Characterization of Banach Lattices in Terms of Quasi-Interior Points

Abstract EN

In terms of quasi-interior points, criteria that a Banach lattice E has order continuous norm or is an AM-space with a unit are given.

For example, if E is Dedekind complete and has a weak order unit, then E has order continuous norm if and only if the set of quasi-interior points of E coincides with the set of weak order units of E; a Banach lattice E is an AM-space with a unit x if and only if the set of all quasi-interior points of E coincides with the set {z:x≤λz for some λ>0}.

Analogous questions are considered for the case of an ordered Banach space Z with a cone K.

Moreover, it is shown that every nonzero point of a cone K≠{0} is quasi-interior if and only if dim Z=1.

We also study various ?-properties of a cone K; in particular, the conditions for which the relation a∧b=0 with a>0 implies that b is not a quasi-interior point are considered.