By V.S. Sunder

Why This e-book: the idea of von Neumann algebras has been starting to be in leaps and limits within the final twenty years. It has continually had powerful connections with ergodic thought and mathematical physics. it really is now starting to make touch with different components comparable to differential geometry and K-Theory. There looks a robust case for placing jointly a ebook which (a) introduces a reader to a couple of the elemental thought had to get pleasure from the new advances, with no getting slowed down via an excessive amount of technical element; (b) makes minimum assumptions at the reader's history; and (c) is sufficiently small in dimension not to attempt the stamina and endurance of the reader. This e-book attempts to fulfill those specifications. as a minimum, it's only what its name pronounces it to be -- a call for participation to the intriguing international of von Neumann algebras. it truly is was hoping that once perusing this booklet, the reader can be tempted to fill within the a variety of (and technically, capacious) gaps during this exposition, and to delve extra into the depths of the speculation. For the specialist, it suffices to say the following that once a few preliminaries, the publication 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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Acurg ' ( o ) * x . loN ' g ' g ' 1 ' d o r 4 u r s € ' . €-I uolrlulJeo t ry/wl lal uollsunc uolsualur(eqJ '€'I 6Z 30 l. The Murray-von Neumann Classification of Factors then the l,l's can be inductively defined. s finitenessof B. If t N/B I ) 2, set N' = 8;-if . 8. 7 will be U called a fundamentalsequencefor the type II tactor M' The following bit of notation will facilitate some of the ptools: let us agreeto write kN for any subspaceof the subsequent form N, e ... e Nn,with N i - N for all i.

FqonBJ eql sorJsll€s0 (q) tti uo ruroJ olrurJop-rluos = [rf'r] uoylenba eql (e) reourpnbsos ? Z) sesrcjaxg q l/4.. repunoq Sn) aql pellec 'uorrelrrJ IEcruqcel InJ3sn l(Jr^ € ol sulBUad uollcos lxeu eql 'uolloes srql ul ecuereadde uE so{€ru osle [111erouaE uollerEalulalll€lnruuocuoN'I'Z tc 2. xy)= O(yx) for all x,y in M: Q@xu*) = 0(x) for all x e M and unitary u e 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 .

Rorrrrri '"N '6" o ru 'E"eu-'N ti"t*=* ( ' * ' N - N - ' N o c u l s") , '("''g'z'll = l€rll osllou ueql 1 lEql etunss€ feur om ? uroz ol l€oclde,u€ '6'I'I uolllsodord Jo Joorcl eql ul sV '0N - N leql q c n s 1 5 0 1 s l s r x a e r s q l l B q l o s 1 , rT r u l € q l 6 ' I ' I u o l l r s o d o r 4 u r o r S s^\olloJ ll 'es€c e^rleuroll€ or{l uI 1,1= d 'Q = I los 'N } W JI 'JooJd s u o l l c e f o . t go l I u I C ' Z ' l EZ l. The Murray-von Neumann Classification of Factors 24 The next few lemmaslead up to a proof of the main result in this section-- that a supremumof two finite projectionsis again a finite projection.