Continuous Geometry

by Von Neumann, John

• ISBN13: 9780691058931
• ISBN10: 0691058938
• Format: Paperback
• Copyright: 1998-04-20
• Publisher: Princeton Univ Pr

Summary

In his work on rings of operators in Hilbert space, John von Neumann discovered a new mathematical structure that resembled the lattice systemLn. 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

FOREWORD	v
Israel Halperin
PART I	1	(62)
CHAPTER I. Foundations and Elementary Properties	1	(7)
CHAPTER II. Independence	8	(8)
CHAPTER III. Perspectivity and Projectivity Fundamental Properties	16	(8)
CHAPTER IV. Perspectivity by Decomposition	24	(8)
CHAPTER V. Distributivity Equivalence of Perspectivity and Projectivity	32	(10)
CHAPTER VI. Properties of the Equivalence Classes	42	(12)
CHAPTER VII. Dimensionality	54	(9)
PART II	63	(177)
CHAPTER I. Theory of Ideals and Coordinates in Projective Geometry	63	(6)
CHAPTER II. Theory of Regular Rings	69	(24)
Appendix 1	82	(2)
Appendix 2	84	(6)
Appendix 3	90	(3)
CHAPTER III. Order of a Lattice and of a Regular Ring	93	(10)
CHAPTER IV. Isomorphism Theorems	103	(14)
CHAPTER V. Projective Isomorphisms in a Complemented Modular Lattice	117	(13)
CHAPTER VI. Definition of L-Numbers; Multiplication	130	(6)
Appendix	133	(3)
CHAPTER VII. Addition of L-Numbers	136	(15)
Appendix	148	(3)
CHAPTER VIII. The Distributive Laws, Subtraction; and Proof that the L-Numbers form a Ring	151	(9)
Appendix	158	(2)
CHAPTER IX. Relations Between the Lattice and its Auxiliary Ring	160	(8)
CHAPTER X. Further Properties of the Auxiliary Ring of the Lattice	168	(9)
CHAPTER XI. Special Considerations. Statement of the Induction to be Proved	177	(14)
CHAPTER XII. Treatment of Case I	191	(6)
CHAPTER XIII. Preliminary Lemmas for the Treatment of Case II	197	(2)
CHAPTER XIV. Completion of Treatment of Case II. The Fundamental Theorem	199	(10)
CHAPTER XV. Perspectivities and Projectivities	209	(8)
CHAPTER XVI. Inner Automorphisms	217	(5)
CHAPTER XVII. Properties of Continuous Rings	222	(9)
CHAPTER XVIII. Rank-Rings and Characterization of Continuous Rings	231	(9)
PART III	240	(43)
CHAPTER I. Center of a Continuous Geometry	240	(24)
Appendix 1	245	(14)
Appendix 2	259	(5)
CHAPTER II. Transitivity of Perspectivity and Properties of Equivalence Classes	264	(13)
CHAPTER III. Minimal Elements	277	(6)
LIST OF CHANGES from the 1935-37 Edition and comments on the text	283	(14)
Israel Halperin
INDEX	297