Algebras of distributions suitable for phase-space quantum mechanics. II. Topologies on the Moyal algebra
Joseph C. Várilly, José M. Gracia-Bondía
TL;DR
The paper establishes three equivalent topologies on the Moyal $*$-algebra $\mathcal{M}$, arising from operator-topologies on configuration/phase-space test spaces, operator-topologies from filtrations of Banach spaces, and filtrations of tempered distributions, and proves their equivalence. It constructs left/right operator-algebra realizations $\mathcal{M}_L$ and $\mathcal{M}_R$, links them to $\mathcal{L}_b(\mathcal{S}_1)$ via the Wigner transform, and uses the kernel theorem to realize $\mathcal{M}$ as an intersection of topologies; it also develops two filtrations $\mathcal{G}_{s,t}$ of $\mathcal{S}'_2$ to characterize membership in $\mathcal{M}$ and to derive criteria for phase-space functions to correspond to trace-class or bounded operators under the Weyl correspondence. These results yield concrete inclusion relations and norm estimates that connect phase-space distributions with operator classes, extending the mathematical foundations of phase-space quantum mechanics and informing Schrödinger dynamics within the Moyal framework. The work sets the stage for spectral theory and semigroup approaches in $\mathcal{M}$ and outlines future directions (e.g., the dual space representation) to broaden applicability.
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.