Twisted pseudo-differential operator on type I locally compact groups. (English) Zbl 1368.81115

Bustos, H.; Măntoiu, M.

Ill. J. Math. 60, No. 2, 365-390 (2016).

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

MSC:

81S30	Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
47G30	Pseudodifferential operators
22D10	Unitary representations of locally compact groups
22D25	\(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
81S10	Geometry and quantization, symplectic methods