Deformation quantization and operator algebras. (English) Zbl 0715.46038

Operator theory, operator algebras and applications, Proc. Summer Res. Inst., Durham/NH (USA) 1988, Proc. Symp. Pure Math. 51, Pt. 1, 411-423 (1990).
[For the entire collection see Zbl 0699.00027.]
A definition of a strict deformation quantization of a manifold with Poison bracket and its invariance relative to a group of diffeomorphisms of the manifold preserving the Poisson structure is proposed and the following examples are given,
(1) The Moyal product,
(2) noncommutative tori,
(3) the generalization of (2) to its product with a Lie group H, divided by a cocompact subgroup of H with an appropriate action on the tori,
(4) semidirect products by \({\mathbb{R}}^ d,\)
(5) semidirect products by \(T^ d,\)
(6) Heisenberg manifolds,
(7) nilpotent Lie algebras,
(8) general Lie algebras, and
(9) the sphere.
Reviewer: H.Araki


46L87 Noncommutative differential geometry
46L60 Applications of selfadjoint operator algebras to physics
81S10 Geometry and quantization, symplectic methods


Zbl 0699.00027