Involutions on certain Banach algebras related to locally compact groups. (English) Zbl 1363.46030
Summary: Let \(\mathcal A\) be a Banach algebra and let \(\mathcal X\) be an introverted closed subspace of \(\mathcal A^\ast\). Here, we give necessary and sufficient conditions for that the dual algebra \(\mathcal X^\ast\) of \(\mathcal X\) or the topological centers \(\mathfrak Z_t^{(1)} (\mathcal X^\ast)\) and \(\mathfrak Z_t^{(2)} (\mathcal X^\ast)\) of \(\mathcal X^\ast\) are Banach \(\ast\)-algebras. We finally apply these results to the Banach space \(L_0^\infty (G)\) of all equivalence classes of essentially bounded functions vanishing at infinity on a locally compact group \(G\).
MSC:
| 46H05 | General theory of topological algebras |
| 43A60 | Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions |
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