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.

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Title

Representations of relations with orthogonality condition and their deformations

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},
}