Pseudo-differential operators, Wigner transform and Weyl systems on type\n I locally compact groups

2015/06/18 by Marius Măntoiu, Michael Ruzhansky, Mantoiu, Marius +1 · 2 citations
Mathematics · #22D25 #47G30 #Advanced Algebra and Geometry #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Operator Algebras (math.OA) #Primary 46L65 #Representation Theory (math.RT) #Secondary 22D10 #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1506.05854

openalex publication_date 2015/06/18 · openalex created_date 2023/01/10 · openalex updated_date 2026/07/28

Abstract

Let G be a unimodular type I second countable locally compact group and\n G its unitary dual. We introduce and study a global pseudo-differential\ncalculus for operator-valued symbols defined on G\× G, and its\nrelations to suitably defined Wigner transforms and Weyl systems. We also\nunveil its connections with crossed products C^*-algebras associated to\ncertain C^*-dynamical systems, and apply it to the spectral analysis of\ncovariant families of operators. Applications are given to nilpotent Lie\ngroups, in which case we relate quantizations with operator-valued and\nscalar-valued symbols.\n

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• The fibre operators in the Bloch-Floquet decomposition of periodic magnetic pseudo-differential operators
• The Hudson theorem in LCA groups and infinite quantum spin systems

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