Stability of properties α and β under absolute sums of Banach spaces

Abstract

We study the stability of properties α and β under absolute sums of two real Banach spaces. We prove that if N is an absolute normalised norm on R² whose unit sphere is polygonal, then X ⊕_N Y has property α whenever both X and Y have property α, and similarly for property β. We also obtain partial converse results. For property α, if (1, 0) is an extreme point of B_{(R², N)} and X ⊕_N Y has property α, then X has property α. For property β, if (1, 0) is not an extreme point of B_{(R², N)} and X ⊕_N Y has property β, then X has property β. As corollaries, we recover the finite ℓ₁- and ℓ_∞-cases and obtain corresponding equivalence results.