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).
Rezensionen / Stimmen
A.C. Mewborn 1972 in Zentralblatt für Mathematik, 242.Band, p. 97: "This book is a systematic exposition of Baer*-rings, i.e. rings with involution in which every annihilator one-sided ideal is generated by a projection. In some respects it is an extension of I. Kaplansky's book "Rings of operators" [.]. The study of Baer*-rings is motivated primarily by certain kinds of algebras of linear operators on a Hilbert space and by certain aspects of lattice theory. This book has some of the flavor of both of these but the treatment is almost entirely algebraic. Motivating examples from operator algebras are used extensively. The book is divided into three parts. The first part is a general discussion of *-rings with various conditions on the partially ordered set of projections. The second part is devoted to the structure of Baer *-rings, including the decomposition into types (Types I, II, III, and finite and infinite). Part III occupies about one half of the book and is devoted to the study of finite Baer*-rings (x*x=1 implies x x* = 1). Chapter 6 introduces the notion of a dimension function and shows that every finite Baer *-ring satisfying a generalized comparability condition for projections has a unique dimension function. Chapter 7 contains a representation of a finite Baer *-ring as a subdirect product of finite Baer *-factors. The last two chapters are devoted to the construction of a 'regular hull' of a finite Baer *-ring satisfying certain conditions and to the study of matrix rings over Baer *-rings. This book is well-written and suitable for use both as a text and as a reference on the subject of Baer *-rings. It contains a large number of excellent exercises and problems of varying degrees of difficulty, including a number of open problems."
Reihe
Auflage
1st ed. 1972. Corr. 3rd printing 2010
Sprache
Verlagsort
Verlagsgruppe
Zielgruppe
Für Beruf und Forschung
Research
Illustrationen
Maße
Höhe: 241 mm
Breite: 160 mm
Dicke: 22 mm
Gewicht
ISBN-13
978-3-540-05751-2 (9783540057512)
DOI
10.1007/978-3-642-15071-5
Schweitzer Klassifikation
General Theory.- Rickart ?-Rings, Baer ?-Rings, AW*-algebras: Generalities and Examples.- Comparability of Projections.- Structure Theory.- Structure Theory of Baer ?-Rings.- Additivity of Equivalence.- Ideals and Projections.- Finite Rings.- Dimension in Finite Baer ?-Rings.- Reduction of Finite Baer ?-Rings.- The Regular Ring of a Finite Baer ?-Ring.- Matrix Rings over Baer ?-Rings.- Errata and Comments for Baer ?-Rings.- Errata and Comments for Baer ?-Rings.
