@MISC{Shim01anegative, author = {Jae-kwan Shim}, title = {A negative answer to a question by Rieffel}, year = {2001} }

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Abstract

Abstract. 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 ∥ √ − {f, g} ∥ → 0