Measures and Hilbert Lattices

261 Pages·1986·6.889 MB·English

by  G. Kalmbach

MEASURES AND HILBEKT LATTICES
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MEASURES AND HILBERT
LATTICES
G. Kalmbach
Universitdt Ulm
World Scientific
Published by
World Scientific Publishing Co. Pte. Ltd.
P. 0. Box 128, Farrer Road, Singapore 9128
Library of Congress Cataloging-in-Publication data is available.
MEASURES AND HILBERT LATTICES
Copyright © 1986 by World Scientific Publishing Co Pte Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced
in any form or by any means, electronic or mechanical, including photo­
copying, recording or any information storage and retrieval system now
known or to be invented, without written permission from the Publisher.
ISBN 9971-50-009-4
Printed in Singapore by Singapore National Printers (Pte) Ltd.
TO S. NAEDA
in admiration.
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Vll
P R E F A CE
This book has two parts. In section 1 to 6 clas­
sical measure theory is extended to the theory of
measures and states on orthomodular lattices.
Examples of such structures are given by the
Boolean a-algebras and the "Hilbert lattices" whose
elements are the closed subspaces of a Hilbert space.
In section 2 the basic definitions and properties
of measures, probabilities and states are given, to­
gether with Maeda's characterization of probabilities
with a support.
Maeda's result is used in section 3 to prove
Gleason's theorem. In this theorem a one-to-one cor­
respondence between probabilities and von Neumann op­
erators is established on a separable Hilbert space.
In section 4 Jordan- and Hahn-decompositions are
studied for the generalized measure concept.
Section 5 contains a spectral theorem for obser-
vables. The occupation with sets of states which have
prescribed properties is the connecting idea between
section 5 and 6. There are 2 ° different equational
classes of orthomodular lattices which have a large
set of states.
In section 7 to 12 of this book special properties
of complete orthomodular lattices are investigated
and characterizations of factors and Hilbert lattices
are given.
A complete orthomodular lattice has maximal factors
which are Boolean, locally modular, atomfree or mul­
tiplicity-free (Section 7).
viii Preface
Hilbert lattices are special dimension lattices.
They are studied in sections 8, 9 and 11. Among com­
plete orthomodular lattices the dimension lattices
are characterized by MacLanes's geometrical exchange
axiom.
Hilbert lattices L have a modular ideal of finite-
dimensional subspaces whose suprema generate all ele­
ments of L. Locally modular lattices have axiomati-
cally this property. They are characterized in sec­
tion 8 as the locally finite dimension lattices.
The Birkhoff-von Neumann theorem is the central
part of section 9. It establishes a one-to-one cor­
respondence between inner products f of a finite-
dimensional vector space V over a skew field D and
orthocomplementations on the lattice of subspaces
of V.
Varadarajan' s coordinatiz'ation theorem is outlined
in section 10 (Varadarajan 1968). It describes the
connection between certain dimension lattices L and
vector spaces V over skew fields D. From a line in L
and an independent set of atoms C in L the coordina­
tes D and the vector space V of functions from C to D
are constructed such that V carries an inner product
f and L is isomorphic to the lattice of f-closed sub-
spaces of V.
Section 11 contains Rakutani-Mackey•s lattice char­
acterization of real or complex Hilbert spaces. Hil­
bert spaces are characterized among Banach spaces by
the property that the lattice of closed subspaces
carries an orthocomplementation. Pre-Hilbert spaces
always have such an orthocomplementation on their
subspace-lattice L. The orthomodularity of L
Preface
IX
characterizes Hilbert spaces among pre-Hilbert
spaces.
Gross has defined orthomodular spaces (V,f) as in­
ner product spaces which satisfy the projection theo­
rem V = U +U for UcV. A particular orthomodular
space constructed by Keller shows that Hilbert lat­
tices do not force the coordinatizing skew field of
section 10 to be R, the complex numbers or the qua­
ternions. This was a long outstanding conjecture in
the attempt to characterize Hilbert spaces by their
lattices of closed subspaces. Keller's example of a
non-classical Hilbert space gives a new perspective
to the whole theory.
This book is a continuation of the author's book
on orthomodular lattices. Orthomodular lattice theory
is reviewed briefly in the introduction. Definitions
are repeated, but for longer proofs the reference is
Kalmbach 1983. The reader may observe that "7.3" or
"proposition 4" as reference means the theorem (or
lemma or definition) 3 in section 7 or proposition 4
in the same section 8 while "Kalmbach 1983, 6.9" re­
fers to theorem 9 in section 6 of the 1983-book on
orthomodular lattices.
A summary is printed in italics at the beginning of
each section and bibliographical references are at
the end with the name of the author and the publica­
tion date of his or her article attached. A newly de­
fined concept mostly is printed in italics,
I wish to thank the following persons for their
help and stimulating interest in getting this book-
project completed:
H. Gensheimer, -from his unpublished articles, quo-