Arens semi-regular Banach algebras

Abstract

For some important Banach algebras, the first and the second Arens product on their biduals are different, i. e. these algebras are not (Arens) regular. Arens semi-regularity is a property strictly weaker than regularity; it characterizes those non-regular algebras (having a bounded two-sided approximate identity) for which the Arens products, though different, still behave in a reasonable way. The definition of semi-regularity is based on the relation of two natural embeddings of the space of double multipliers into the bidual of the Banach algebra. It is shown that each commutative Banach algebra is semi-regular and that semi-regularity is equivalent to the equality of the Arens products on certain subspaces of the bidual. Among others, group algebras and algebras of compact operators are treated as examples.