Continuous Geometry / / John von Neumann.

• Superior document: Title is part of eBook package: De Gruyter Princeton University Press eBook-Package Archive 1927-1999
• VerfasserIn: von Neumann, John,
• MitwirkendeR: Halperin, Israel,
• TeilnehmendeR: Halperin, Israel.
• Place / Publishing House: Princeton, NJ : : Princeton University Press, , [2016]; ©1998
• Year of Publication: 2016
• Language: English
• Series: Princeton Landmarks in Mathematics and Physics ; 46
• Physical Description: 1 online resource (312 p.)

In his work on rings of operators in Hilbert space, John von Neumann discovered a new mathematical structure that resembled the lattice system Ln. In characterizing its properties, von Neumann founded the field of continuous geometry. This book, based on von Neumann's lecture notes, begins with the development of the axioms of continuous geometry, dimension theory, and--for the irreducible case--the function D(a). The properties of regular rings are then discussed, and a variety of results are presented for lattices that are continuous geometries, for which irreducibility is not assumed. For students and researchers interested in ring theory or projective geometries, this book is required reading.

Table of Contents

Frontmatter -- Foreword -- Table of Contents -- Part I -- Chapter I. Foundations and Elementary Properties -- Chapter II. Independence -- Chapter III. Perspectivity and Projectivity. Fundamental Properties -- Chapter IV. Perspectivity by Decomposition -- Chapter V. Distributivity. Equivalence of Perspectivity and Projectivity -- Chapter VI. Properties of the Equivalence Classes -- Chapter VII. Dimensionality -- Part II -- Chapter I. Theory of Ideals and Coordinates in Projective Geometry -- Chapter II. Theory of Regular Rings -- Chapter III. Order of a Lattice and of a Regular Ring -- Chapter IV. Isomorphism Theorems -- Chapter V. Projective Isomorphisms in a Complemented Modular Lattice -- Chapter VI. Definition of L-Numbers; Multiplication -- Chapter VII. Addition of L-Numbers -- Chapter VIII. The Distributive Laws, Subtraction; and Proof that the L-Numbers form a Ring -- Chapter IX. Relations Between the Lattice and its Auxiliary Ring -- Chapter X. Further Properties of the Auxiliary Ring of the Lattice -- Chapter XI. Special Considerations. Statement of the Induction to be Proved -- Chapter XII. Treatment of Case I -- Chapter XIII. Preliminary Lemmas for the Treatment of Case II -- Chapter XIV. Completion of Treatment of Case II. The Fundamental Theorem -- Chapter XV. Perspectivities and Projectivities -- Chapter XVI. Inner Automorphism -- Chapter XVII. Properties of Continuous Rings -- Chapter XVIII. Rank-Rings and Characterization of Continuous Rings -- Part III -- Chapter I. Center of a Continuous Geometry -- Chapter II. Transitivity of Perspectivity and Properties of Equivalence Classes -- Chapter III. Minimal Elements -- List of Changes from the 1935-37 Edition and comments on the text -- Index