On Fréchet–Hilbert algebras

Măntoiu, Marius; Purice, Radu

doi:10.1007/s00013-014-0675-8

Cited by 5 publications

(5 citation statements)

References 22 publications

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“…Techniques from [27] could be applied to define and study large Moyal algebras of vectorvalued symbols corresponding to the spaces B D(G) and B D ′ (G) of operators.…”

Section: Restrictions and Extensions Of The Pseudo-differential Calculusmentioning

confidence: 99%

Pseudo-differential Operators Associated to General Type I Locally Compact Groups

Măntoiu¹,

Sandoval²

2019

Springer Proceedings in Mathematics &Amp; Statistics

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Let G be a unimodular type I second countable locally compact group and G its unitary dual. We introduce and study a global pseudo-differential calculus for operator-valued symbols defined on G × G , and its relations to suitably defined Wigner transforms and Weyl systems. We also unveil its connections with crossed products C * -algebras associated to certain C * -dynamical systems, and apply it to the spectral analysis of covariant families of operators. Applications are given to nilpotent Lie groups, in which case we relate quantizations with operator-valued and scalar-valued symbols. * 2010 Mathematics Subject Classification: Primary 46L65, 47G30, Secondary 22D10, 22D25.

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“…Techniques from [27] could be applied to define and study large Moyal algebras of vectorvalued symbols corresponding to the spaces B D(G) and B D ′ (G) of operators.…”

Section: Restrictions and Extensions Of The Pseudo-differential Calculusmentioning

confidence: 99%

Pseudo-differential Operators Associated to General Type I Locally Compact Groups

Măntoiu¹,

Sandoval²

2019

Springer Proceedings in Mathematics &Amp; Statistics

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“…By capital letters we denote distributions. We are going to skip the easy justifications and refer to [33] for an abstract approach.…”

Section: The Product Between a Multiplication Operator And A Left Conmentioning

confidence: 99%

Quantization and Coorbit Spaces for Nilpotent Groups

Măntoiu

2020

Applied and Numerical Harmonic Analysis

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We reconsider the quantization of symbols defined on the product between a nilpotent Lie algebra and its dual. To keep track of the non-commutative group background, the Lie algebra is endowed with the Baker-Campbell-Hausdorff product, making it via the exponential diffeomorphism a copy of its unique connected simply connected nilpotent Lie group. Using harmonic analysis tools, we emphasize the role of a Weyl system, of the associated Fourier-Wigner transformation and, at the level of symbols, of an important family of exponential functions. Such notions also serve to introduce a family of phase-space shifts. These are used to define and briefly study a new class of coorbit spaces of symbols and its relationship with coorbit spaces of vectors, defined via the Fourier-Wigner transform. *

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“…In general they are not comparable with B 2 L 2 (G) or B 2 L 2 (G) . Behind these operator algebras there are algebras of symbols with composition and involution given by All these may be recast in the framework of Fréchet-Hilbert algebras and their associated Moyal algebras [29], but we shall not do this here.…”

Section: D(g) and Extends To A Topological Isomorphismmentioning

confidence: 99%

“…All these may be recast in the framework of Fréchet-Hilbert algebras and their associated Moyal algebras [29], but we shall not do this here.…”

Section: Taking Into Account the Strong Dual One Gets A Gelfand Triplementioning

confidence: 99%

Global and concrete quantizations on general type I groups

Măntoiu

Sandoval

2019

Monatsh Math

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In recent papers and books, a global quantization has been developed for unimodular groups of type I . It involves operator-valued symbols defined on the product between the group G and its unitary dual G , composed of equivalence classes of irreducible representations. For compact or for graded Lie groups, this has already been developed into a powerful pseudo-differential calculus. In the present article we extend the formalism to arbitrary locally compact groups of type I , making use of the Fourier theory of non-unimodular second countable groups. The unitary dual and its Plancherel measure being quite abstract in general, we put into evidence situations in which concrete forms are available. Kirillov theory and parametrizations of large parts of G allow rewriting the basic formulae in a manageable form. Some examples of completely solvable groups are worked out. * 2010 Mathematics Subject Classification: Primary 46L65, 47G30, Secondary 22D10, 22D25.

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On Fréchet–Hilbert algebras

Cited by 5 publications

References 22 publications

Pseudo-differential Operators Associated to General Type I Locally Compact Groups

Pseudo-differential Operators Associated to General Type I Locally Compact Groups

Quantization and Coorbit Spaces for Nilpotent Groups

Global and concrete quantizations on general type I groups

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