By V.S. Sunder

Why This publication: the idea of von Neumann algebras has been growing to be in leaps and limits within the final twenty years. It has continually had robust connections with ergodic concept and mathematical physics. it really is now starting to make touch with different components akin to differential geometry and K-Theory. There looks a powerful case for placing jointly a ebook which (a) introduces a reader to a few of the fundamental idea had to take pleasure in the hot advances, with out getting slowed down via an excessive amount of technical aspect; (b) makes minimum assumptions at the reader's heritage; and (c) is sufficiently small in measurement not to try the stamina and endurance of the reader. This publication attempts to fulfill those necessities. as a minimum, it is only what its name broadcasts it to be -- a call for participation to the fascinating international of von Neumann algebras. it truly is was hoping that when perusing this publication, the reader can be tempted to fill within the various (and technically, capacious) gaps during this exposition, and to delve additional into the depths of the idea. For the specialist, it suffices to say the following that when a few preliminaries, the e-book commences with the Murray - von Neumann category of things, proceeds throughout the easy modular thought 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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T B / N " l +I D ( B ) 1 B l N" l ' let n - - , recall that I B/N"] ' o and conclude that D(X) = D(8 )D3 (m). t) = DS (lv1)= o for infinite l( conclude that D =D(x)Ds. o Proposition l3-f f. , t,f (Jtl")is a sequenceof pairwise (nl4)and if 14= 6t4n,then D(M)= ID(M"). orthogonalsubspaces Proof. , D(l't) > -N and M N are mutually } { N,x D ( N ) ) . S i n c et h e p o s s i b i l i t i e s M = exclusiveand exhaustive,as are the possibilitiesD(J't)< D( N )' D(1"1) (a) in follows. D(N ) and D(X) > D(N ), the reverseimplication For finite sequences,the assertion (b) is a consequenceof the assumedfinite additivity (cf.

Clearly, then, the net (xt: d e 1) converges weakly to x. Exercises ( 0 . 4 . 1 2 ) . I f [ . , . ,,n]= t i\q + ikn, q,+ ikn1. 13). Let (x,: i e I\ be a monotone increasing net of self-adjoint operatorson lf and let x = lim xt (as in Prop. 4'll). ) '2 uer = p '[44r] araqn n = (a)c uer uaql '(74/)d t a pus erqaEle uu€runeN uol € sI l{ JI ( ' p u o c e so q l _ s e l l d r y lp u e . e sr setqoEle uueruneN uo^ Jo flrrue3 fuu go uollcosrelul aql leql uaroeql luelnrutuoc elqnop oql Jo ecuenbosuoc ,{see uB sl lI 'Q[ (n)d.

N It follows from the double commutant theorem that if l'1is a closed subspace, then I\nM if and only if py e M. In general, there exists s e v e r a l n o n - c l o s e d s u b s p a c e sa f f i l i a t i d t o M ; i f , f o r i n s t a n c e , t h e r e e x i s t s a i n M s u c h t h a t r a n a i s n o t c l o s e d ,t h e n r a n a w o u l d b e s u c h an example. To deal with such subspaces,it becomes necessary to deal with unbounded operators. In this context, the following d e f i n i t i o n s u p p l e m e n t sD e f i n i t i o n 1 .