In this paper we study the Fock representation of a certain * -algebra which appears naturally in the framework of quantum group theory. It is also a generalization of the twisted CCR-algebra introduced by W. Pusz and S. Woronowicz. We prove that the Fock representation is a faithful irreducible representation of the algebra by bounded operators in a Hilbert space, and, moreover, it is the only (up to unitary equivalence) representation possessing these properties.
We introduce and study, in the framework of a theory of quantum Cartan domains, a q-analogue of the Berezin transform on the unit ball. We construct q-analogues of weighted Bergman spaces, Toeplitz operators and covariant symbol calculus. In studying the analytical properties of the Berezin transform we introduce also the q-analogue of the SU (n, 1)-invariant Laplace operator (the Laplace-Beltrami operator) and present related results on harmonic analysis on the quantum ball.. These are applied to obtain an analogue of one result by A. Unterberger and H. Upmeier. An explicit asymptotic formula expressing the q-Berezin transform via the q-Laplace-Beltrami operator is also derived. At the end of the paper, we give an application of our results to basic hypergeometric q-orthogonal polynomials.
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