Weak-polynomial convergence on a Banach space

Abstract

We show that any super-reflexive Banach space is a Λ \Lambda -space (i.e., the weak-polynomial convergence for sequences implies the norm convergence). We introduce the notion of κ \kappa -space (i.e., a Banach space where the weak-polynomial convergence for sequences is different from the weak convergence) and we prove that if a dual Banach space Z Z is a κ \kappa -space with the approximation property, then the uniform algebra A ( B ) A(B) on the unit ball of Z Z generated by the weak-star continuous polynomials is not tight.