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Applications of Group-Like Construct...

Patchell, Gregory David,

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Applications of Group-Like Constructions to the Structure Theory of Tracial Von Neumann Algebras /

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Applications of Group-Like Constructions to the Structure Theory of Tracial Von Neumann Algebras // Gregory David Patchell.
作者:	Patchell, Gregory David,
面頁冊數:	1 electronic resource (151 pages)
附註:	Source: Dissertations Abstracts International, Volume: 86-12, Section: B.
Contained By:	Dissertations Abstracts International86-12B.
標題:	Mathematics. -
電子資源:	https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=31940155
ISBN:	9798286444205

Applications of Group-Like Constructions to the Structure Theory of Tracial Von Neumann Algebras /
Patchell, Gregory David,

Applications of Group-Like Constructions to the Structure Theory of Tracial Von Neumann Algebras /Gregory David Patchell. - 1 electronic resource (151 pages)

Source: Dissertations Abstracts International, Volume: 86-12, Section: B.

Von Neumann algebras give us a way to represent infinite groups and dynamical systems in ways matrix algebras simply cannot. However, the structure and classification of von Neumann algebras, especially II1 factors, has always been challenging. The results in this dissertation take concepts from group theory and ergodic theory and use them to study the structure of tracial von Neumann algebras.In Chapter 1, primeness, or lack of tensor product decomposition, is studied for a class of II1 factors known as generalized wreath products. Generalized wreath products are the operator algebras that arise from generalized Bernoulli actions. We see here that non amenability of the action combined with an amenability assumption on the stabilizers of the action gives primeness.In Chapter 2, we introduce a new equivalence relation on tracial von Neumann algebras as well as two new cardinal invariants. These invariants are all related to sequentially commuting unitaries. Our invariants interact well with known examples, such as the free group factors and factors with property Gamma. We also study graph products and relate our diameter to the diameter of the underlying graph. We introduce the ``uniform-flattening'' strategy which allows us to compute relative commutants in ultrapowers.In Chapter 3, we show that the existence of sequentially commuting diffuse unitaries in the ultrapower is equivalent to the existence of sequentially commuting almost diffuse unitaries in the original algebra. Using this, we show that a large class of II1 factors is singly generated.In Chapter 4, we foray into classification of nonseparable II1 factors. We demonstrate the existence of exotic II1 factors which have no known decomposition except as an inductive limit. These factors have large families of separable subalgebras which are isomorphic to each other only if they are unitarily conjugate. Using similar ideas, we construct separable subalgebras where property (T) subalgebras are isomorphic only if they are uniformly approximately unitarily conjugate. We also show that any finite set of elements in a II1 factor can be perturbed to generate a factor.

English

ISBN: 9798286444205Subjects--Topical Terms:

515831
Mathematics.
Subjects--Index Terms:

Ergodic theory

Applications of Group-Like Constructions to the Structure Theory of Tracial Von Neumann Algebras /
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520 $a Von Neumann algebras give us a way to represent infinite groups and dynamical systems in ways matrix algebras simply cannot. However, the structure and classification of von Neumann algebras, especially II1 factors, has always been challenging. The results in this dissertation take concepts from group theory and ergodic theory and use them to study the structure of tracial von Neumann algebras.In Chapter 1, primeness, or lack of tensor product decomposition, is studied for a class of II1 factors known as generalized wreath products. Generalized wreath products are the operator algebras that arise from generalized Bernoulli actions. We see here that non amenability of the action combined with an amenability assumption on the stabilizers of the action gives primeness.In Chapter 2, we introduce a new equivalence relation on tracial von Neumann algebras as well as two new cardinal invariants. These invariants are all related to sequentially commuting unitaries. Our invariants interact well with known examples, such as the free group factors and factors with property Gamma. We also study graph products and relate our diameter to the diameter of the underlying graph. We introduce the ``uniform-flattening'' strategy which allows us to compute relative commutants in ultrapowers.In Chapter 3, we show that the existence of sequentially commuting diffuse unitaries in the ultrapower is equivalent to the existence of sequentially commuting almost diffuse unitaries in the original algebra. Using this, we show that a large class of II1 factors is singly generated.In Chapter 4, we foray into classification of nonseparable II1 factors. We demonstrate the existence of exotic II1 factors which have no known decomposition except as an inductive limit. These factors have large families of separable subalgebras which are isomorphic to each other only if they are unitarily conjugate. Using similar ideas, we construct separable subalgebras where property (T) subalgebras are isomorphic only if they are uniformly approximately unitarily conjugate. We also show that any finite set of elements in a II1 factor can be perturbed to generate a factor.
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650 4 $a Theoretical mathematics. $3 3173530
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653 $a Functional analysis
653 $a Group theory
653 $a Model theory
653 $a Operator algebras
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