  Using the notions of frame transform and of square integrable projective
representation of a locally compact group $G$, we introduce a class of
isometries (tight frame transforms) from the space of Hilbert-Schmidt operators
in the carrier Hilbert space of the representation into the space of square
integrable functions on the direct product group $G\times G$. These transforms
have remarkable properties. In particular, their ranges are reproducing kernel
Hilbert spaces endowed with a suitable 'star product' which mimics, at the
level of functions, the original product of operators. A 'phase space
formulation' of quantum mechanics relying on the frame transforms introduced in
the present paper, and the link of these maps with both the Wigner transform
and the wavelet transform are discussed.
