Completely continuous Banach algebras

 For 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.

Keywords:

Amenability , Completely continuous , Banach algebra