Completely continuous Banach algebras

Document Type : Research Paper


Department of Mathematics, Malayer University, P.O. Box 16846-13114, Malayer, Iran


 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.