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University of Illinois – Urbana-Champaign

1. Rezvani, Sepideh. Approximating rotation algebras and inclusions of C*-algebras.

Degree: PhD, Mathematics, 2017, University of Illinois – Urbana-Champaign

URL: http://hdl.handle.net/2142/97307

In the first part of this thesis, we will follow Kirchberg’s categorical perspective to establish new notions of WEP and QWEP relative to a C∗-algebra, and develop similar properties as in the classical WEP and QWEP. Also we will show some examples of relative WEP and QWEP to illustrate the relations with the classical cases.
The focus of the second part of this thesis is the approximation of rotation algebras in the quantum Gromov–Hausdorff distance. We introduce the completely bounded quantum Gromov–Hausdorff distance and show that for even dimensions, the higher dimensional rotation algebras can be approximated by matrix algebras in this sense. Finally, we show that for even dimensions, matrix algebras converge to the rotation algebras in the strongest form of Gromov–Hausdorff distance, namely in the sense of Latrémolière’s Gromov–Hausdorff propinquity.
*Advisors/Committee Members: Junge, Marius (advisor), Boca, Florin (Committee Chair), Ruan, Zhong-Jin (committee member), Oikhberg, Timur (committee member).*

Subjects/Keywords: C*-algebras; Weak expectation property (WEP); Quotient weak expectation property (QWEP); A-WEP; A-QWEP; Relatively weak injectivity; Order-unit space; Noncommutative tori; Compact quantum metric space; Conditionally negative length function; Heat semigroup; Poisson semigroup; Rotation algebra; Continuous field of compact quantum metric spaces; Gromov–Hausdorff distance; Completely bounded quantum Gromov–Hausdorff distance; Gromov–Hausdorff propinquity

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APA (6^{th} Edition):

Rezvani, S. (2017). Approximating rotation algebras and inclusions of C*-algebras. (Doctoral Dissertation). University of Illinois – Urbana-Champaign. Retrieved from http://hdl.handle.net/2142/97307

Chicago Manual of Style (16^{th} Edition):

Rezvani, Sepideh. “Approximating rotation algebras and inclusions of C*-algebras.” 2017. Doctoral Dissertation, University of Illinois – Urbana-Champaign. Accessed September 26, 2020. http://hdl.handle.net/2142/97307.

MLA Handbook (7^{th} Edition):

Rezvani, Sepideh. “Approximating rotation algebras and inclusions of C*-algebras.” 2017. Web. 26 Sep 2020.

Vancouver:

Rezvani S. Approximating rotation algebras and inclusions of C*-algebras. [Internet] [Doctoral dissertation]. University of Illinois – Urbana-Champaign; 2017. [cited 2020 Sep 26]. Available from: http://hdl.handle.net/2142/97307.

Council of Science Editors:

Rezvani S. Approximating rotation algebras and inclusions of C*-algebras. [Doctoral Dissertation]. University of Illinois – Urbana-Champaign; 2017. Available from: http://hdl.handle.net/2142/97307

2. Zhao, Mingyu. Smoothing estimates for non commutative spaces.

Degree: PhD, Mathematics, 2018, University of Illinois – Urbana-Champaign

URL: http://hdl.handle.net/2142/101570

In the first part of this thesis, we follow Varopoulos's perspective to establish the noncommutaive Sobolev inequaties (namely, Hardy-Littlewood-Sobolev inequalites), and extend the Sobolev embedding from noncommutative L_{p} spaces to general Orlicz function spaces related with Cowling and Meda's work. Also we will show some examples to illustract the relation between the Orlicz function, dispersive estimate on semigroup T_{t} and general resolvent formula on the generator A of the semigroup (i.e. Ax= \lim_{t → 0} \frac{T_{t} x - x}{t}). And we prove a borderline case of noncommutaive Sobolev inequality, namely the noncommutative Trudinger Moser's inequality.
The focus of the second part of the thesis is the completely bounded version of noncommutative Sobolev inequalities. We prove a cb version of the Sobolev inequality for noncommutative L_{p} spaces. As a tool, we further develop a general embedding theory for von Neumann algebra, continuing the work for . Finally we prove the cb version of Varopolous's theorem and provide some examples and applications.
The third part of the thesis proves the existence of abstract Strichartz estimates on \rx_{\ta} for operators that satisfies ultracontractivity and energy estimate. And we show the abstract Strichartz estimates are applicable to the Schrödinger equation problem on quantum Euclidean spaces \rx_{\ta}^{n}.
*Advisors/Committee Members: Junge, Marius (advisor), Ruan, Zhong-Jin (Committee Chair), Boca, Florin (committee member), Oikhberg, Timur (committee member).*

Subjects/Keywords: harmonic analysis; Hardy-Littlewood-Sobolev inequalities; Functional analysis; Operator space; Operator algebras; non-commutaive $L_p$ spaces

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Zhao, M. (2018). Smoothing estimates for non commutative spaces. (Doctoral Dissertation). University of Illinois – Urbana-Champaign. Retrieved from http://hdl.handle.net/2142/101570

Chicago Manual of Style (16^{th} Edition):

Zhao, Mingyu. “Smoothing estimates for non commutative spaces.” 2018. Doctoral Dissertation, University of Illinois – Urbana-Champaign. Accessed September 26, 2020. http://hdl.handle.net/2142/101570.

MLA Handbook (7^{th} Edition):

Zhao, Mingyu. “Smoothing estimates for non commutative spaces.” 2018. Web. 26 Sep 2020.

Vancouver:

Zhao M. Smoothing estimates for non commutative spaces. [Internet] [Doctoral dissertation]. University of Illinois – Urbana-Champaign; 2018. [cited 2020 Sep 26]. Available from: http://hdl.handle.net/2142/101570.

Council of Science Editors:

Zhao M. Smoothing estimates for non commutative spaces. [Doctoral Dissertation]. University of Illinois – Urbana-Champaign; 2018. Available from: http://hdl.handle.net/2142/101570

3. Gao, Li. On quantum Euclidean spaces: Continuous deformation and pseudo-differential operators.

Degree: PhD, Mathematics, 2018, University of Illinois – Urbana-Champaign

URL: http://hdl.handle.net/2142/101546

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.
*Advisors/Committee Members: Junge, Marius (advisor), Ruan, Zhong-Jin (Committee Chair), Boca, Florin P. (committee member), Oikhberg, Timur (committee member).*

Subjects/Keywords: Noncommutative Euclidean spaces; Moyal Deformation; Pseudo-differential operators

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Gao, L. (2018). On quantum Euclidean spaces: Continuous deformation and pseudo-differential operators. (Doctoral Dissertation). University of Illinois – Urbana-Champaign. Retrieved from http://hdl.handle.net/2142/101546

Chicago Manual of Style (16^{th} Edition):

Gao, Li. “On quantum Euclidean spaces: Continuous deformation and pseudo-differential operators.” 2018. Doctoral Dissertation, University of Illinois – Urbana-Champaign. Accessed September 26, 2020. http://hdl.handle.net/2142/101546.

MLA Handbook (7^{th} Edition):

Gao, Li. “On quantum Euclidean spaces: Continuous deformation and pseudo-differential operators.” 2018. Web. 26 Sep 2020.

Vancouver:

Gao L. On quantum Euclidean spaces: Continuous deformation and pseudo-differential operators. [Internet] [Doctoral dissertation]. University of Illinois – Urbana-Champaign; 2018. [cited 2020 Sep 26]. Available from: http://hdl.handle.net/2142/101546.

Council of Science Editors:

Gao L. On quantum Euclidean spaces: Continuous deformation and pseudo-differential operators. [Doctoral Dissertation]. University of Illinois – Urbana-Champaign; 2018. Available from: http://hdl.handle.net/2142/101546