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Title On quantum Euclidean spaces: Continuous deformation and pseudo-differential operators
Publication Date
Date Accessioned
Degree PhD
Discipline/Department Mathematics
Degree Level doctoral
University/Publisher University of Illinois – Urbana-Champaign
Abstract Quantum Euclidean spaces are noncommutative deformations of Euclidean spaces. They are prototypes of locally compact noncommutative manifolds in Noncommutative Geometry. In this thesis, we study the continuous deformation and Pseudo-differential calculus of quantum Euclidean spaces. After reviewing the basic definitions and representation theory of quantum Euclidean spaces in Chapter 1, we prove in Chapter 2 a Lip^(1/2) continuous embedding of the family of quantum Euclidean spaces. This result is the locally compact analog of U. Haagerup and M. R\o rdom's work on Lip^(1/2) continuous embedding for quantum 2-torus. As a corollary, we also obtained Lip^(1/2) embedding for quantum tori of all dimensions. In Chapter 3, we developed a Pseudo-differential calculus for noncommuting covariant derivatives satisfying the Canonical Commutation Relations. Based on some basic analysis on quantum Euclidean spaces, we introduce abstract symbol classs following the idea of abstract pseudo-differential operators introduced by A. Connes and H. Moscovici. We proved the two main ingredients pseudo-differential calculus  – the L2-boundedness of 0-order operators and the composition identity. We also identify the principal symbol map in our setting. Chapter 4 is devoted to application in the local index formula in noncommutative Geometry. We show that our setting with noncommuting covariant derivatives is an example of locally compact noncommutative manifold. After developed the Getzler super-symmetric symbol calculus, we calculate the local index formula for the a noncommutative analog of Bott projection.
Subjects/Keywords Noncommutative Euclidean spaces; Moyal Deformation; Pseudo-differential operators;
Contributors Junge, Marius (advisor); Ruan, Zhong-Jin (Committee Chair); Boca, Florin P. (committee member); Oikhberg, Timur (committee member)
Language en
Rights Copyright 2018 by Li Gao
Country of Publication us
Record ID handle:2142/101546
Repository uiuc
Date Indexed 2020-03-09
Grantor University of Illinois at Urbana-Champaign
Issued Date 2018-07-10 00:00:00

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…It facilitates the recognition of quantum mechanics as a deformation of classical mechanics, with the Planck constant on the parameter h, h being a deformation parameter. The Moyal product, depending gives a continuous family of deformations from…

…x5B;43]) and deformation of C ∗ -algebras by actions of Rd [39]. In the rst part of this thesis, we study the continuity of Moyal deformation, or more precisely, the continuity of the quantization map λh depending on h…

…algebraically and then allows the commutative function algebras to be generalized to possibly non-commutative algebras. From this point of view, the Heisenberg relation and the CCR algebra are noncommutative deformation of Euclidean space, which are called Moyal

…x28;ξ)dξ , f ∈ S(Rd ), Rd is the Fourier transform of f. the quantized Schwartz class. The matrix multiplication of product ?θ associated to the parameter We denote by Sθ is equivalent to the Moyal θ, −d Z Z λθ (f…

…fˆ(ξ − η)ĝ(η)e 2 ξ·θη λθ (ξ)dξdη , 2d (2π) d d Z RZ R i 1 f\ ?θ g(ξ) = fˆ(ξ − η)ĝ(η)e 2 ξ·θη dξdη d (2π) Rd Rd 7 The Moyal product is bilinear, associative and…

…reversed under complex conjugation f ?θ g = g ?θ f , which makes (S(Rd ), ?θ ) a ∗-algebra. We refer to [17, 47] for more information about Moyal analysis. The algebra C ∗ -quantum (S(Rd ), ?θ )…

…equivalent denitions: i) the C ∗ -enveloping of the Moyal product algebra (S(Rd ), ?θ ); ii) the full twisted group C ∗ -algebra C ∗ (Rd , σθ ) iii) the reduced twisted group C ∗ -algebra Cr∗ (Rd , σθ )…

…coordinate (xj g)(x) = xj g(x) 2. the Moyal product 3. E0 = C0 (Rd ) Rθ is the usual point-wise multiplication of functions; is the space of continuous functions on {R0 = L∞ (Rd )} Thus ?0…