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

الأصل · EN

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 a menable Banach algebra whose underlying Banach space is an Lᵖ-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.