By V.S. Sunder

Why This publication: the idea of von Neumann algebras has been becoming in leaps and limits within the final twenty years. It has consistently had robust connections with ergodic idea and mathematical physics. it's now commencing to make touch with different parts reminiscent of differential geometry and K-Theory. There seems a robust case for placing jointly a publication which (a) introduces a reader to a couple of the elemental idea had to relish the hot advances, with out getting slowed down through an excessive amount of technical aspect; (b) makes minimum assumptions at the reader's heritage; and (c) is sufficiently small in measurement not to try out the stamina and endurance of the reader. This e-book attempts to satisfy those requisites. at the least, it's only what its name announces it to be -- a call for participation to the fascinating international of von Neumann algebras. it really is was hoping that when perusing this e-book, the reader will 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 idea. For the specialist, it suffices to say right here that when a few preliminaries, the e-book commences with the Murray - von Neumann category of things, proceeds during the simple modular concept 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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So, there exists :Ro C N! ,. :Ro. ,. CM, J = contradicting the finiteness of M. Both assertions follow from the non-existence of a T as above for any pair of admissible decompositions. 3. 6. If 11, N TI M are both non-zero and finite, let [M/ N ] denote the uniquely determined integer card I, as in Prop. 5. 0 Note that in the example M = l(Jf), [M/ N] is the greatest integer which does not exceed dim M/dim N, so the similarity with the notation for the greatest integer function ({t] = n iff n , t < n + 1) is not an accident.

If x e M, then x e M+ if and only If CP(x) "'0 for all cP in M. +, and similarly, the dual statement (with the roles of cP and x interChanged) is also valid. (As above, we shall think of the elements of M. ,+' let x e M and let Mo = (x}", the von Neumann algebra generated by x and 1. (a) If x is normal, the equation vX

Before doing that, however, it will help to examine the quantitative aspects of the Euclidean algorithm established earlier (cf. Prop. 3). 5. Let M, N T) M; suppose N '" (0) and M is finite. ,. N for all i E I and :R ~ N (as in Prop. 3), the index set I is finite and its cardinality is independent of the particular decomposition chosen. Proof. ) $:R' is another such decomposition and suppose, if possible, Jthat J there exists a map T: I .. J which is injective but not surjective. ' . So, there exists :Ro C N!