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