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Title:Noncommutative Sobolev type inequalities
Author(s):Li, Haojian
Director of Research:Junge, Marius
Doctoral Committee Chair(s):Boca, Florin
Doctoral Committee Member(s):Oikhberg, Timur; Leditzky, Felix
Department / Program:Mathematics
Discipline:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Degree:Ph.D.
Genre:Dissertation
Subject(s):Sobolev inequalities
noncommutative analysis
open quantum system
Abstract:In this thesis, we study Sobolev type inequalities in the noncommutative (quantum) settings. We establish the abstract Bakry-\'Emery criterion for the operator-valued $f$-Sobolev inequalities on the derivation triple. We recapture the celebrated Bakry-\'Emery theorem for operator-valued functions on Riemannian manifolds. We discuss examples including noncommutative $f$-Sobolev inequalities on the intervals, Lindblad operators of finite dimensional matrix algebras, and discrete graphs. By generalizing the monotone metrics in the space of quantum states, we develop a deeper understanding of $f$-Sobolev inequalities in the noncommutative setting.
Issue Date:2021-07-16
Type:Thesis
URI:http://hdl.handle.net/2142/113210
Rights Information:Copyright 2021 Haojian Li
Date Available in IDEALS:2022-01-12
Date Deposited:2021-08


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