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# Representations of relations with orthogonality condition and their deformations

### 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.

### 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},
}