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.

1.
J. M.
Gracia‐Bondía
and
J. C.
Várilly
,
J. Math. Phys.
29
,
869
(
1988
).
2.
J. P. Amiet and P. Huguenin, Mécaniques classique et quantique dans l ’espace de phase (Universitedé Neuchâtel, Neuchâtel, 1981).
3.
R. F. V.
Anderson
,
J. Funct. Anal.
9
,
423
(
1972
).
4.
J. C. T.
Pool
,
J. Math. Phys.
7
,
66
(
1966
).
5.
E. P.
Wigner
,
Phys. Ref.
40
,
749
(
1932
).
6.
J. C.
Várilly
and
J. M.
Gracia‐Bondía
,
J. Math. Phys.
28
,
2390
(
1987
).
7.
R.
Cressman
,
J. Funct. Anal.
22
,
405
(
1976
).
8.
J. Horváth, Topological Vector Spaces and Distributions I (Addison‐Wesley, Reading, MA, 1966).
9.
F. Treves, Topological Vector Spaces, Distributions and Kernels (Academic, New York, 1967).
10.
L.
Hörmander
,
Commun. Pure Appl. Math.
32
,
359
(
1979
).
11.
A. Grothendieck, Produits tensoriels topologiques et espaces nuckaires (Am. Math. Soc., Providence, RI, 1955).
12.
D. Vogt, in Functional Analysis: Surveys and Recent Results III (North‐Holland, Amsterdam, 1984), p. 349.
13.
F.
Bayen
and
J. M.
Maillard
,
Lett. Math. Phys.
6
,
491
(
1982
).
14.
P.
Huguenin
,
Lett. Math. Phys.
2
,
321
(
1978
).
15.
J. M.
Gracia‐Bondía
,
Phys. Rev. A
30
,
691
(
1984
).
16.
H. H. Schaefer, Topological Vector Spaces (Macmillan, New York, 1966).
17.
J.
Unterberger
,
C. R. Acad. Sci. Ser.
290A
,
1053
(
1984
).
18.
S. J. L.
van Eijndhoven
and
J.
de Graff
,
J. Funct. Anal.
63
,
74
(
1985
);
Lecture Notes in Mathematics, Vol. 1162 (Springer, Heidelberg, 1985).
19.
J. M. Gracia‐Bondía and J. C. Várilly, “The dual space of the algebra Lb (S),” to appear.
20.
J.‐B.
Kammerer
,
J. Math. Phys.
27
,
529
(
1986
).
21.
G. Köthe, Topological Vector Spaces II (Springer, Heidelberg, 1979).
22.
I.
Daubechies
,
J. Math. Phys.
24
,
1453
(
1983
).
23.
I.
Daubechies
,
Commun. Math. Phys.
75
,
229
(
1980
).
24.
J. M. Gracia‐Bondía, M. Gadella, L. M. Nieto‐Calzada, and J. C. Várilly, “Feynman integrals, quadratic Hanultonians, and Poincaré’s generating function in the phase‐space approach to Quantum Mechanics,” to appear.
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