Compact Right Topological Semigroups and Generalizations of Almost Periodicity
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663 J. F. Berglund H. D. Junghenn p. Milnes
Compact Right Topological Semigroups and Generalizations of Almost Periodicity
Springer-Verlag Berlin Heidelberg New York 1978
Authors John F. Berglund Virginia Commonwealth University Richmond, Virginia 23284/USA
Hugo D. Junghenn George Washington University Washington, D.C. 20052/USA Paul Milnes The University of Western Ontario London, Ontario Canada N6A 5B9
AMS Subject Classifications (1970): 22A15, 22A20, 43A07, 43A60
ISBN 3-540-08919-5 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-08919-5 Springer-Verlag New York Heidelberg Berlin 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 the publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin Heidelberg 1978 Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
INTRODUCTION The primary objective of this monograph is to present a reasonably self-contained treatment of the theory of compact right topological semigroups and, in particular, of semigroup compactifications.
By semi group compactification we mean a
compact right topological semigroup which contains a dense continuous homomorphic image of a given semi topological semigroup.
The classical example is the Bohr (or almost periodic)
compactification (a,AR) of the usual additive
Here AR is a compact topological group and a: R tinuous homomorphism with dense image.
numbers R.
~eal
+
AR is a con-
An important feature of
the Bohr compactification is the following universal mapping property which it enjoys:
given any compact topological group
G and any continuous homomorphism tinuous homomorphism ¢: AR
+
~:
R
+
G such that
G there exists a con~
=
¢
0
a.
Such
universal mapping properties are central to the theory of semigroup compactifications. Compactifications of semigroups can be produced in a variety of ways.
One way is by the use of operator theory,
a technique employed by deLeeuw and Glicksberg in their now classic 1961 paper on applications of almost periodic compactifications.
In this setting, AR appears as the strong operator
closure of the group of all translation operators on the C*algebra AP{R) of almost periodic functions on R.
More
IV
generally, but using essentially the same ideas, deLeeuw and Glicksberg were able to construct the almost periodic and weakly almost periodic compactifications of any semi topological semigroup with identity. Another method of obtaining compactifications is based on the Adjoint Functor Theorem of category theory.
The first
systematic use of this technique appeared in the 19.6 7 monograph of Berglund and Hofmann, where it was shown that any semitopological semigrou
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