New examples of non-commutative Valdivia compact spaces

Abstract

The aim of this note is to characterize trees, endowed with coarse wedge topology, that have a retractional skeleton. We use this characterization to provide new examples of non-commutative Valdivia compact spaces that are not Valdivia.MSC: 54D30, 54C15, 54G20, 54F05

On compact trees with the coarse wedge topology

“…The equivalence between (i) and (ii) follows from [21,Proposition 3.2], while the equivalence between (ii) and (iii) follows by [10,Theorem 19.11], observing that a tree T endowed with the coarse wedge topology is a zero-dimensional space. We conclude this section providing a description of the Radon measures on trees, these results are useful to investigate the spaces of continuous functions on trees.…”

“…It is more difficult to prove that the Banach space C([0, ω 2 ]), which has a projectional skeleton, is not Plichko [11]. In [21] we studied the class of trees endowed with the coarse wedge topology, providing new examples of non Valdivia compact spaces with retractional skeletons. In the same paper it was proved that every tree with height less or equal than ω 1 + 1 is Valdivia and no Valdivia tree has height greater than ω 2 .…”

Projectional skeletons and Markushevich bases

“…Similarly as in [45], choose for any x ∈ S(T ) \ I(T ) a countable set φ(x) ⊂ x ∩ I(T ) with supremum x. Now we are ready to provide a proof of (i).…”

“…A retractional skeleton constructed in is formed by the retractions where runs through a carefully chosen subfamily of . Using similar ideas, we present a simplified more canonical approach.…”

“…So, we can work with densities and, moreover, the density of a subset is not larger than the density of the original set. Similarly as in , choose for any a countable set with supremum . Now we are ready to provide a proof of (i).…”