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Completely continuous Banach algebras. (English) Zbl 1449.46038
Summary: For a Banach algebra $$\mathfrak A$$, we introduce $$c.c(\mathfrak A)$$, the set of all $$\phi\in \mathfrak A^*$$ such that $$\theta_{\phi}:\mathfrak A \to\mathfrak A^*$$ is a completely continuous operator, where $$\theta_{\phi}$$ is defined by $$\theta_{\phi}(a)=a \cdot \phi$$ for all $$a \in \mathfrak A$$. We call $$\mathfrak A$$, a completely continuous Banach algebra if $$c.c(\mathfrak A)=\mathfrak A^*$$. We give some examples of completely continuous Banach algebras and a sufficient condition for an open problem raised for the first time by J. E. Galé et al. [Trans. Am. Math. Soc. 331, No. 2, 815–824 (1992; Zbl 0761.46037)]: does there exist an infinite dimensional amenable Banach algebra whose underlying Banach space is reflexive? We prove that a reflexive, amenable, completely continuous Banach algebra with the approximation property is trivial.
##### MSC:
 46H20 Structure, classification of topological algebras 46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) 46B10 Duality and reflexivity in normed linear and Banach spaces
##### Keywords:
amenability; complete continuity; Banach algebra
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##### References:
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