  We generalize some earlier results on a Berezin-Toeplitz type of quantization
on Hilbert spaces built over certain matrix domains. In the present, wider
setting, the theory could be applied to systems possessing several kinematic
and internal degrees of freedom. Our analysis leads to an identification of
those observables, in this general context, which admit a semi-classical limit
and those for which no such limit exists. It turns out that the latter class of
observables involve the internal degrees of freedom in an intrinsic way.
Mathematically, the theory, being a generalization of the standard
Berezin-Toeplitz quantization, points the way to applying such a quantization
technique to possibly non-commutative spaces, to the extent that points in
phase space are now replaced by $N\times N$ matrices.
