Star product algebras of test functions

Abstract

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 in quantum field theory on noncommutative spacetime. 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 $\beta<1/2$.