Twisted Pseudo-differential Operators on Type I Locally Compact Groups

Let $\text{G}$ be a locally compact group satisfying some technical requirements and $\text{wG}$ its unitary dual. Using the theory of twisted crossed product $C^{\ast }$-algebras, we develop a twisted global quantization for symbols defined on $\text{G}\times \text{wG}$ 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 $T^{\ast }(\text{G})$ and with a Quantum Mechanical theory of observables in the presence of variable magnetic fields.