Semigroup compactifications of semidirect products

Junghenn, H. D.; Lerner, B. T.

doi:10.1090/S0002-9947-1981-0610956-2

Semigroup compactifications of semidirect products
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by H. D. Junghenn and B. T. Lerner PDF

Trans. Amer. Math. Soc. 265 (1981), 393-404 Request permission

Abstract:

Let $S$ and $T$ be semigroups, $S\text {\textcircled {$𝜏$}} T$ a semidirect product, and $F$ a ${C^ \ast }$-algebra of bounded, complex-valued functions on $S\text {\textcircled {$𝜏$}} T$. Necessary and sufficient conditions are given for the $F$-compactification of $S\text {\textcircled {$𝜏$}} T$ 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 $S\text {\textcircled {$𝜏$}} T$ 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 $S\text {\textcircled {$𝜏$}} T$. Applications are made to wreath products.

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