Mazur Intersection Property for Asplund Spaces

The main result of the present note states that it is consistent with the ZFC axioms of set theory (relying on Martin’s Maximum MM axiom), that every Asplund space of density character ω1 has a renorming with the Mazur intersection property. Combined with the previous result of Jiménez and Moreno (based upon the work of Kunen under the continuum hypothesis) we obtain that the MIP renormability of Asplund spaces of density ω1 is undecidable in ZFC. The Mazur intersection property (MIP for short) was first investigated by S. Mazur in [10] as a purely geometrical isometric property of a Banach space, and has since been studied extensively over the years. An early result of Mazur claims that a Banach space with a Fréchet differentiable norm (necessarily an Asplund space) has the MIP ([10]). Phelps [12] proved that a separable Banach space has a MIP renorming if and only if its dual is separable, or equivalently, if it is an Asplund space. Much of the further progress in the theory depended on an important characterization of MIP, due to Giles, Gregory and Sims [6], by the property that w∗-denting points of BX∗ are norm dense in SX∗ . This result again suggests a close connection of MIP to Asplund spaces, as the latter can be characterized in a similar way as spaces such that bounded subsets of their dual are w∗-dentable. It has opened a way to applying biorthogonal systems to the MIP. Namely, Jiménez and Moreno [9] have proved that if a Banach space X∗ admits a fundamental biorthogonal system {(xα, fα)}, where fα belong to X ⊂ X∗∗, then X has a MIP renorming. As a corollary to this criterion ([9]), they get that every Banach space can be embedded into a Banach space which is MIP renormable, a rather surprising result which in particular strongly shows that MIP and Asplund properties, although closely Date: March 2008. 2000 Mathematics Subject Classification. 46B03.