Spaces of Type $S$ as Topological Algebras under Twisted Convolution and Star Product

The properties of the generalized Gelfand–Shilov spaces $S_{b_n}^{a_k}$ are studied from the viewpoint of deformation quantization. We specify the conditions on the defining sequences $(a_k)$ and $(b_n)$ under which $S_{b_n}^{a_k}$ is an algebra with respect to the twisted convolution and, as a consequence, its Fourier transformed space $S^{b_n}_{a_k}$ is an algebra with respect to the Moyal star product. We also consider a general family of translation-invariant star products. We define and characterize the corresponding algebras of multipliers and prove the basic inclusion relations between these algebras and the duals of the spaces of ordinary pointwise and convolution multipliers. Analogous relations are proved for the projective counterpart of the Gelfand–Shilov spaces. 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 $Q$.