Noncommutative Valdivia compacta. (English) Zbl 1313.46026

Cúth, Marek

Commentat. Math. Univ. Carol. 55, No. 1, 53-72 (2014).

The paper under review nontrivially pushes ahead results of O. F. K. Kalenda [Extr. Math. 14, No. 3, 355–371 (1999; Zbl 0983.46020); Fundam. Math. 162, No. 2, 181–192 (1999; Zbl 0989.54019); Stud. Math. 138, No. 2, 179–191 (2000; Zbl 1073.46009)].
Let \(K\) be a compact space. Let \((\Gamma ,\leq)\) be a partially ordered set which is moreover up-directed and \(\sigma \)-complete, which means that every increasing sequence in \(\Gamma \) admits a supremum. According to W. Kubiś, by a retractional skeleton on \(K\) we understand any system \(\{r_s:\;s\in \Gamma \}\) of retractions \(r_s:K\rightarrow K\) such that:
(i) for every \(s\in \Gamma \) the range \(r_s[K]\) is metrizable,
(ii) \(x=\lim _{s\in \Gamma } r_s(x)\) for every \(x\in K\),
(iii) whenever \(s,t\in \Gamma \) and \(s\leq t\), then \(r_s\circ r_t =r_s\;(=r_t\circ r_s)\), and
(iv) whenever \(s_1, s_2, \ldots \in \Gamma \) is an increasing sequence, with supremum \(t\), then \(r_t(x)=\lim _{n\to \infty }r_{s_n}(x)\) for every \(x\in K\).
A compact space is called Corson if it is homeomorphic to a compact subspace of the so-called \(\Sigma \)-product of real lines.
Among several results obtained we mention two. The dual unit ball of a Banach space \(X\), endowed with the weak\(^*\) topology, is Corson if and only if for every equivalent norm on \(X\) the corresponding dual ball admits a retractional skeleton. A compact space is Corson if and only if every continuous image of it admits a retractional skeleton.
The proofs on the one hand imitate the methods from the papers of O. Kalenda mentioned above, and on the other hand they use a nowadays trendy and powerful technology of elementary submodels from logic.

Reviewer: Marián Fabian (Praha)

MSC:

46B26	Nonseparable Banach spaces
54D30	Compactness

Keywords:

retractional skeleton; projectional skeleton; Valdivia compacta; Plichko spaces

Citations:

Zbl 0983.46020; Zbl 0989.54019; Zbl 1073.46009