Closed star products and cyclic cohomology. (English) Zbl 0767.55005

From the authors’ abstract: We define the notion of a closed star product. A (generalized) star product (deformation of the associative product of functions on a symplectic manifold \(W\)) is closed iff integration over \(W\) is a trace on the deformed algebra. We show that for these products the cyclic cohomology replaces the Hochschild cohomology in the usual star products. We then define the character of a closed star product as the cohomology class (in the cyclic bicomplex) of a well- defined cocycle, and show that, in the case of pseudodifferential operators (standard ordering on the cotangent bundle to a compact Riemannian manifold), the character is defined and given by the Todd class, while in general it fails to satisfy the integrality condition.
Reviewer: P.J.Kahn (Ithaca)


55N35 Other homology theories in algebraic topology
19D55 \(K\)-theory and homology; cyclic homology and cohomology
19K56 Index theory
17B37 Quantum groups (quantized enveloping algebras) and related deformations
81S30 Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics
58J40 Pseudodifferential and Fourier integral operators on manifolds
57R20 Characteristic classes and numbers in differential topology
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