Quantum Differentiability on Noncommutative Euclidean Spaces
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Communications in
Mathematical Physics
Quantum Differentiability on Noncommutative Euclidean Spaces Edward McDonald1 , Fedor Sukochev1 , Xiao Xiong1,2 1 School of Mathematics and Statistics, UNSW, Kensington, NSW 2052, Australia.
E-mail: [email protected]; [email protected]
2 Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin 150001, China.
E-mail: [email protected] Received: 31 May 2019 / Accepted: 8 September 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract: We study the topic of quantum differentiability on quantum Euclidean ddimensional spaces (otherwise known as Moyal d-spaces), and we find conditions that are necessary and sufficient for the singular values of the quantised differential d¯x to have decay O(n −α ) for 0 < α ≤ d1 . This result is substantially more difficult than the analogous problems for Euclidean space and for quantum d-tori.
1. Introduction Quantum Euclidean spaces were first introduced by a number of authors, including Groenewold [28] and Moyal [47], for the study of quantum mechanics in phase space. The constructions of Groenewold and Moyal were later abstracted into more general canonical commutation relation (CCR) algebras, and have since become fundamental in mathematical physics. Under the names Moyal planes or Moyal-Groenewold planes, these algebras play the role of a central and motivating example in noncommutative geometry [5,22]. As geometrical spaces with noncommutating spatial coordinates, noncommutative Euclidean spaces have appeared frequently in the mathematical physics literature [21], in the contexts of string theory [61] and noncommutative field theory [48]. Quantum Euclidean spaces have also been studied as an interesting noncommutative setting for classical and harmonic analysis, and for this we refer the reader to recent work such as [24,39,46,67]. Connes introduced the quantised calculus in [8] as a replacement for the algebra of differential forms for applications in a noncommutative setting, and afterwards this point of view found application to mathematical physics [9]. Connes successfully applied his quantised calculus in providing a formula for the Hausdorff measure of Julia sets and for limit sets of Quasi-Fuchsian groups in the plane [10, Chapter 4, Sect. 3.γ ] (for a more recent exposition see [14,17]).
E. McDonald, F. Sukochev, X. Xiong
Following [8], quantised calculus may be defined defined in terms of a Fredholm module. The idea behind a Fredholm module has its origins with Atiyah’s work on K -homology [2], and further details can be found in, for example, [33, Chapter 8]. A Fredholm module can be defined with the following data: a separable Hilbert space H , a unitary self-adjoint operator F on H and a C ∗ -algebra A represented on H such that the commutator [F, a] is a compact operator on H for all a in A. The quantised differential of a ∈ A is then defined to be the operator da ¯ = i[F, a]. It is suggestive to think of the compact operators on H as being analogous to “inf
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