## 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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