Semnan UniversityInternational Journal of Nonlinear Analysis and Applications2008-68227120160601Completely continuous Banach algebras30130838310.22075/ijnaa.2016.383ENBahmanHayatiDepartment of Mathematics, Malayer University, P.O. Box 16846-13114, Malayer, Iran0000-0002-6283-3935Journal Article20150417 For a Banach algebra $mathfrak{A}$, we introduce ~$c.c(mathfrak{A})$, the set of all $phiin mathfrak{A}^*$ such that $theta_phi:mathfrak{A}to mathfrak{A}^*$ is a completely continuous operator, where $theta_phi$ is defined by $theta_phi(a)=acdotphi$~~ for all $ain mathfrak{A}$. 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.https://ijnaa.semnan.ac.ir/article_383_3bb2ce040cb0b5b1b2133ec62d0d7465.pdf