An Invitation to von Neumann Algebras - download pdf or read online

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 transforming into in leaps and boundaries within the final twenty years. It has constantly had robust connections with ergodic conception and mathematical physics. it's now starting to make touch with different components reminiscent of 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 couple of the fundamental idea had to take pleasure in 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 endurance of the reader. This booklet attempts to fulfill those standards. at the least, it's only what its name declares it to be -- a call for participation to the interesting international of von Neumann algebras. it really is was hoping that when perusing this booklet, the reader should be tempted to fill within the a number 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 when a few preliminaries, the booklet commences with the Murray - von Neumann category of things, proceeds in the course of 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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The first inequality, of both (a) and (b), is an immediate consequence. Turn to the second: (a) The inequality [B / N] ~ ([ B /M] + 1)([M/ N] + 1) would imply the existence, inside B, of ([ B /M] + 1)([M/ N] + 1) pairwise orthogonal copies of N, and consequently, if ([ B /M] + 1) pairwise orthogonal copies of M (since M { ([M/ N] + I)N) which is a con tradiction. ) Since finiteness is inherited by M $ B from that of M and B, the desired inequality cannot be false. 10. 3. The Dimension Function 31 let 14, B n M be finite and non-zero.

We shall complete the proof by showing that if 11: M ... sm of M onto 11(M). To prove 11 is isometric it suffices thanks to the C*-identi ty to verify that 111(x) II = IIx I when x = x* E M. Fix such an x; since the norm of a self-adjoint operator is equal to its spectral radius, it is more than sufficient to show that sp l1(x) = sp x. For this, if >. r l E M, by the double commutant theorem; hence l1(x) - ). r l ). Thus, sp l1(x) f sp x. Suppose this inclusion is strict; then there exists a continuous real function I on sp x, which vanishes on sp l1(x) but not everywhere in sp x.

Conclude that D = D( B)DS. 0 . 11. 1. Then. (a) (b) P(M) .... • if {Mn} is a sequence of pairwise orthogonal subspaces (nM) and if X = eMn• then D(M) = I:D(Mn). { N # Proof. { N coupled with (a). this yields: X < N :9 D(M) < D( N ); thus. • D(M) > D( N». Since the possibilities M ~ N. X Nand M ~ N are mutually exclusive and exhaustive. as are the possibilities D(M) < D( N ). D(M) = D( N ) and D(M) > D( N ). the reverse implication in (a) follows. For finite sequences. the assertion (b) is a consequence of the assumed finite additivity (cf.

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

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