Continuous Geometry / John von Neumann.

Format:

Book

Author/Creator:

von Neumann, John, author.

Contributor:

Halperin, Israel.

Series:

Princeton landmarks in mathematics and physics.

Princeton Landmarks in Mathematics and Physics

Language:

English

Subjects (All):

Topology.

Continuous groups.

Geometry, Projective.

Physical Description:

1 online resource (316 p.)

Place of Publication:

Princeton, NJ : Princeton University Press, [2016]

Language Note:

In English.

Summary:

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.

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 / Halperin, Israel

Index

Notes:

Includes index.

Description based on online resource; title from PDF title page (publisher's Web site, viewed 08. Jul 2019)

ISBN:

9781400883950

1400883954

OCLC:

948779995