W-Algebras - ISBN:0677006756

W-Algebras

Date: 23 May 2011, 15:34

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The present set of notes treats the general theory of von Neumann algebras. I have confined the discussion to algebras acting in separable Hilbert spaces and have systematically emphasized the use of direct integral decompositions. The integral decomposition theory reduces the study of W-* algebras to the study of factors. In developing the von Neumann-Murray analysis of factors I have used the straightforward and relatively pedestrian methods of the original papers rather than the elegant but somewhat more abstract approach which one finds in the well-known work of Dixmier on this subject. The main remaining problem in the field is undoubtedly that of the structural classification of factors. Proofs of the existence of at least three non-isomorphic factors of types II and III are given in the present notes. The results presented have recently been extended very considerably by Powers in the type III case; he has succeeded in showing that there are uncountably many distinct factors of type III. Strikingly, these examples have been suggested by the quantum theory of fields, thereby verifying von Neumann's earlier feeling that W-* algebras would exhibit interesting relationships to quantum theory.

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