Abstract
Let S S and T T be semigroups, S\text {\textcircled {𝜏 }} T a semidirect product, and F F a C ∗ {C^ \ast } -algebra of bounded, complex-valued functions on S\text {\textcircled {𝜏 }} T. Necessary and sufficient conditions are given for the F F -compactification of S\text {\textcircled {𝜏 }} T to be expressible as a semidirect product of compactifications of S S and T 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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