an:07176975
Zbl 1449.46038
Hayati, Bahman
Completely continuous Banach algebras
EN
Int. J. Nonlinear Anal. Appl. 10, No. 1, 55-62 (2019).
00447395
2019
j
46H20 46H25 46B10
amenability; complete continuity; Banach algebra
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 \textit{J. E. Gal??} 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.
Zbl 0761.46037