Continuous Geometry

By: John von Neumann

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Paperback | 10 May 1998

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312 Pages

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25.4 x 19.7 x 254

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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.

Industry Reviews

"This historic book should be in the hands of everyone interested in rings and projective geometry."--R. J. Smith, The Australian Journal of Science "Much in this book is still of great value, partly because it cannot be found elsewhere ... partly because of the very clear and comprehensible presentation. This makes the book valuable for a first study of continuous geometry as well as for research in this field."--F. D. Veldkamp, Nieuw Archief voor Wiskunde

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Foreword
Foundations and Elementary Properties	p. 1
Independence	p. 8
Perspectivity and Projectivity. Fundamental Properties	p. 16
Perspectivity by Decomposition	p. 24
Distributivity. Equivalence of Perspectivity and Projectivity	p. 32
Properties of the Equivalence Classes	p. 42
Dimensionality	p. 54
Theory of Ideals and Coordinates in Projective Geometry	p. 63
Theory of Regular Rings	p. 69
	p. 82
	p. 84
	p. 90
Order of a Lattice and of a Regular Ring	p. 93
Isomorphism Theorems	p. 103
Projective Isomorphisms in a Complemented Modular Lattice	p. 117
Definition of L-Numbers; Multiplication	p. 130
Appendix	p. 133
Addition of L-Numbers	p. 136
Appendix	p. 148
The Distributive Laws, Subtraction; and Proof that the L-Numbers form a Ring	p. 151
Appendix	p. 158
Relations Between the Lattice and its Auxiliary Ring	p. 160
Further Properties of the Auxiliary Ring of the Lattice	p. 168
Special Considerations. Statement of the Induction to be Proved	p. 177
Treatment of Case I	p. 191
Preliminary Lemmas for the Treatment of Case II	p. 197
Completion of Treatment of Case II. The Fundamental Theorem	p. 199
Perspectivities and Projectivities	p. 209
Inner Automorphisms	p. 217
Properties of Continuous Rings	p. 222
Rank-Rings and Characterization of Continuous Rings	p. 231
Center of a Continuous Geometry	p. 240
Appendix 1	p. 245
Appendix 2	p. 259
Transitivity of Perspectivity and Properties of Equivalence Classes	p. 264
Minimal Elements	p. 277
List of Changes from the 1935-37 Edition and comments on the text by Israel Halperin	p. 283
Index	p. 297
Table of Contents provided by Publisher. All Rights Reserved.

Continuous Geometry

At a Glance

Paperback

Industry Reviews

Shipping

How to return your order

Defective items

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