By V.S. Sunder

Why This booklet: the speculation of von Neumann algebras has been turning out to be in leaps and limits within the final two decades. It has constantly had powerful connections with ergodic concept and mathematical physics. it truly is now commencing to make touch with different components similar to differential geometry and K-Theory. There seems a robust case for placing jointly a e-book which (a) introduces a reader to a couple of the elemental concept had to have fun with the hot advances, with out getting slowed down via an excessive amount of technical element; (b) makes minimum assumptions at the reader's heritage; and (c) is sufficiently small in measurement not to try the stamina and persistence of the reader. This e-book attempts to satisfy those specifications. at least, it's only what its name declares it to be -- a call for participation to the interesting international of von Neumann algebras. it's was hoping that when perusing this booklet, the reader may be tempted to fill within the various (and technically, capacious) gaps during this exposition, and to delve additional into the depths of the speculation. For the professional, it suffices to say right here that once a few preliminaries, the publication commences with the Murray - von Neumann category of things, proceeds throughout the simple modular concept 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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**Sample text**

3(c)]. Suppose, then, that 4J is a faithful normal state on a (not necessarily factorial) von Neumann algebra M. According to Ex. 7), 4J is tracial if and only if 111lq,(x)nq,11 = 111lq,(x*)nq,1I for all x in M. So, in order to study infinite factors, it might be instructive to examine (the lack of isometry of) the operator ll4J(x)n 4J .... ll4J(x*)nq,. The advisability of such an investigation is convincingly demonstrated by the celebrated Tomita-Takesaki theorem, which provides a powerful toul for the study of infinite factors (and 46 2.

The Murray-von Neumann Classification of Factors 24 The next few lemmas lead up to a proof of the main result in this section -- that a supremum of two finite projections is again a finite projection. 5 -- are interesting in their own right. 25. 1 Nand B f X $ N. Then 11, N and B admit decompo~itions X = X~ $ X2 $ X3, N. l N3) $ + A~: ~ E dom A}, where A is a closed operator affiliated to M such that dom A = Ml' ran A = Nl , ker A = (0). Further Xl "" Bo"" Nr Proof. Simply define Xl' X2' M3, Nl , N 2' N 3 as above, and let B0.

Y*x*xy) ~ IlxIl 2dJ(y*y) = IIxll2 IITJ(y) 11 2, where we have used x*x ~ IIx1121, and so y*x*xy ~ y*(ilxlb 21)y. there exists a unique bounded operator TldJ(x) on lfdJ such TldJ(x)TJ(y) = TJ(xy) for all y in M. It is easily verified that TldJ *-algebra homomorphism of Minto :e(lf¢). As a sample, we show TldJ preserves adjoints. For this, note that TJ(M) is dense in lfq" that if x E M, then for all y,z in M,