Derivations and reflection positivity on the quantum cylinder
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https://doi.org/10.1007/s11425-019-9533-3
Derivations and reflection positivity on the quantum cylinder Slawomir Klimek1 & Matt McBride2,∗ 1Department
of Mathematical Sciences, Indiana University-Purdue University Indianapolis, Indianapolis, IN 46202, USA; 2Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762, USA Email: [email protected], [email protected] Received January 1, 2019; accepted April 12, 2019
Abstract
We describe the general structure of unbounded derivations in the quantum cylinder. We prove a
noncommutative analog of reflection positivity for Laplace-type operators in a noncommutative cylinder following the ideas of the proof by Jaffe and Ritter (2008) of reflection positivity for Laplace operators on manifolds equipped with a reflection. Keywords MSC(2010)
derivations, reflection positivity, noncommutative cylinder 46L57
Citation: Klimek S, McBride M. Derivations and reflection positivity on the quantum cylinder. Sci China Math, 2020, 63, https://doi.org/10.1007/s11425-019-9533-3
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Introduction
Part of this work is a continuation of the program started in [9,11] on studying unbounded derivations in quantum domains, their implementations, and possible spectral triples associated to them. Another part of this work was inspired by Glimm and Jaffe’s note [2] on reflection positivity for the Laplace operator in Rn . Additionally, we were influenced by Jaffe and Ritter’s paper [5], which considered reflection positivity for Laplace operators on manifolds equipped with a reflection. Reflection positivity in the Euclidean space is the following remarkable inequality in L2 (Rn ): ⟨Θf, (−∆ + m2 )−1 f ⟩ > 0 for all f ∈ H+ = {f ∈ L2 (Rn ) : f (x1 , . . . , xn ) = 0, for x1 < 0}. Here, ∆ is the Laplace operator in Rn , m is a positive constant and Θ : L2 (Rn ) → L2 (Rn ) is the reflection in the first coordinate Θf (x1 , . . . , xn ) = f (−x1 , . . . , xn ). This inequality is a key step in proving the reflection positivity axiom of Osterwalder-Schrader for the free field [3]. Reflection positivity has been generalized in many directions, of particular interest for this paper is the already-mentioned manifold generalization in [5]. * Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
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Klimek S et al.
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A natural question is whether such ideas can be extended to noncommutative geometry to include examples of Laplace-type operators on noncommutative manifolds. One of the simplest possibilities, studied in detail in this paper, is a quantum cylinder which classically has a natural reflection through the middle. The noncommutative cylinder (quantum annulus) was constructed in [7] and further studied in [9, 10]. It has a natural rotational symmetry as well as a reflection, as will be shown later, and it also has an analog of the Lebesgue measure. To define a class of interesting, reflection invariant, Laplace-type operators in the corres
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