### Abstract

Irreducible representations of $*$-algebras $A_q$ generated by relations of the form $a_i^*a_i+a_ia_i^*=1$, $i=1,2$, $a_1^*a_2=qa_2a_1^*$, where $q\in (0,1)$ is fixed, are classified up to the unitary equivalence. The case $q=0$ is considered separately. It is shown that the $C^*$-algebras $\mathcal{A}_q^F$ and $\mathcal{A}_0^F$ generated by operators of Fock representations of $A_q$ and $A_0$ are isomorphic for any $q\in (0,1)$. A realisation of the universal $C^*$-algebra $\mathcal{A}_0$ generated by $A_0$ as an algebra of continuous operator-valued functions is given.

### Full Text

### Article Information

Title | Representations of relations with orthogonality condition and their deformations |

Source | Methods Funct. Anal. Topology, Vol. 18 (2012), no. 4, 373-386 |

MathSciNet |
MR3058463 |

zbMATH |
1289.46096 |

Copyright | The Author(s) 2012 (CC BY-SA) |

### Authors Information

*V. L. Ostrovskyi*

Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine

*D. P. Proskurin*

Department of Cybernetics, Kyiv National Taras Shevchenko University, 64 Volodymyrs'ka, Kyiv, 01033, Ukraine

*R. Y. Yakymiv*

National University of Life and Environmental Sciences, 15 Heroyiv Oborony, Kyiv, 03041, Ukraine

### Citation Example

V. L. Ostrovskyi, D. P. Proskurin, and R. Y. Yakymiv, *Representations of relations with orthogonality condition and their deformations*, Methods Funct. Anal. Topology **18**
(2012), no. 4, 373-386.

### BibTex

@article {MFAT649,
AUTHOR = {Ostrovskyi, V. L. and Proskurin, D. P. and Yakymiv, R. Y.},
TITLE = {Representations of relations with orthogonality condition and their deformations},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {18},
YEAR = {2012},
NUMBER = {4},
PAGES = {373-386},
ISSN = {1029-3531},
MRNUMBER = {MR3058463},
ZBLNUMBER = {1289.46096},
URL = {http://mfat.imath.kiev.ua/article/?id=649},
}

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