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Author: Iain Raeburn Publisher: American Mathematical Soc. ISBN: 0821808605 Category : Mathematics Languages : en Pages : 345
Book Description
A modern treatment of this complex mathematical topic for students beginning research in operator algebras as well as mathematical physicists. Topics include the algebra of compact operators, sheaves, cohomology, the Brauer group and group actions, and the imprimivity theorem. The authors assume a knowledge of C*-algebras, the Gelfand-Naimark Theorem, continuous functional calculus, positivity, and the GNS- construction. Annotation copyrighted by Book News, Inc., Portland, OR
Author: Iain Raeburn Publisher: American Mathematical Soc. ISBN: 0821808605 Category : Mathematics Languages : en Pages : 345
Book Description
A modern treatment of this complex mathematical topic for students beginning research in operator algebras as well as mathematical physicists. Topics include the algebra of compact operators, sheaves, cohomology, the Brauer group and group actions, and the imprimivity theorem. The authors assume a knowledge of C*-algebras, the Gelfand-Naimark Theorem, continuous functional calculus, positivity, and the GNS- construction. Annotation copyrighted by Book News, Inc., Portland, OR
Author: Shoichiro Sakai Publisher: Springer Science & Business Media ISBN: 3642619932 Category : Mathematics Languages : en Pages : 271
Book Description
From the reviews: "This book is an excellent and comprehensive survey of the theory of von Neumann algebras. It includes all the fundamental results of the subject, and is a valuable reference for both the beginner and the expert." Mathematical Reviews
Author: Se-Jin Kim Publisher: ISBN: Category : Languages : en Pages : 118
Book Description
The purpose of this thesis is to give an exposition of two topics, mostly following the books \cite{R & W} and \cite{Wil}. First, we wish to investigate crossed product $C^*$-algebras in its most general form. Crossed product $C^*$-algebras are $C^*$-algebras which encode information about the action of a locally compact Hausdorff group $G$ as automorphisms on a $C^*$-algebra $A$. One of the prettiest example of such a dynamical system that I have observed in the wild arises in the gauge-invariant uniqueness theorem \cite{Rae}, which assigns to every $C^*$-algebra $C^*(E)$ associated with a graph $E$ a \emph{gauge action} of the unit circle $\T$ to automorphisms on $C^*(E)$. Group $C^*$-algebras also arise as a crossed product of a dynamical system. I found crossed products in its most general form very abstract and much of its constructions motivated by phenomena in a simpler case. Because of this, much of the initial portion of this exposition is dedicated to the action of a discrete group on a unital $C^*$-algebra, where most of the examples are given. I must admit that I find calculations of crossed products when one has an indiscrete group $G$ acting on our $C^*$-algebra daunting except under very simple cases. This leads to our second topic, on imprimitivity theorems of crossed product $C^*$-algebras. Imprimitivity theorems are machines that output (strong) Morita equivalences between crossed products. Morita equivalence is an invariant on $C^*$-algebras which preserve properties like the ideal structure and the associated $K$-groups. For example, no two commutative $C^*$-algebras are Morita equivalent, but $C(X) \otimes M_n$ is Morita equivalent to $C(X)$ whenever $n$ is a positive integer and $X$ is a compact Hausdorff space. Notice that Morita equivalence can be used to prove that a given $C^*$-algebra is simple. All this leads to our concluding application: Takai duality. The set-up is as follows: we have an action $\alpha$ of an abelian group $G$ on a $C^*$-algebra $A$. On the associated crossed product $A \rtimes_\alpha G$, there is a dual action $\Hat{\alpha}$ from the Pontryagin dual $\Hat{G}$. Takai duality states that the iterated crossed product $(A \rtimes_\alpha G) \rtimes \Hat{G}$ is isomorphic to $A \otimes \calK(L^2(G))$ in a canonical way. This theorem is used to show for example that all graph $C^*$-algebras are nuclear or to establish theorems on the $K$-theory on crossed product $C^*$-algebras.
Author: K. R. Goodearl Publisher: American Mathematical Soc. ISBN: 082182435X Category : Mathematics Languages : en Pages : 161
Book Description
Motivated by (i) Elliott's classification of direct limits of countable sequences of finite-dimensional semisimple complex algebras and complex AF C*-algebras, (ii) classical results classifying involutions on finite-dimensional semisimple complex algebras, and (iii) the classification by Handelman and Rossmann of automorphisms of period two on the algebras appearing in (i) we study the real algebras described above and completely classify them, up to isomorphism, Morita equivalence, or stable isomorphism. We also show how our classification easily distinguishes various types of algebras within the given classes, and we partially solve the problem of determining exactly which values are attained by the invariants used in classifying these algebras.
Book Description
Muhly and Solel developed a notion of Morita equivalence for C*- correspondences, which they used to show that if two C*-correspondences E and F are Morita equivalent then their tensor algebras $\mathcal{T}_{+}(E)$ and $\mathcal{T}_{+}(F)$ are (strongly) Morita equivalent operator algebras. We give the weak* version of this result by considering (weak) Morita equivalence of W*-correspondences and employing Blecher and Kashyap's notion of Morita equivalence for dual operator algebras. More precisely, we show that weak Morita equivalence of W*-correspondences E and F implies weak Morita equivalence of their Hardy algebras $H^{\infty}(E)$ and $H^{\infty}(F)$. We give special attention to W*-graph correspondences and show a number of results related to their Morita equivalence. We study how different representations of a W*-algebra give rise to Morita equivalent objects. For example, we show that if (E,A) is a W*-graph correspondence and we have two faithful normal representations $\sigma$ and $\tau$ of A, then the commutants of the induced representions $\sigma ^{\ms{F}(E)}(H^{\infty}(E))$ and $\tau ^{\ms{F}(E)}(H^{\infty}(E))$ are weakly Morita equivalent dual operator algebras. We also develop a categorical approach to Morita equivalence of W*- correspondences. This involves building categories of covariant representations and studying the groups $Aut(\mathbb{D}({(E^{\sigma}})^*)$ and $Aut(H^{\infty}(E))$ (the automorphism groups of the unit ball of intertwiners and the Hardy algebra). In this regard, we advance the work of Muhly and Solel by showing new results about these groups, their matrix representation and their algebraic properties.
Author: Dana P. Williams Publisher: American Mathematical Soc. ISBN: 0821842420 Category : Mathematics Languages : en Pages : 546
Book Description
The theory of crossed products is extremely rich and intriguing. There are applications not only to operator algebras, but to subjects as varied as noncommutative geometry and mathematical physics. This book provides a detailed introduction to this vast subject suitable for graduate students and others whose research has contact with crossed product $C*$-algebras. in addition to providing the basic definitions and results, the main focus of this book is the fine ideal structure of crossed products as revealed by the study of induced representations via the Green-Mackey-Rieffel machine. in particular, there is an in-depth analysis of the imprimitivity theorems on which Rieffel's theory of induced representations and Morita equivalence of $C*$-algebras are based. There is also a detailed treatment of the generalized Effros-Hahn conjecture and its proof due to Gootman, Rosenberg, and Sauvageot. This book is meant to be self-contained and accessible to any graduate student coming out of a first course on operator algebras. There are appendices that deal with ancillary subjects, which while not central to the subject, are nevertheless crucial for a complete understanding of the material. Some of the appendices will be of independent interest. to view another book by this author, please visit Morita Equivalence and Continuous-Trace $C*$-Algebras.
Author: Gene Abrams Publisher: Springer ISBN: 1447173449 Category : Mathematics Languages : en Pages : 296
Book Description
This book offers a comprehensive introduction by three of the leading experts in the field, collecting fundamental results and open problems in a single volume. Since Leavitt path algebras were first defined in 2005, interest in these algebras has grown substantially, with ring theorists as well as researchers working in graph C*-algebras, group theory and symbolic dynamics attracted to the topic. Providing a historical perspective on the subject, the authors review existing arguments, establish new results, and outline the major themes and ring-theoretic concepts, such as the ideal structure, Z-grading and the close link between Leavitt path algebras and graph C*-algebras. The book also presents key lines of current research, including the Algebraic Kirchberg Phillips Question, various additional classification questions, and connections to noncommutative algebraic geometry. Leavitt Path Algebras will appeal to graduate students and researchers working in the field and related areas, such as C*-algebras and symbolic dynamics. With its descriptive writing style, this book is highly accessible.
Author: Iain Raeburn
Author: Iain Raeburn
Author: Walter Josef Beer
Author: Marc Aristide Rieffel
Author: Shoichiro Sakai
Author: Mingyi Wang
Author: Se-Jin Kim
Author: K. R. Goodearl
Author: René Ardila
Author: Dana P. Williams
Author: Gene Abrams