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

Abstract.

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 \( {\frak A} \) is a reflexive, amenable Banach algebra such that for each maximal left ideal L of \( {\frak A} \) (i) the quotient \( {\frak A}/L \) has the approximation property and (ii) the canonical map from \( {\frak A} \check{\otimes} L^\perp $ to $({\frak A} / L) \check{\otimes} L^\perp \) is open, then \( {\frak A} \) is finite-dimensional. As an application, we show that, if \( {\frak A} \) is an amenable Banach algebra whose underlying Banach space is an \( {\scr L}^p \)-space with \( p\in (1,\infty) \) such that for each maximal left ideal L the quotient \( {\frak A}/L \) has the approximation property, then \( {\frak A} \) is finite-dimensional.