Let μg and μp denote the Gaussian and Poisson measures on R, respectively. We show that there exists a unique measure μ̃g on C such that under the Segal-Bargmann transform Sμg the space L (R, μg) is isomorphic to the space HL(C, μ̃g) of analytic L-functions on C with respect to μ̃g . We also introduce the Segal-Bargmann transform Sμp for the Poisson measure μp and prove the corresponding result. A...

A symplectic variety X is a normal algebraic variety (defined over C) which admits an everywhere non-degenerate d-closed 2-form ω on the regular locus Xreg of X such that, for any resolution f : X̃ → X with f (Xreg) ∼= Xreg, the 2-form ω extends to a regular closed 2-form on X̃ (cf. [Be]). There is a natural Poisson structure { , } on X determined by ω. Then we can introduce the notion of a Poiss...

A symplectic variety X is a normal algebraic variety (defined over C) which admits an everywhere non-degenerate d-closed 2-form ω on the regular locus Xreg of X such that, for any resolution f : X̃ → X with f (Xreg) ∼= Xreg, the 2-form ω extends to a regular closed 2-form on X̃ (cf. [Be]). There is a natural Poisson structure { , } on X determined by ω. Then we can introduce the notion of a Poiss...

Let π be a Poisson structure on Rn vanishing at 0. It leads to a Kontsevich type star product ⋆π on C∞(Rn)[[ǫ]]. We show that 1. The evaluation map at 0 ev0 : C ∞(Rn) → C can in general not be quantized to a character of (C∞(Rn)[[ǫ]], ⋆π). 2. A given Poisson structure π vanishing at zero can in general not be extended to a formal Poisson structure πǫ also vanishing at zero, such that ev0 can be...

In this article, we address one of the questions raised by Marc Rieffel in his collection of questions on deformation quantization. The question is whether the K-groups remain the same under flabby strict deformation quantizations. By “deforming” the question slightly, we produce a negative answer to the question. In his collection of questions on deformation quantization [8], Marc Rieffel aske...

A strict quantization of a Poisson manifold P on a subset I ⊆ R containing 0 as an accumulation point is defined as a continuous field of C∗-algebras {Ah̄}h̄∈I , with A0 = C0(P ), a dense subalgebra Ã0 of C0(P ) on which the Poisson bracket is defined, and a set of continuous cross-sections {Q(f )} f∈Ã0 for which Q0(f ) = f . Here Qh̄(f ∗) = Qh̄(f )∗ for all h̄ ∈ I , whereas for h̄ → 0 one requires t...