2 Banach space properties forcing a reflexive , amenable Banach algebra to be trivial

It is an open problem whether an infinite-dimensional amenable Banach algebra exists whose underlying Banach space is reflexive. We give sufficient conditions for a reflexive, amenable Banach algebra to be finite-dimensional (and thus a finite direct sum of full matrix algebras). If A is a reflexive, amenable Banach algebra such that for each maximal left ideal L of A (i) the quotient A/L has the approximation property and (ii) the canonical map from A⊗̌L to (A/L)⊗̌L is open, then A is finite-dimensional. As an application, we show that, if A is an amenable Banach algebra whose underlying Banach space is an L-space with p ∈ (1,∞) such that for each maximal left ideal L the quotient A/L has the approximation property, then A is finite-dimensional. 2000 Mathematics Subject Classification: 46B10, 46B20, 46H20 (primary), 46H25, 46M18.