Star product algebras of test functions

We prove that the Gelfand–Shilov spaces $S^{\beta}_{\alpha}$ are topological algebras under the Moyal $\star$-product if and only if $\alpha\ge\beta$. These spaces of test functions can be used to construct a noncommutative field theory. The star product depends on the noncommutativity parameter continuously in their topology. We also prove that the series expansion of the Moyal product converges absolutely in $S^{\beta}_{\alpha}$ if and only if $\beta<1/2$.