Deformation quantization and the spaces of type S

We study properties of the Gelfand-Shilov spaces of type S in the context of deformation quantization, considering them as topological algebras under the twisted convolution and under star products induced by the Weyl correspondence and other quantization maps. We define their associated algebras of multipliers and prove the inclusion relations between these multiplier algebras and the duals of the spaces of ordinary multipliers and convolutors. We also consider Hilbert space representations of the multiplier algebras and describe their properties. The obtained results extend the Weyl symbol calculus beyond the traditional framework of tempered distributions. A key role in our analysis is played by a theorem characterizing those spaces of type S for which the function exp(iQ(x)) is a pointwise multiplier for any real quadratic form Q.