Completely Continuous Banach Algebras

Document Type: Research Paper

Author

Malayer University

Abstract

For a Banach algebra $\fA$, we introduce ~$c.c(\fA)$, the set of all $\phi\in \fA^*$ such that $\theta_\phi:\fA\to
\fA^*$ is a completely continuous operator, where $\theta_\phi$ is defined by
$\theta_\phi(a)=a\cdot\phi$~~ for all $a\in \fA$. We call $\fA$, a completely
continuous Banach algebra if $c.c(\fA)=\fA^*$. 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 Gal\`{e},
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.

Keywords