Algebras of distributions suitable for phase-space quantum mechanics. II. Topologies on the Moyal algebra

TL;DR

This paper investigates different topologies on the Moyal algebra in phase-space quantum mechanics and proves their equivalence, providing new conditions for phase-space functions to correspond to trace-class operators.

Contribution

It establishes the equivalence of three topological frameworks for the Moyal algebra and derives new criteria for trace-class operator correspondence.

Findings

Proved the equivalence of three topologies on the Moyal algebra.

Provided new sufficient conditions for phase-space functions to be trace-class.

Linked topological structures with operator correspondence in quantum mechanics.

Abstract

The topology of the Moyal $*$-algebra may be defined in three ways: the algebra may be regarded as an operator algebra over the space of smooth declining functions either on the configuration space or on the phase space itself; or one may construct the $*$-algebra via a filtration of Hilbert spaces (or other Banach spaces) of distributions. We prove the equivalence of the three topologies thereby obtained. As a consequence, by filtrating the space of tempered distributions by Banach subspaces, we give new sufficient conditions for a phase-space function to correspond to a trace-class operator via the Weyl correspondence rule.