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Author: Jan Michael Cameron Publisher: ISBN: Category : Languages : en Pages :
Book Description
For an inclusion N [subset of or equal to] M of finite von Neumann algebras, we study the group of normalizers N_M(B) = {u: uBu^* = B} and the von Neumann algebra it generates. In the first part of the dissertation, we focus on the special case in which N [subset of or equal to] M is an inclusion of separable II1 factors. We show that N_M(B) imposes a certain "discrete" structure on the generated von Neumann algebra. An analyzing the bimodule structure of certain subalgebras of N_M(B)", then yieds to a "Galois-type" theorem for normalizers, in which we find a description of the subalgebras of N_M(B)" in terms of a unique countable subgroup of N_M(B). We then apply these general techniques to obtain results for inclusions B [subset of or equal to] M arising from the crossed product, group von Neumann algebra, and tensor product constructions. Our work also leads to a construction of new examples of norming subalgebras in finite von Neumann algebras: If N [subset of or equal to] M is a regular inclusion of II1 factors, then N norms M: These new results and techniques develop further the study of normalizers of subfactors of II1 factors. The second part of the dissertation is devoted to studying normalizers of maximal abelian self-adjoint subalgebras (masas) in nonseparable II1 factors. We obtain a characterization of masas in separable II1 subfactors of nonseparable II1 factors, with a view toward computing cohomology groups. We prove that for a type II1 factor N with a Cartan masa, the Hochschild cohomology groups H^n(N, N)=0, for all n [greater than or equal to] 1. This generalizes the result of Sinclair and Smith, who proved this for all N having separable predual. The techniques and results in this part of the thesis represent new progress on the Hochschild cohomology problem for von Neumann algebras.
Author: Jan Michael Cameron Publisher: ISBN: Category : Languages : en Pages :
Book Description
For an inclusion N [subset of or equal to] M of finite von Neumann algebras, we study the group of normalizers N_M(B) = {u: uBu^* = B} and the von Neumann algebra it generates. In the first part of the dissertation, we focus on the special case in which N [subset of or equal to] M is an inclusion of separable II1 factors. We show that N_M(B) imposes a certain "discrete" structure on the generated von Neumann algebra. An analyzing the bimodule structure of certain subalgebras of N_M(B)", then yieds to a "Galois-type" theorem for normalizers, in which we find a description of the subalgebras of N_M(B)" in terms of a unique countable subgroup of N_M(B). We then apply these general techniques to obtain results for inclusions B [subset of or equal to] M arising from the crossed product, group von Neumann algebra, and tensor product constructions. Our work also leads to a construction of new examples of norming subalgebras in finite von Neumann algebras: If N [subset of or equal to] M is a regular inclusion of II1 factors, then N norms M: These new results and techniques develop further the study of normalizers of subfactors of II1 factors. The second part of the dissertation is devoted to studying normalizers of maximal abelian self-adjoint subalgebras (masas) in nonseparable II1 factors. We obtain a characterization of masas in separable II1 subfactors of nonseparable II1 factors, with a view toward computing cohomology groups. We prove that for a type II1 factor N with a Cartan masa, the Hochschild cohomology groups H^n(N, N)=0, for all n [greater than or equal to] 1. This generalizes the result of Sinclair and Smith, who proved this for all N having separable predual. The techniques and results in this part of the thesis represent new progress on the Hochschild cohomology problem for von Neumann algebras.
Author: Frederick M. Goodman Publisher: Springer Science & Business Media ISBN: 1461396417 Category : Mathematics Languages : en Pages : 297
Book Description
A recent paper on subfactors of von Neumann factors has stimulated much research in von Neumann algebras. It was discovered soon after the appearance of this paper that certain algebras which are used there for the analysis of subfactors could also be used to define a new polynomial invariant for links. Recent efforts to understand the fundamental nature of the new link invariants has led to connections with invariant theory, statistical mechanics and quantum theory. In turn, the link invariants, the notion of a quantum group, and the quantum Yang-Baxter equation have had a great impact on the study of subfactors. Our subject is certain algebraic and von Neumann algebraic topics closely related to the original paper. However, in order to promote, in a modest way, the contact between diverse fields of mathematics, we have tried to make this work accessible to the broadest audience. Consequently, this book contains much elementary expository material.
Author: J. Dixmier Publisher: Elsevier ISBN: 0080960154 Category : Mathematics Languages : en Pages : 479
Book Description
In this book, we study, under the name of von Neumann algebras, those algebras generally known as “rings of operators“ or “W*-algebras.“ The new terminology, suggested by J. Dieudonng, is fully justified from the historical point of view. Certain of the results are valid for more general algebras. We have, however systematically avoided this kind of generalization, except when it would facilitate the study of von Neumann algebras themselves. Parts I and I1 comprise those results which at present appear to’be the most useful for applications, although we do not embark on the study of those applications. Part 111, which is more technical, is primarily intended for specialists; it is virtually independent of Part 11.
Author: V.S. Sunder Publisher: Springer Science & Business Media ISBN: 1461386691 Category : Mathematics Languages : en Pages : 184
Book Description
Why This Book: The theory of von Neumann algebras has been growing in leaps and bounds in the last 20 years. It has always had strong connections with ergodic theory and mathematical physics. It is now beginning to make contact with other areas such as differential geometry and K-Theory. There seems to be a strong case for putting together a book which (a) introduces a reader to some of the basic theory needed to appreciate the recent advances, without getting bogged down by too much technical detail; (b) makes minimal assumptions on the reader's background; and (c) is small enough in size to not test the stamina and patience of the reader. This book tries to meet these requirements. In any case, it is just what its title proclaims it to be -- an invitation to the exciting world of von Neumann algebras. It is hoped that after perusing this book, the reader might be tempted to fill in the numerous (and technically, capacious) gaps in this exposition, and to delve further into the depths of the theory. For the expert, it suffices to mention here that after some preliminaries, the book commences with the Murray - von Neumann classification of factors, proceeds through the basic modular theory to the III). classification of Connes, and concludes with a discussion of crossed-products, Krieger's ratio set, examples of factors, and Takesaki's duality theorem.
Author: Șerban Strătilă Publisher: Cambridge University Press ISBN: 1108496849 Category : Mathematics Languages : en Pages : 441
Book Description
The text covers fundamentals of von Neumann algebras, including the Tomita's theory of von Neumann algebras and the latest developments.
Author: Serban-Valentin Stratila Publisher: Cambridge University Press ISBN: 1108750222 Category : Mathematics Languages : en Pages : 442
Book Description
Written in lucid language, this valuable text discusses fundamental concepts of von Neumann algebras including bounded linear operators in Hilbert spaces, finite von Neumann algebras, linear forms on algebra of operators, geometry of projections and classification of von Neumann algebras in an easy to understand manner. The revised text covers new material including the first two examples of factors of type II^1, an example of factor of type III and theorems for von Neumann algebras with a cyclic and separating vector. Pedagogical features including solved problems and exercises are interspersed throughout the book.
Author: Igor Fulman Publisher: American Mathematical Soc. ISBN: 0821805576 Category : Mathematics Languages : en Pages : 122
Book Description
In this book, the author introduces and studies the construction of the crossed product of a von Neumann algebra. This construction is the generalization of the construction of the crossed product of an abelian von Neumann algebra by an equivalence relation introduced by J. Feldman and C. C. Moore. Many properties of this construction are proved in the general case. In addition, the generalizations of the Spectral Theorem on Bimodules and of the theorem on dilations are proved.