March 7, 2017

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

By V.S. Sunder

Why This booklet: the idea of von Neumann algebras has been growing to be in leaps and limits within the final two decades. It has regularly had robust connections with ergodic concept and mathematical physics. it's now starting to make touch with different parts comparable to differential geometry and K-Theory. There appears to be like a powerful case for placing jointly a e-book which (a) introduces a reader to a couple of the fundamental concept had to take pleasure in the new advances, with no getting slowed down through an excessive amount of technical element; (b) makes minimum assumptions at the reader's history; and (c) is sufficiently small in measurement not to attempt the stamina and endurance of the reader. This ebook attempts to satisfy those requisites. at the least, it's only what its name publicizes it to be -- a call for participation to the interesting global of von Neumann algebras. it's was hoping that once perusing this ebook, the reader will be tempted to fill within the quite a few (and technically, capacious) gaps during this exposition, and to delve extra into the depths of the idea. For the professional, it suffices to say the following that once a few preliminaries, the publication commences with the Murray - von Neumann type of things, proceeds throughout the uncomplicated modular idea to the III). category 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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Example text

Till further notice, assume that M is a von Neumann algebra acting on lf and that O is a cyclic and separating vector f or M. -l- Let So and Fobe the conjugate-linear operators,with domains MQ and MtCl respectively,defined (unambiguously)by So(xo) = x*$ Fo(x'a) = at*q Then So and Fo are densely defined closable operators: their closures, denoted by S and F, respectively, satisfy S = F*=FSandF=S*=S3. Proof. Since o is cyclic and separating for M as well as for M' (cf . Ex. 6)), it is clear that both ^90 and Fo are densely and u n a m b i g u o u s l y d e f i n e d .

Casc (ii): M is of type II. S i n c e M h a s a f u n d a m e n t a l s e q u e n c e ,i t f o l l o w s t h a t A d o e s n o t contain a smallest positive number. Let d = D(Xf). Infer from L e m m-

4) Let M = L-(X,T,p), and let 6 e W. with m,y, in the notation of Ex. ) (We have identified 0 ( a ) T h e e q u a t i o n v ( E ) = 0 ( l o ) d e f i n e s a f i n i t e l y a d d i t i v e m e a s u r eo n ( X , F ) , w h i c h i s a b s o l u t e l y c o n t i n u o u s w i t h r e s p e c tt o p . only if v (as above) is countably a9ditive, in which case 0(,f) = f lc dpfor some non-negative g e Lr(X,v); (c) 0 is faithful if ind only if p is absolutely continuous with D respect to v. 4) (b) is a special case of a more general fact: a positive linear functional on M is normal if and only if it is o-weakly continuous.

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