Representations of a \(q\)-analogue of the \(*\)-algebra \(\text{Pol(Mat}_{2,2})\) (Q2729426)

The \(*\)-algebra Pol(Mat\({}_{2,2}\))\({}_q\), a \(q\)-analogue of the algebra of polynomials on the space of complex \(2\times 2\) matrices, was introduced by \textit{D. L. Shklyarov, S. D. Sinel'shchikov} and \textit{L. L. Vaksman} [Czech. J. Phys. 50, 175-180 (2000; Zbl 0979.17008)]. The author describes all irreducible bounded Hilbert space \(*\)-representations of this algebra. She also finds those representations of Pol(Mat\({}_{2,2}\))\({}_q\) which are induced by representations of the \(*\)-algebra Pol\((S(\mathbb U))_q\), a \(q\)-analogue of the polynomial algebra on the Shilov boundary of the matrix unit ball. The technique is based on the study of a dynamical system arising on the spectrum of a commutative \(*\)-subalgebra of Pol(Mat\({}_{2,2}\))\({}_q\) [see \textit{V. Ostrovskyi} and \textit{Yu. Samoilenko}, Rev. Math. Math. Phys. 11, 1-261 (1999; Zbl 0947.46037)].