For a Banach algebra A, we introduce c:c(A), the set of all 2 A such that : A ! A is a completely continuous operator, where is de ned by (a) = a for all a 2 A. We call A, a completely continuous Banach algebra if c:c(A) = A. We give some examples of completely continuous Banach algebras and a sucient condition for an open problem raised for the rst time by J.E Gale, T.J. Ransford and M. C. White: Is there exist an in nite dimensional amenable Banach algebra whose underlying Banach space is re exive? We prove that a re exive, amenable, completely continuous Banach algebra with the approximation property is trivial.
