University of Illinois Urbana-Champaign

Non commutative version of arithmetic geometric mean inequality and crossed product of ternary ring of operators

Albar, Wafaa Abdullah

Content Files
ALBAR-DISSERTATION-2017.pdf

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https://hdl.handle.net/2142/98206

Description

Title
Non commutative version of arithmetic geometric mean inequality and crossed product of ternary ring of operators
Author(s)
Albar, Wafaa Abdullah
Issue Date
2017-07-11
Director of Research (if dissertation) or Advisor (if thesis)
Ruan, Zhong-Jin
Doctoral Committee Chair(s)
Boca, Florin
Committee Member(s)
  • Junge, Marius
  • Li, Xiaochun
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Date of Ingest
2017-09-29T17:45:09Z
Keyword(s)
  • Arithmetic geometric mean inequality (AGM)
  • Random matrices
  • Ternary ring of operators (TRO)
  • Crossed product of ternary ring of operators (TROs)
Abstract
"This thesis is structured into two parts. In the first two chapters, we prove the non commutative version of the Arithmetic Geometric Mean (AGM) inequality (this is a joint work with Mingyue Zhao and Maruis Junge). We start Chapter 2 by giving some background about the partition and M{\""o}bius function. We then prove the two main theorems: The AGM inequality for the norm and for the order. In Chapter 3, we provide some applications from random matrices such as Wishart random matrices, vector-valued moments of convex bodies, and freely independent operators. The second part is about a ternary ring of operators (TRO). After giving a quick survey for the work of Todorov on the operator space version of Zettl's decomposition theorem, we introduce crossed products of ternary ring of operators (the full crossed product and the reduced crossed product). We also prove that $V\rtimes_{\alpha^{V}}G$ as the off-diagonal corner of the $C^*$-algebra $A(V)\rtimes_{\alpha^{A(V)}}G$. Equivalently, we have the $*$-isomorphism between the two linking $C^*$-algebras, i.e. $A(V\rtimes_{\alpha^{V}}G)=A(V)\rtimes_{\alpha^{A(V)}}G.$ By using this identity, we obtain that if the group $G$ is amenable, some local properties for TRO's preserve with the crossed product. We also provide a counter example which shows that if the linking $C^*$-algebras $A(V)$ and $A(W)$ are $*$-isomorphic or if their diagonal components are $*$-isomorphic, then their TRO's are not isomorphic. Similar example will be applied for $W^*$-TRO's."
Graduation Semester
2017-08
Type of Resource
text
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http://hdl.handle.net/2142/98206
Copyright and License Information
Copyright 2017 Wafaa Albar

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