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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Chapter 0. Introduction

Abstract
As the title suggests, this chapter is devoted to developing some of the basic technical results needed in the theory, and may be safely omitted by the expert. The first section establishes some of the notation employed throughout the book and lists, without proof, the basic facts concerning operators on Hilbert space. The next section establishes the “non-commutative analogue” of the classical results \( c_0^* = \ell ^1 \,{\text{and}}\,(\ell ^1 )* = \ell ^\infty \) -- null-convergent, summable, and bounded sequences being replaced by compact, trace-class and bounded operators respectively.
V. S. Sunder

Chapter 1. The Murray — Von Neumann Classification of Factors

Abstract
The notion of unitary equivalence, while being most natural, has the disadvantage of not being additive in the following sense: if e1, e2, f1 and f2 are Projections such that ei is unitarily equivalent to fi, for i = 1,2, and if e1e2 and f1f2, it is not necessarily true that e1 + e2 is unitarily equivalent to f1 + f2. This problem disappears if, more generally, one considers two projections as being equivalent if their ranges are the initial and final spaces of a partial isometry. This equivalence, when all the operators concerned -- the projections as well as the partial isometry -- are required to come from a given factor M, is the subject of Section 1.1, where the crucial result is that, with respect to a natural order, the set of equivalence classes of the projections in a factor is totally ordered. The next section examines finite projections -- those not equivalent to proper subprojections; the main result being that finiteness is preserved under taking finite suprema. The final section, via a quantitative analysis of the order relation discussed earlier, effects a primary classification of factors into three types. The principal tool used is called a ‘relative dimension function’ by Murray and von Neumann.
V. S. Sunder

Chapter 2. The Tomita-Takesaki Theory

Abstract
Section 2.1 discusses the following question (which, in the case M = L (X, F) with μ finite, is answered affirmatively by the existence of the Lebesgue integral): if m: P(M) → [0,1] is countably additive in the sense that
$$ m\left[ {\mathop{V}\limits_{{n = 1}}^{\infty } {e_n}} \right] = \sum\limits_{{n = 1}}^{\infty } {m({e_n})} $$
for any countable collection of pairwise orthogonal projections in M, does m extend to a linear functional on M which is well-behaved under monotone convergence?
V. S. Sunder

Chapter 3. The Connes Classification of Type III Factors

Abstract
The first section discusses the extent to which the modular group σ ϕ depends upon the fns weight σ ϕ . The precise description is the unitary cocycle theorem of Connes, which says, loosely, that modulo the group of inner automorphisms of M, the modular group σ ϕ is independent of σ ϕ .
V. S. Sunder

Chapter 4. Crossed-Products

Abstract
The crossed-product construction was first employed by Murray and von Neumann to exhibit examples of factors of types I, II and III. The set-up is as follows: one starts with a dynamical system (M,G,α) -- with G not necessarily abelian -- and constructs an associated von Neumann algebra M (usually denoted by Mα G) on a larger Hilbert space H.
V. S. Sunder

Backmatter

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