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Boundaries, Martin's Axiom, and (P)-properties in dual Banach spaces

Granero A. S., Hernández J. M.

Commentationes Mathematicae

2016

Vol 56, No. 1

1--16

Let X be a~Banach space and Seq(X∗∗) (resp., Xℵ0) the subset of elements ψ∈X∗∗ such that there exists a~sequence (xn)n≥1⊂X such that xn→ψ in the ω∗-topology of X∗∗ (resp., there exists a~separable subspace Y⊂X such that ψ∈Y¯¯¯¯ω∗). Then: (i) if Dens(X)≥ℵ1, the property X∗∗=Xℵ0 (resp., X∗∗=Seq(X∗∗)) is ℵ1-determined, i.e., X~has this property iff Y has, for every subspace Y⊂X with Dens(Y)=ℵ1; (ii) if X∗∗=Xℵ0, (B(X∗∗),ω∗) has countable tightness; (iii) under the Martin's axiom MA(ω1) we have X∗∗=Seq(X>∗∗) iff (B(X∗),ω∗) has countable tightness and overlineco(B)=co¯¯¯¯¯ω∗(K) for every subspace Y⊂X, every ω∗-compact subset K of Y∗, and every boundary B⊂K .