Pseudo-Differential Operators, Wigner Transform and Weyl Systems on Type I Locally Compact Groups

Let $\G$ be a unimodular type I second countable locally compact group and let $\wG$ be its unitary dual. We introduce and study a global pseudo-differential calculus for operator-valued symbols defined on $\G\times\wG$, 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.

Keywords and Phrases: locally compact group, nilpotent Lie group, noncommutative Plancherel theorem, pseudo-differential operator, C^*-algebra, dynamical system.

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