Completely Continuous Banach Algebras

Document Type : Research Paper

Author

Faculty of Mathematical Sciences, Malayer University, P. O. Box 16846-13114, Malayer, Iran

Abstract

For a Banach algebra $\mathfrak{A}$, we introduce ~$c.c(\mathfrak{A})$, the set of all $\phi\in \mathfrak{A}^*$ such that $\theta_\phi:\mathfrak{A}\to \mathfrak{A}^*$ 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 Gale, T.J. Ransford and M. C. White: Is 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.

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