Completely Continuous Banach Algebras

For a Banach algebra 21, we introduce c.c(21), the set of all ϕ∈21∗ such that θϕ:21→21 is a completely continuous operator, where θϕ is defined by θϕ(a)=a⋅ϕ for all a∈21. We call 21, a completely continuous Banach algebra if c.c(21)=21. 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.