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 Lp-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.