Download e-book for kindle: An Invitation to von Neumann Algebras by V.S. Sunder

By V.S. Sunder

ISBN-10: 0387963561

ISBN-13: 9780387963563

ISBN-10: 1461386691

ISBN-13: 9781461386698

Why This booklet: the speculation of von Neumann algebras has been starting to be in leaps and limits within the final twenty years. It has consistently had robust connections with ergodic idea and mathematical physics. it really is now starting to make touch with different parts akin to differential geometry and K-Theory. There appears to be like a robust case for placing jointly a booklet which (a) introduces a reader to a few of the fundamental concept had to delight in the new advances, with out getting slowed down by way of an excessive amount of technical aspect; (b) makes minimum assumptions at the reader's heritage; and (c) is sufficiently small in dimension not to attempt the stamina and endurance of the reader. This publication attempts to satisfy those necessities. as a minimum, it's only what its name broadcasts it to be -- a call for participation to the interesting international of von Neumann algebras. it really is was hoping that when perusing this ebook, the reader will be tempted to fill within the various (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 when a few preliminaries, the publication commences with the Murray - von Neumann type of things, proceeds during the easy modular thought to the III). class 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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CO [~ ] /' + co ; and [B INn] exists and is a finite positive number. Proof. 9 (a), for any n ;.. 1, we have [t ] ;. [t ] [:] ~ 2n - l [ } ]; n since { N l/X] is a fixed finite integer it follows that [N niX] = 0 for all sufficiently large n; thus, there exists an integer no such that X 1 N n for all n ;.. no; so, if n ;.. no' Nn 1 11; in other words [XI Nn ] ;.. 1. Then, for any integer k, [N X N [Nno [ro-] ;. 2k-l, no+k X ];.. no+k ] and (a) is proved. 9 (a), we see that if n is so large that [ B INn] ;..

Since N is not minimal (M being of type II), there exists B TIM such that (0) '# BeN; the finiteness of N ensures finiteness of B. If [ N/B ] ~ 2, set N' = B; if [N/B]=I--notethat[N/B]>O--then N= B $:Rwith:R{ B; further :R '# (0) since B '# N; note that [ N /:R] ~ 2 and set N' =:R. 8. 7 will be called a fundamental sequence for the type II factor M. 0 The following bit of notation will facilitate some of the subsequent proofs: let us agree to write kN for any subspace of the form Ni $ ... $ Nk , with Ni"'" N for all i.

Since ball M = {x € M: Ix II ~ I} is a-weakly compact and since 11 is isometric, infer that ball l1(M) = l1(ball M) is a-weakly compact. , for instance, [Yos]) states that a linear subspace of a (Banach-) dual space is weak*-closed if and only if its unit ball is weak*-closed. Conclude that Tr(M) is a-weakly closed and consequently a von Neumann algebra. ) Since 11- 1: Tr(M) ... ) complete. 0 2. The Tomita-Takesaki Theory 42 It may be relevant to point out here that, more generally than was established in the above proof, it is true that an Injective *-homomorphism between two (possibly abstract) C*-algebras is isometric; it may be inferred from this -- by passing through an appropriate quotient algebra -- that a *-homomorphic image of a C*-algebra is norm-closed.

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

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