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Algebras of symbols associated with the Weyl calculus for Lie group representations. (English) Zbl 1256.47031

This paper is a continuation of I. Beltiţă and D. Beltiţă [J. Funct. Anal. 260, No. 7, 1944–1968 (2011; Zbl 1217.22007)]. The approach to the Weyl calculus for representations of infinite-dimensional Lie groups is developed by establishing continuity properties of the Moyal product for symbols belonging to various modulation spaces. Several interesting results are presented. For instance, it is shown that the modulation space of symbols \(M^{\infty,1}\) is an associative Banach algebra and that the corresponding operators are bounded. These results are applied to two classes of representations: to the unitary irreducible representations of nilpotent Lie groups and to the natural representations of the semidirect product groups that govern the magnetic Weyl calculus.

MSC:

47G30 Pseudodifferential operators
22E25 Nilpotent and solvable Lie groups
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
47G10 Integral operators

Citations:

Zbl 1217.22007
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References:

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