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Completely continuous Banach algebras. (English) Zbl 1370.46028
Author’s abstract: For a Banach algebra $$\mathfrak A$$, we introduce $$c.c({\mathfrak A})$$, the set of all $$\phi\in{\mathfrak A}^*$$ such that $$\theta_{\phi}:{\mathfrak A}\rightarrow{\mathfrak A^*}$$ 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:
 46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) 46H20 Structure, classification of topological algebras 46B10 Duality and reflexivity in normed linear and Banach spaces
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