Semnan UniversityInternational Journal of Nonlinear Analysis and Applications2008-682210120191101Completely Continuous Banach Algebras5562405210.22075/ijnaa.2019.1184.1268ENBahmanHayatiFaculty of Mathematical Sciences, Malayer University, P. O. Box 16846-13114, Malayer, IranJournal Article20181231For 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.https://ijnaa.semnan.ac.ir/article_4052_98d6bd83d5d5695668206e7b12bb7f92.pdf