Denseness of norm-attaining operators into strictly convex spaces

Abstract

We show that no infinite-dimensional Banach space provided with a strictly convex norm satisfies Lindenstrauss's property B. This is a generalization of previous results by Lindenstrauss for rotund spaces isomorphic to C 0 and by Gowers for ℓ p (1 < p < ∞). Also, there is an appropriate complex version of the announced result that works for all the C-strictly convex spaces. As a consequence, the Hardy space H 1 , any infinite-dimensional complex L 1 (μ) , and, in general, any infinite-dimensional predual of a von Neumann algebra lacks Lindenstrauss's property B.