ON λ-STRICT IDEALS IN BANACH SPACES

We define and study A-strict ideals in Banach spaces, which for lambda = 1 means strict ideals. Strict u-ideals in their biduals are known to have the unique ideal property; we prove that so also do A-strict u-ideals in their biduals, at least for lambda > 1/2. An open question, posed by Godefroy etal. ['Unconditional ideals in Banach spaces', Studia Math. 104 (1993), 13-59] is whether the Banach space X is a u-ideal in Ba(X), the Baire-one functions in X**, exactly when kappa(u) (X) = 1; we prove that if kappa(u)(X) = I then X is a strict u-ideal in Ba(X), and we establish the converse in the separable case.