Compact Semitopological Semigroups and Weakly Almost Periodic Functions
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42
J. F. Berglund
K. H. Hofmann
Tulane University, New Orleans
1967
Compact Semitopological Semigroups and Weakly Almost Periodic Functions
Springer-Verlag· Berlin· Heidelberg· New York
This work was supported in part by NSF GrantGP6219. The second author is a Fellow of the Alfred P. Sloan Foundation
All rights, especially that of translation into foreign languages, reserved. It is also forbidden to reproduce this book, either whole or in part, by photomechanlcal means (photostat, microfilm and/or microcard) or by other procedure without written permission from Springer Verlag. C by Springer·Verlag Berlin' Heidelberg 1967. Ubruy of Congress Cltalog Card Number 67 -29251. Printed in Germany. Title No. 7362
TABLE OF CONTENTS
INTRODUCT ION
CHAPTER I.
1
PRELIMINARIES
1. Compactness Criteria ••••••••••.•••••••••••••.••••• 12 Theorem 1.8
Equivalent conditions for compactness in function spaces.
16
2. Equicontinuous Semigroups of Linear Operators and Affine Transformations. Affine Semigroups •••• 21 Theorem 2.9
26
Theorem 2.10
27
The almost periodic subspace. The weakly almost periodic subspace.
Proposition 2.13 .......•...............•........ 30 Kakutani fixed point theorem. Theorem 2.16
34
3.. Ellis' Theorem
36
Ryll-Nardzewski fixed point theorem.
4. Actions of Compact Groups on Topological Vee tor Spaces
Proposition 4.5 A Banach weak G-module is a strong G-module if G is a locally compact group.
CHAPTER II.
37
41
COMPACT SEMITOPOLOGICAL SElJIIGROUPS
1. Algebraic Background Material ••••••••••••••••••••• 44 Proposition 1.9 The Rees Theorem.
47
Propos 1tlon 1.23
57
The group supporting subspace.
Proposition 1.26 ....•........................... 59 The semigroup with zero supporting SUbspace.
2. Locally Compact Paragroups •••••••••••••••••••••••• 60 Proposition 2.4 .......•..•..•................... 61 The structure of a minimal ideal in a locally compact semitopological semigroup.
3. Compact Semitopological Semigroups •••••••••••••••• 65 3.5 The first fundamental theorem of compact semi topological semlgroups.
67
Theorem
Proposition 3.12 ....•.......................•.•. 71 The strongly almost periodic subspace. Theorem 3 . 23
The main theorem on semigroups of operators on a Banach space.
80
4. Invariant Measures on Locally Compact Semigroups •• 88 Theorem 4.14 Necessary and sufficient conditions for invariance of a measure.
CHAPTER III.
97
PERIODIC AND WEAKLY ALMOST PERIODIC FUNCTIONS ON SEMITOPOLOGICAL SEMIGROUPS
1. Various Universal Functors ••••••••••••••••••••••• 112 2. The Definition of Almost Periodic Functions .••••• 120 Proposition 2.10 ...........•.....•..•.......... 126 Decomposition of weakly almost periodic functions.
3.. Invariant Means Theorem
3 .. 2
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127 127
4. Locally Compact Semitopological Semigroups ••••••• 130 Proposition 4.5 132 Necessary and sufficient conditions for the embedding into the weakly almost periodic compactiflcation to be topological.
Proposition 4.6 •••••
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