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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
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