## Continuous GeometryIn 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."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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### Contents

Foundations and Elementary Properties | 1 |

Independence | 8 |

Perspectivity and Projectivity Fundamental Properties | 16 |

Perspectivity by Decomposition | 24 |

Distributivity Equivalence of Perspectivity and Projectivity | 32 |

Properties of the Equivalence Classes | 42 |

Dimensionality | 54 |

PART II | 61 |

Relations Between the Lattice and its Auxiliary Ring | 160 |

Further Properties of the Auxiliary Ring of the Lattice | 168 |

Special Considerations Statement of the Induction to be Proved | 177 |

Treatment of Case I | 191 |

Preliminary Lemmas for the Treatment of Case II | 197 |

Completion of Treatment of Case II The Fundamental Theorem | 199 |

Perspectivities and Projectivities | 209 |

Inner Automorphisms | 217 |

Theory of Ideals and Coordinates in Projective Geometry | 63 |

Theory of Regular Rings | 69 |

APPENDIX 1 | 82 |

APPENDIX 2 | 84 |

APPENDIX 3 | 90 |

Order of a Lattice and of a Regular Ring | 93 |

Isomorphism Theorems | 103 |

Projective Isomorphisms in a Complemented Modular Lattice | 117 |

Definition of LNumbers Multiplication | 130 |

Addition of LNumbers | 136 |

The Distributive Laws Subtraction and Proof that the LNumbers Form a Ring | 151 |

Properties of Continuous Rings | 222 |

RankRings and Characterization of Continuous Rings | 231 |

PART III | 239 |

Center of a Continuous Geometry | 240 |

APPENDIX 1 | 245 |

APPENDIX 2 | 259 |

Transitivity of Perspectivity and Properties of Equivalence Classes | 264 |

Minimal Elements | 277 |

LIST OF CHANGES from the 193537 Edition and comments on the text | 283 |

297 | |

### Common terms and phrases

a u b a.el anti-automorphism Appendix applies automorphism Axioms I—VI bounded operators Chapter Clearly complemented lattice complemented modular lattice completes the proof continuous Boolean algebra continuous geometry Conversely Corollary to Definition Corollary to Theorem decomposition define denote dimension function division algebra division ring established factor factor-correspondence factor-isomorphism finite follows given Hilbert space holds homogeneous basis hypothesis idempotent implies induction integer irreducible Israel Halperin L-Numbers lattice-isomorphism least upper bound left ideal Lemma matrix ring matrix units minimal Moreover Neumann original Notes perspective isomorphism principal right ideals projective geometry proof of Theorem properties prove R(ei regular ring relation replaced respectively right member rn(A satisfies Axioms semi-simple sequence Similarly subsets Suppose Theorem 2.1 unique inverse vector set whence yields Z(aa