Baer *-Rings (Grundlehren der mathematischen Wissenschaften, 195) - Hardcover

Berberian, Sterling K.

Hardcover

ISBN 10: 354005751X ISBN 13: 9783540057512

Publisher: Springer, 1972

Synopsis

This book is an elaboration of ideas of Irving Kaplansky introduced in his book Rings of operators ([52], [54]). The subject of Baer *-rings has its roots in von Neumann's theory of 'rings of operators' (now called von Neumann algebras), that is, *-algebras of operators on a Hilbert space, containing the identity op- ator, that are closed in the weak operator topology (hence also the name W*-algebra). Von Neumann algebras are blessed with an excess of structure-algebraic, geometric, topological-so much, that one can easily obscure, through proof by overkill, what makes a particular theorem work. The urge to axiomatize at least portions of the theory of von N- mann algebras surfaced early, notably in work of S. W. P. Steen [84], I. M. Gel'fand and M. A. Naimark [30], C. E. Rickart 1741, and von Neumann himself [53]. A culmination was reached in Kaplansky's AW*-algebras [47], proposed as a largely algebraic setting for the - trinsic (nonspatial) theory of von Neumann algebras (i. e., the parts of the theory that do not refer to the action of the elements of the algebra on the vectors of a Hilbert space). Other, more algebraic developments had occurred in lattice theory and ring theory. Von Neumann's study of the projection lattices of certain operator algebras led him to introduce continuous geometries (a kind of lattice) and regular rings (which he used to 'coordinatize' certain continuous geometries, in a manner analogous to the introd- tion of division ring coordinates in projective geometry).

About this title

From the Back Cover

A systematic exposition of Baer *-Rings, with emphasis on the ring-theoretic and lattice-theoretic foundations of von Neumann algebras. Equivalence of projections, decompositio into types; connections with AW*-algebras, *-regular rings, continuous geometries. Special topics include the theory of finite Baer *-rings (dimension theory, reduction theory, embedding in *-regular rings) and matrix rings over Baer *-rings. Written to be used as a textbook as well as a reference, the book includes more than 400 exercises, accompanied by notes, hints, and references to the literature. Errata and comments from the author have been added at the end of the present reprint (2nd printing 2010).

Edition details

• Publisher: Springer
• Publication date: 1972
• Language: English
• ISBN 10: 354005751X
• ISBN 13: 9783540057512
• Binding: Hardcover
• Number of pages: 314