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

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.

© 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