Completely Continuous Banach Algebras

For a banach algebra a, we introduce c:c(a), the set of all  φ ∈a* such that θ φ: a → a* is a completely continuous operator, where θ φ is defined by θ φ(a) = a . φ  for all a ∈ 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 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 reexive? we prove that a reexive, amenable, completely continuous banach algebra with the approximation property is trivial.