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.

Zbl 0859.43001
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