiv2019

On $q$-tensor product of Cuntz algebras

We consider $C^*$-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, thus showing that the isomorphism class of $\mathcal{E}_{n,m}^q$ does not depend of $q$. For the case $|q|=1$ we give an explicit description of all ideals in $\mathcal{E}_{n,m}^q$. In particular $\mathcal{E}_{n,m}^q$ contains unique largest ideal $\mathcal{M}_q$. Then 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 $q$.

math.OA↗