On $w^{*}$-binormality of the dual space $C_k(X)'$

Arkady Leiderman (Ben-Gurion U, Beer-Sheva)

Mar 21. 2025, 09:00 — 09:20

If $E$ is a lcs (locally convex space) with its dual $E'$, by $\beta(E',E)$ and $w^{*}=\sigma(E',E)$ we mean the strong  and the weak$^{*}$  topology  of $E'$, respectively.

{\bf Definition.} \cite{KKuL}
For a lcs $E$ we say that its dual $E'$ is {\it $w^{*}$-binormal} if for every disjoint $\beta(E',E)$-closed $A\subset E'$ and $w^{*}$-closed $B\subset E'$ there exist disjoint $\beta(E',E)$-open $D\subset E'$ and $w^{*}$-open $C\subset E'$ such that $A\subset C$ and $B\subset D$.

For a Tychonoff space $X$ by $C_k(X)$ we denote lcs of continuous real-valued  functions  on $X$  with the compact-open topology. $\Delta$-spaces have been defined and investigated in the recentpapers \cite{KL1} and \cite{KL2}.

{\bf Theorem 1.} \cite{KKuL}
Let $X$ be a pseudocompact space and assume that the strong dual $C_k(X)'$ of $C_k(X)$ is $w^{*}$-binormal.  Then $X$ is a $\Delta$-space.

We proved also that for a Corson compact space $X$ the converse of Theorem 1 is true.
A new property which formally is stronger than being a $ w^{*} $-binormal has been defined.

{\bf Theorem 2.} \cite{KKuL}
Let $X$ be a compact space. Then  $C(X)'$ is effectively $ w^{*} $-binormal if and only if $ X $ is an effectively $\Delta$-space.

Examples of compact effectively $\Delta$-spaces include all scattered Eberlein compact spaces and one-point compactification of Isbell-Mr\'owka spaces $\Psi(\mathcal{A})$. We will present some other results about the class of compact effectively $\Delta$-spaces and pose open problems.

\begin{thebibliography}{99}
\bibitem{KKuL} J. K\c akol, O. Kurka, A. Leiderman, \textit{On Asplund spaces $C_k(X)$ and $w^{*}$-binormality}, Results Math.,  \textbf{78:203} (2023), 19 pp.

\bibitem{KL1} J. K\c akol, A. Leiderman,
{\it A  characterization of $X$ for  which  spaces $C_p(X)$ are distinguished and its applications}, Proc. Amer. Math. Soc., series B, {\bf 8} (2021), 86--99.

\bibitem{KL2} J. K\c akol, A. Leiderman,
{\it Basic properties of $X$ for which the space $C_p(X)$ is distinguished}, Proc. Amer. Math. Soc., series B, \textbf{8} (2021), 267--280. \end{thebibliography}

Further Information
Venue:
ESI Boltzmann Lecture Hall
Associated Event:
Structures in Banach Spaces (Workshop)
Organizer(s):
Antonio Aviles (U Murcia)
Vera Fischer (U of Vienna)
Grzegorz Plebanek (U of Wroclaw)
Damian Sobota (U of Vienna)