Moyal multiplier algebras of the test function spaces of type S

TL;DR

This paper explores the Moyal multiplier algebras of Gel'fand-Shilov spaces of type S, extending the Moyal product to broader classes of functions and distributions, with implications for quantum field theory on noncommutative spacetime.

Contribution

It characterizes Moyal multipliers for Gel'fand-Shilov spaces and identifies the smallest Fourier-invariant space with maximal multiplier algebra, extending the Moyal product to ultradistributions and hyperfunctions.

Findings

Characterization of Moyal multipliers for S-type spaces.

Identification of the smallest Fourier-invariant space with maximal algebra.

Application to causality conditions in noncommutative quantum field theory.

Abstract

The Gel'fand-Shilov spaces of type S are considered as topological algebras with respect to the Moyal star product and their corresponding algebras of multipliers are defined and investigated. In contrast to the well-studied case of Schwartz's space S, these multipliers are allowed to have nonpolynomial growth or infinite order singularities. The Moyal multiplication is thereby extended to certain classes of ultradistributions, hyperfunctions, and analytic functionals. The main theorem of the paper characterizes those elements of the dual of a given test function space that are the Moyal multipliers of this space. The smallest nontrivial Fourier-invariant space in the scale of S-type spaces is shown to play a special role, because its corresponding Moyal multiplier algebra contains the largest algebra of functions for which the power series defining their star products are absolutely convergent. Furthermore, it contains analogous algebras associated with cone-shaped regions, which can be used to formulate a causality condition in quantum field theory on noncommutative space-time.