On *-representations of polynomial algebras in quantum matrix spaces of rank 2

In this paper we study of *-representations for polynomial algebras on quantum matrix spaces. We deal with two special cases of the polynomial algebras, namely the algebra of polynomials on quantum complex matrices $\mathrm{Mat_2}$ and on quantum complex symmetric matrices $\mathrm{Mat_2^{sym}}$. For the second algebra we classify all irreducible *-representations by bounded operators in a Hilbert space (up to a unitary equivalence). Moreover, we present a construction of *-representations of the above algebras which enables to obtain the full list of *-representations (sometimes by passing to subrepresentations).