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The second conjugate algebras of Banach algebras. (English) Zbl 0818.46055

Arens regularity for a Banach algebra \(A\) is defined by the identity of the two natural Arens multiplications on \(A^{**}\). The author shows, among other characterizations of Arens regularity, that, if each continuous linear map of \(A\) into \(A^*\) is weakly compact, then \(A\) is Arens regular. From this it is deduced that a \(B^*\)-algebra is Arens regular. It is also shown that a closed subalgebra inherits Arens regularity.

MSC:

46H10 Ideals and subalgebras
46H05 General theory of topological algebras
46H99 Topological algebras, normed rings and algebras, Banach algebras
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