An Invitation to von Neumann Algebras by V.S. Sunder

By V.S. Sunder

Why This ebook: the speculation of von Neumann algebras has been transforming into in leaps and boundaries within the final two decades. It has consistently had powerful connections with ergodic thought and mathematical physics. it's now commencing to make touch with different components reminiscent of differential geometry and K-Theory. There appears to be like a powerful case for placing jointly a publication which (a) introduces a reader to a few of the elemental conception had to savour the new advances, with no getting slowed down by way of an excessive amount of technical aspect; (b) makes minimum assumptions at the reader's history; and (c) is sufficiently small in dimension not to attempt the stamina and persistence of the reader. This e-book attempts to fulfill those necessities. at least, it is only what its name broadcasts it to be -- a call for participation to the fascinating global of von Neumann algebras. it's was hoping that once perusing this ebook, the reader should be tempted to fill within the a variety of (and technically, capacious) gaps during this exposition, and to delve additional into the depths of the idea. For the professional, it suffices to say right here that when a few preliminaries, the e-book commences with the Murray - von Neumann class of things, proceeds in the course of the easy modular thought to the III). class of Connes, and concludes with a dialogue of crossed-products, Krieger's ratio set, examples of things, and Takesaki's duality theorem.

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1. Noncommutative Integration generality also makes an appearance in this section. The next section pertains to a very useful technical criterion, called the KMS boundary condition, which gives an intrinsic characterisation (that does not appeal to the GNS construction) of the modular group associated with a weight. The chapter ends with a discussion of (a) the noncommutative Radon-Nikodym theorem of Pedersen and Takesaki, and (b) conditional expectations and Takesaki's theorem which identifies those situations in which normal conditional expectations exist.

0 For the rest of this chapter, the symbols 11, N, Band :R will, unless otherwise specified, always denote closed subspaces affiliated to M. 7. The relation ... '" ... (reI M) is countably additive in the following sense: if Mn..... Nn for n = 1,2, ... 1 Nn for m f. n, then $ Mn '" $ Nn . Proof. First observe that $ Mn, $ Nn n M since Mn' Nn n M. If un: Mn ..... Nn' it is easy to see. under the hypothesis. that the sequence O::~=1um}:=1 converges strongly to a partial isometry u such that u: Mn '" $ Nn .

Then, (a) [~] 'I- 0 eventually; in fact n (b) n lim [XI Nu ] n .... CO [~ ] /' + co ; and [B INn] exists and is a finite positive number. Proof. 9 (a), for any n ;.. 1, we have [t ] ;. [t ] [:] ~ 2n - l [ } ]; n since { N l/X] is a fixed finite integer it follows that [N niX] = 0 for all sufficiently large n; thus, there exists an integer no such that X 1 N n for all n ;.. no; so, if n ;.. no' Nn 1 11; in other words [XI Nn ] ;.. 1. Then, for any integer k, [N X N [Nno [ro-] ;. 2k-l, no+k X ];..

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