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Arens regularity of module actions. (English) Zbl 1165.46024

Richard Arens showed that given a continuous bilinear map \(f:X\times Y\to Z\), where \(X,Y,Z\) are normed linear spaces, \(f\) can be extended uniquely to a bilinear map \(f^{***}:X^{**}\times Y^{**}\to Z^{**}\) such that the map \(X^{**}\to Z^{**}\), \(x''\to f^{***}(x'',y'')\) is weak\(^*\)-weak\(^*\) continuous for each \(y''\in Y^{**}\). The first topological centre of \(f=Z(f)\) is the set \(\{x''\in X^{**}: y''\to f^{***} (x'',y'')\) is weak\(^*\)-weak\(^*\) continuous from \(Y^{**}\to Z^{**}\}\). An important special case of this concept is when \(A\) is a Banach algebra and \(f:A\times A\to A\) is the multiplication map. In this paper, the author studies Arens regularity on Banach module actions of Banach algebras. For example, it is shown that if \(A\) has a bounded right approximate identity, then \(A\) is reflexive if and only if the right module action of \(A\) on \(A^*\) is Arens regular, i.e., \(Z(f)=A^{***}\), when \(f: A^*\times A\to A^*\) is the right module action. They study \(Z(A^{**}), Z^t(A^{**})\) (the first and second topological centres of \(A^{**})\) when \(A\) is a triangular algebra. Using these, they provide examples to show that (i) \(A^*A=AA^*\) is not sufficient for \(Z(A^{**})=Z^t (A^{**})\); (ii) \(Z(A^{**})=Z^t(A^{**})\) and \(A^*A=A^*\) are not sufficient for \(AA^*=A^*\); (iii) there exists a Banach algebra \(A\) that is strongly irregular but not weakly sequentially complete. The results (ii) and (iii) are new examples answering Problems \(6d\) and \(6j\) in [A.T.–M.Lau and A.Ülger, Trans.Am.Math.Soc.348, No.3, 1191–1212 (1996; Zbl 0859.43001)].

MSC:

46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
46H20 Structure, classification of topological algebras
43A20 \(L^1\)-algebras on groups, semigroups, etc.

Citations:

Zbl 0859.43001
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