Completely Continuous Banach Algebras
For a Banach algebra A, we introduce c.c(A), the set of all φ ∈ A ∗ such that θφ : A → A ∗ is a completely continuous operator, where θφ is defined by θφ(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 Gal`e, 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.
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