Completely Continuous Banach Algebras

Abstract

For a Banach algebra $fA$, we introduce ~$c.c(fA)$, the set of all $phiin fA^*$ such that $theta_phi:fAto<br /> fA^*$ is a completely continuous operator, where $theta_phi$ is defined by<br /> $theta_phi(a)=acdotphi$~~ for all $ain fA$. We call $fA$, a completely<br /> continuous Banach algebra if $c.c(fA)=fA^*$. We give some examples of completely<br /> continuous Banach algebras and a sufficient condition for an open problem raised for the first time by J.E Gal`{e},<br /> T.J. Ransford and M. C. White: Is there exist an infinite dimensional amenable Banach algebra whose underlying Banach space is<br /> reflexive? We prove that a reflexive, amenable, completely continuous Banach algebra with the approximation property is<br /> trivial.