Star product algebras of test functions

Abstract

We prove that the Gelfand-Shilov spaces $S_{\alpha }^{\beta }$ 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 $S_{\alpha }^{\beta }$ if and only if $\beta <1/2$.

AMS-LaTeX, 12 pages, no figures; typos corrected, minor changes to agree with published version

References in corpus (4)

• Algebras of distributions suitable for phase-space quantum mechanics. I
• On "full" twisted Poincare' symmetry and QFT on Moyal-Weyl spaces
• PCT, spin and statistics, and analytic wave front set
• Axiomatic formulations of nonlocal and noncommutative field theories

Cited by in corpus (13)

• Higher Spin Fields in Siegel Space, Currents and Theta Functions
• Noncommutativity and theta-locality
• Moyal multiplier algebras of the test function spaces of type S
• Duality covariant quantum field theory on noncommutative Minkowski space
• Twisted convolution and Moyal star product of generalized functions
• Divergences in quantum field theory on the noncommutative two-dimensional Minkowski space with Grosse-Wulkenhaar potential
• UV/IR duality in noncommutative quantum field theory
• Noncommutative deformations of quantum field theories, locality, and causality
• Wedge locality and asymptotic commutativity
• Failure of microcausality in noncommutative field theories
• Propagators and Matrix Basis on Noncommutative Minkowski Space
• Haag's Theorem in Noncommutative Quantum Field Theory
• Towards an Axiomatic Formulation of Noncommutative Quantum Field Theory. II