CONTINUOUS GEOMETRY PRINCETON LANDMARKS IN MATHEMATICS AND PHYSICS Non-standard Analysis, by Abraham Robinson General Theory of Relativity, by P.A.M. Dirac Angular Momentum in Quantum Mechanics, by A. R. Edmonds Mathematical Foundations of Quantum Mechanics, by John von Neumann Introduction to Mathematical Logic, by Alonzo Church Convex Analysis, by R. Tyrrell Rockafellar Riemannian Geometry, by Luther Pfahler Eisenhart The Classical Groups, by Hermann Weyl Topology from the Differentiable Viewpoint, by John W. Milnor Algebraic Theory of Numbers, by Hermann Weyl Continuous Geometry, by John von Neumann Linear Programming and Extensions, by George B. D antzig Operator Techniques in Atomic Spectroscopy, by Brian R. Judd CONTINUOUS GEOMETRY BY John von Neumann FOREWORD BY ISRAEL HALPERIN PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY Copyright © 1960 by Princeton University Press Copyright renewed © 1988 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, Chichester, West Sussex L.C. Card: 59-11084 ISBN 0-691-05893-8 Princeton University Press books are printed on acid-free paper and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources First printing, in the Princeton Landmarks in Mathematics and Physics series, 1998 http://pup.princeton.edu Printed in the United States of America 1 3 5 7 9 10 8 6 4 2 FOREWORD This book reproduces the notes of lectures on Continuous Geometry given by John von Neumann at Princeton. Part I was given during the academic year 1935—36, and Parts II and III were given during the academic year 1936—37. The notes were prepared, while the lectures were in progress, by L. Roy Wilcox, and multigraphed copies were distributed by the Institute for Advanced Study. The supply was soon exhausted, and the notes have not been reproduced until now. In the present edition many slips in typing have been corrected, in Part I with the help of Wallace Givens. I have inserted a few editorial remarks and I have made a small number of changes in the text. These changes, together with some comments, are listed at the back of the book. Of these changes, only one is essential and it was authorized explicitly by von Neumann. Continuous geometry was invented by von Neumann in the fall of 1935. His previous work on rings of operators in Hilbert space, partly in collabo­ ration with F. J. Murray, had led to the discovery of a new mathematical structure which possessed a dimension function (cf. F. J. Murray and J. von Neumann, On Rings of Operators, Annals of Mathematics, vol. 37 (1936), page 172, Case (Hi)). The new structure had incidence properties resembling those of the system Ln (Ln denotes the lattice of all linear sub­ sets of an n —1 dimensional projective geometry), but its dimension func­ tion assumed as values all real numbers in the interval (0, 1). Von Neumann set out to formulate suitable axioms to characterise the new structure. It happened that just previously, K. Menger and G. Birk- hoff had characterised Ln by lattice-type axioms; in particular, Birkhoff had shown the structures Ln could be characterised as the complemented modular irreducible lattices which satisfy a chain condition. Von Neu­ mann dropped the chain condition and replaced it by two of its weak consequences: (i) order completeness of the lattice (Axiom II on page 1 of this book), and (ii) continuity of the lattice operations (Axiom III on page 2 of this book). Lattices which are complemented, modular, irreducible, satisfy (i) and (ii), but do not satisfy a chain condition, were called by von Neumann: continuous geometries (reducible continuous geometries were considered later, in Part III). It is easy to see that in a continuous geometry there can be no minimal element — that is, no atomic element or point. The structure previously discovered by Murray and von Neumann in their research on rings of operators was an example, the first, of a con­ tinuous geometry. Von Neumann’s first fundamental result was the construction, for an arbitrary continuous geometry, of a dimension function with values rang­ ing over the interval (0, 1). The construction was based on the definition: x and y are to be called equidimensional if x and y are in perspective rela­ tion, that is: for some w the lattice join and meet of x with w are identical with those of y with w. The essential difficulty is to prove that the per­ spective relation is transitive. Part I of this book discusses the axioms for continuous geometry and gives the construction of the dimension function. These results were sum­ marized in von Neumann’s note, Continuous Geometry, Proceedings of the National Academy of Sciences, vol. 22 (1936), pages 92—100. In a second note, Examples of Continuous Geometries, Proceedings of the National Academy of Sciences, vol. 22 (1936), pages 101—108, von Neumann used a simple device to construct an extensive class of new con­ tinuous geometries. He started with finite dimensional vector spaces over an arbitrary but fixed division ring 3 and showed how to imbed an n- dimensional vector space in a ^-dimensional one with k ^ 2; a countable number of repetitions of this imbedding procedure, followed by a metric completion, gives a continuous geometry. This construction uses a coun­ table number of factors k, each ^ 2, but the final result is independent of the particular choice of the k. This was shown by von Neumann in a manuscript (still unpublished): Independence of From The Sequence v (MS written in 1936—37). If 3 is the ring of all complex numbers, the continuous geometry obtained by the imbedding procedure is non-isomorphic to, and simpler than, the continuous geometry obtained from a ring of operators. This non-isomorphism is established in von Neumann’s paper The Non-Iso­ morphism Of Certain Continuous Rings, Annals of Mathematics, vol. 67 (1958), pages 485—496 (MS actually written in 1936—37). The imbedding procedure, the new examples, and the non-isomorphism theorem are not mentioned in the present book. Von Neumann’s next fundamental result was a deep and technically remarkable generalization of the classical Hilbert - Veblen and Young coordinatization theorem. This classical theorem asserts that if n ^ 4, then the points of Ln can be coordinatized with homogeneous coordinates, vi the coordinates to be taken in some suitable division ring.®. Von Neumann first expressed this classical theorem in the following two equivalent forms which do not mention points explicitly and which coordinatize all the linear subsets in Ln: Ln, as a lattice, is isomorphic to (i) the lattice of all right ideals in ®n («®n denotes the ring of n-th order matrices with elements in Sf) and to (ii) the lattice of all right submodules of Q)n (<®n denotes the right module over ® of all w-tuples (a;1, • • *, xn) with all x* in ®). Then he established these coordinatization-isomorphism theorems for every complemented modular lattice of order n ^ 4 (this restriction is a lattice generalization of the condition n ^ 4 for Ln). But in formulating the general coordinatization theorem, von Neumann made some changes: first, the division ring ® was replaced by a more general ring © with the properties: © has an identity element and for each x in @, xyx = x for some y in © (such rings © were called by von Neumann: regular)] second­ ly, in the isomorphism with right ideals in ©n only principal right ideals were to be used; finally, in the isomorphism with right submodules of ©n only those right submodules were to be used which are spanned by a finite number of vectors. The properties of regular rings and the proof of the coordinatization- isomorphism theorem are the main content of Part II of this book. This material was summarized in two notes: On Regular Rings, Proceedings of the National Academy of Sciences, vol. 22 (1936), pages 707—713; and Algebraic Theory of Continuous Geometries, Proceedings of the National Academy of Sciences, vol. 23 (1937), pages 16—22. The coordinatization is carried out and the ring 3? = ©n (©n is a regular ring along with ©) is obtained, without using any of the following axioms: completeness of the lattice, continuity of the lattice operations, irreducibility. However, if these axioms do hold, the ring 91 has special properties: a rank function R(x) can be defined for all x in 91. This rank function leads to a natural metric in 91 and 91 is topologically complete with respect to this metric. Rings with a rank function were called by von Neumann: rank rings, and when complete with respect to the rank metric: continuous rings. Rank rings and continuous rings are studied in Chapters XVII and XVIII of this book. Chapter IV of Part II considers complemented modular lattices L of order n ^ 3. If L can be coordinatized by a regular ring 91 (the coordinati­ zation theorem shows that n ^ 4 is sufficient), then as von Neumann shows, vii every lattice isomorphism of L is generated by a ring isomorphism of 9t, and every dual-automorphism of L can be generated by an anti-automorph­ ism of 9T If the anti-automorphism of L is an orthogonalization, then the corresponding anti-automorphism of 91 is a Hermitian conjugation: x -> x*, which is non-singular in the sense that x*x = 0 implies x = 0. The continuous geometries obtained from rings of operators do, in fact, possess such orthogonalizations; the Hermitian conjugation in the co- ordinatizing continuous ring coincides with the operation of taking the Hermitian adjoint operator in Hilbert space.