Banach space properties forcing a reflexive, amenable Banach algebra to betrivial

It is an open problem whether an infinite-dimensional amenable Banach algeb ra exists whose underlying Banach space is reflexive. We give sufficient co nditions for a reflexive, amenable Banach algebra to be finite-dimensional (and thus a finite direct sum of full matrix algebras). If U is a reflexive , amenable Banach algebra such that for each maximal left ideal L of U (i) the quotient U/L has the approximation property and (ii) the canonical map from UxL(perpendicular to) to (U/L)xL(perpendicular to) is open, then U is finite-dimensional. As an application, we show that, if U is an amenable Ba nach algebra whose underlying Banach space is an L-p-space with p is an ele ment of (1, infinity) such that for each maximal left ideal L the quotient U/L has the approximation property, then U is finite-dimensional.