Compact Semitopological Semigroups: An Intrinsic Theory
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1079 Wolfgang Ruppert
Compact Semitopological Semigroups: An Intrinsic Theory
Springer-Verlag Berlin Heidelberg New York Tokyo 1984
Author Wolfgang Ruppert Institut fur Mathematik und angewandte Statistik Universitat fur Bodenkultur Gregor-Mendel-Str. 33,1180 Wien, Austria
AMS Subject Classification (1980): 22A15, 22A20, 22A26, 43A60 ISBN 3-540-13387-9 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-13387-9 Springer-Verlag New York Heidelberg Berlin Tokyo
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© by Springer-Verlag Berlin Heidelberg 1984 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach / Bergstr. 2146/3140-543210
PREFACE
The present notes are intended as a complement to the existing literature on compact semi topological semigroups and semigroup compactifications; they try to present the major areas of current research on this subject strictly under the point of view of the topological algebraical strucures involved, avoiding (as much as possible) the use of methods borrowed from functional analysis. The account given is by far not complete, many problems were left open or touched only briefly; the author hopes that this will inspire others to further research.
During the preparation of these notes the author has received the encouragement and the help of many mathematicians. In particular, the author wants to express his gratitude to Professor K.H. Hofmann, who has read large parts of the manuscript; his comments and suggestions were most valuable. Special thanks go also to J.D.Lawson, D. Helmer, J.P.Troallic, J.F.Berglund, H.Junghenn and P.Milnes for many fruitful conversations, to his colleague W.Mliller for his help with the proofreading, and to Mrs. Holy, who patiently typed various earlier versions of the manuscript.
W.R.
Introduction I.
Elementary facts and constructions
1. Preliminaries 2. Idempotents and their order 3. Ideals 4. How to get new semi groups from old ones 5. Summary of the main results of chapter I 6. Notes and references to chapter I Chapter II.
8
12 19 33
41 43
Joint continuity
1. Baire machinery 2. Transport arguments
47 52
3. Consequences for the metrizable case
55
4. The general case
61
5. Getting joint continuity if the underlying space is an interval or the circle 6. Joint continuity in semigroups on suitable subsets of manifolds
80
7. A glimpse of the non-compact case 8. Summary of the main results of chapter II 9. Notes and references to chapter II
86 95 96
Chapter III.
72
Semigroup compactifications of groups
1. Generalities on compactifications of groups and semi groups 2. The associated function spaces 3. Imbedding problems and r
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