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# On isometries satisfying deformed commutation relations

### Abstract

We consider an $C^*$-algebra $\mathcal{E}_{1,n}^q$, $q\le 1$, generated by isometries satisfying $q$-deformed commutation relations. For the case $|q|<1$, we prove that $\mathcal E_{1,n}^q \simeq\mathcal E_{1,n}^0=\mathcal O_{n+1}^0$. For $|q|=1$ we show that $\mathcal E_{1,n}^q$ is nuclear and prove that its Fock representation is faithul. In this case we also discuss the representation theory, in particular construct a commutative model for representations.

Key words: Cuntz-Toeplitz algebra, $q$-deformation, Fock representation, commutative model.

### Article Information

 Title On isometries satisfying deformed commutation relations Source Methods Funct. Anal. Topology, Vol. 25 (2019), no. 2, 152-160 MathSciNet MR3978679 Milestones Received 02/04/2019 Copyright The Author(s) 2019 (CC BY-SA)

### Authors Information

Olha Ostrovska
National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”

Roman Yakymiv
Faculty of Computer Sciences and Cybernetics, Kiev National Taras Shevchenko University

### Citation Example

Olha Ostrovska and Roman Yakymiv, On isometries satisfying deformed commutation relations, Methods Funct. Anal. Topology 25 (2019), no. 2, 152-160.

### BibTex

@article {MFAT1170,
AUTHOR = {Olha Ostrovska and Roman Yakymiv},
TITLE = {On isometries satisfying deformed commutation relations},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {25},
YEAR = {2019},
NUMBER = {2},
PAGES = {152-160},
ISSN = {1029-3531},
MRNUMBER = {MR3978679},
URL = {http://mfat.imath.kiev.ua/article/?id=1170},
}

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