Average locally uniformly rotund dual norms

Sebastián Lajara (Universidad de Castilla La Mancha)

Average locally uniformly rotund norms (ALUR for short) are a generalization of LUR norms. They were introduced by Troyanski, who proved that every rotund norm with the Kadec property is ALUR and that every ALUR space admits an equivalent LUR norm. This result suggests the following problem: If a dual Banach space is ALUR, does it necessarily admit an equivalent dual LUR norm? The dual of the James Tree space provides a negative answer to this question (Schachermayer). Therefore, the class of dual LUR spaces is strictly contained in the class of ALUR dual renormable spaces. Banach spaces with equivalent LUR norms were characterized in terms of

countable decompositions of such spaces by Moltó, Orihuela

and Troyanski, and by Raja, who gave a generalization to the dual case. In this note, we provide a covering type characterization of dual Banach spaces with equivalent ALUR dual norms.