Deformation quantization of Heisenberg manifolds. (English) Zbl 0679.46055

The author studies the deformation quantization for a smooth manifold equipped with a Poisson bracket in the \(C^*\)-algebra framework, as examples of non-commutative manifolds. The much studied non-commutative tori is shown to be an example of the strict form of deformation quantization in author’s formulation.
As main result of the paper, invariant Poisson structures of Heisenberg manifolds under the action of the Heisenberg Lie group are determined and found to fall into two types, leading to different structure of the resulting algebras for deformation quantization. Generalizations to more general situations are studied in detail. Finally, the SO(3) action of the 2-sphere is shown to be rigid, all deformation being commutative.
Reviewer: H.Araki


46L60 Applications of selfadjoint operator algebras to physics
81S10 Geometry and quantization, symplectic methods
46L05 General theory of \(C^*\)-algebras
46L55 Noncommutative dynamical systems
46L80 \(K\)-theory and operator algebras (including cyclic theory)
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