A Negative Answer to a Question by Rieffel

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 asked the following; “Are the K-groups of the C∗-algebra completions of the algebras of any flabby strict deformation quantization all isomorphic?” Up to my knowledge, the question is still open. But this article will show that the answer is negative if we ask the same question for the case of orbifolds. Definition 1 [8]. Let (M, {·, ·}) be a Poisson manifold. A strict deformation quantization of M in the direction of {·, ·} is a dense ∗-algebra A of C∞(M) which is closed under the Poisson bracket, together with a closed subset I of the real line containing 0 as a non-isolated point, and for each ~ ∈ I an associative product ∗~, an involution ∗~ , and a pre-C∗-norm ‖ · ‖~ on A, which for ~ = 0 are the original pointwise multiplication, complex conjugation, and supremum norm respectively, and such that (1) for each f ∈ A, f → ‖f‖~ and ‖f‖∗~ are continuous on I (this implies that {A~}~∈I forms a continuous fields of C∗-algebras over I, where A~ is the C∗completion of A~), and (2) for f, g ∈ A, ∥∥∥∥ f ∗~ g − g ∗~ f √ −1~ − {f, g} ∥∥∥∥ ~ → 0 as ~ → 0. This definition still makes sense for a simple Poisson orbifold M/Γ, where Γ is a finite group acting onM and the action preserves the Poisson bracket. A function on M/Γ is just a function on M which is constant on each orbit. A function f on M/Γ is defined to be smooth if it is smooth as a function on M . 1991 Mathematics Subject Classification. Primary 53D15, 81S10.