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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Semigroup compactifications of semidirect products


Authors: H. D. Junghenn and B. T. Lerner
Journal: Trans. Amer. Math. Soc. 265 (1981), 393-404
MSC: Primary 22A20; Secondary 43A60, 54H15
MathSciNet review: 610956
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Abstract: Let $ S$ and $ T$ be semigroups, $ S\circlebin{\tau} T$ a semidirect product, and $ F$ a $ {C^ \ast }$-algebra of bounded, complex-valued functions on $ S\circlebin{\tau} T$. Necessary and sufficient conditions are given for the $ F$-compactification of $ S\circlebin{\tau} T$ to be expressible as a semidirect product of compactifications of $ S$ and $ T$. This result is used to show that the strongly almost periodic compactification of $ S\circlebin{\tau} T$ is a semidirect product and that, in certain general cases, the analogous statement holds for the almost periodic compactification and the left uniformly continuous compactification of $ S\circlebin{\tau} T$. Applications are made to wreath products.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1981-0610956-2
PII: S 0002-9947(1981)0610956-2
Keywords: Semitopological semigroup, semidirect product, compactification, wreath product, minimal ideal, almost periodic, strongly almost periodic
Article copyright: © Copyright 1981 American Mathematical Society