Download PDF by V.S. Sunder: An Invitation to von Neumann Algebras

By V.S. Sunder

ISBN-10: 0387963561

ISBN-13: 9780387963563

ISBN-10: 1461386691

ISBN-13: 9781461386698

Why This publication: the idea of von Neumann algebras has been turning out to be in leaps and boundaries within the final twenty years. It has consistently had robust connections with ergodic concept and mathematical physics. it's now commencing to make touch with different parts reminiscent of differential geometry and K-Theory. There seems a powerful case for placing jointly a e-book which (a) introduces a reader to a few of the elemental conception had to have fun with the hot advances, with out getting slowed down by way of an excessive amount of technical element; (b) makes minimum assumptions at the reader's historical past; and (c) is sufficiently small in measurement not to try the stamina and endurance of the reader. This booklet attempts to satisfy those standards. as a minimum, it is only what its name pronounces it to be -- a call for participation to the fascinating international of von Neumann algebras. it's was hoping that once perusing this ebook, the reader should be tempted to fill within the quite a few (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 e-book commences with the Murray - von Neumann type of things, proceeds in the course of the uncomplicated modular conception to the III). type 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

The collection of positive operators in M will be denoted by M+. The positive cone M+ defines an order on the real vector space Mh of self-adjoint elements of M, whereby x , y precisely when y - x E M+. Thus, for instance, x e M and x ~ 0 are equivalent, as of course they should be. linear functional ~ on M is said to be positive if ~x*x) ~ 0 for all x in M, or, equivalently, if ~M+) f IR+. The collection of such ~ will be denoted by Aft. An element ~ of ~ is called a state if it is normalized so that ~1) = I.

MIN and N 1 M imply M ..... N . Proof. Let u: M ..... M' f N and v: N ..... N f 11 Set M1 = wM, where w = vu. It is clear that w e M and that w'h is a partial isometry for each n (since w maps M isometrically into itself) such that wn : Mo ..... Mn for each n ;iI O. where Mo = M and Mn = wnMo. Since No f 11, we also have wnPNo: No..... Nn for each n ~ O. where Nn = wn No; consequently. for n wn(pM o ;iI O. - PN 0 ): (Mo Q No) '" (Mn Q Nn). where we write M Q N for M r'I ~. • and so r'lnMn = r'ln N n = :R (say).

The Murray-von Neumann Classification of Factors B = BO$M2$ N2 ",M1 $M2 $ N2 1 M1 $M2 $M3 =M. Case (ii): M3 1 N2' Regarding A as a closed densely defined operator from MI. to Nl' let A+ denote the closed densely defined operator from Nl to M1 which is the adjoint of A. Then A+, viewed as an operator in If, is clearly affiliated to M; further, from the general fact about the graph of the adjoint, it is clear that (M1 $ N1 ) 9 Bo = {-n + A+n: n E dom A+}. 5, it may be seen that N1 '" «M1 $ N1 ) 9 Bo) '" M1; hence, (M $ N) 9 B = «M1 $ N1) 9 Bo) $ M3 $ N3 1 N1 $ N2 N3 $ = N.

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

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