  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.
