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. The equivalence of the three topologies thereby obtained is proved. As a consequence, by filtrating the space of tempered distributions by Banach subspaces, new sufficient conditions are given for a phase‐space function to correspond to a trace‐class operator via the Weyl correspondence rule.
Algebras of distributions suitable for phase‐space quantum mechanics. II. Topologies on the Moyal algebra
Joseph C. Várilly, José M. Gracia‐Bondía; Algebras of distributions suitable for phase‐space quantum mechanics. II. Topologies on the Moyal algebra. J. Math. Phys. 1 April 1988; 29 (4): 880–887. https://doi.org/10.1063/1.527984
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