Twisted pseudo-differential operator on type I locally compact groups

Abstract

Let $\mathsf{G}$ be a locally compact group satisfying some technical requirements and $\widehat{\sf{G}}$ its unitary dual. Using the theory of twisted crossed product $C^{*}$-algebras, we develop a twisted global quantization for symbols defined on $\mathsf{G}\times\widehat{\sf{G}}$ and taking operator values. The emphasis is on the representation-theoretic aspect. For nilpotent Lie groups, the connection is made with a scalar quantization of the cotangent bundle $T^{*}(\mathsf{G})$ and with a Quantum Mechanical theory of observables in the presence of variable magnetic fields.