On linear continuous operators between distinguished spaces $C_p(X)$

TL;DR

This paper investigates the conditions under which linear continuous operators between spaces of continuous functions are open, characterizes spaces admitting such operators, and explores properties of distinguished spaces related to $ abla$-spaces and their subspaces.

Contribution

It provides a complete characterization of spaces $Y$ for which there exists a surjective continuous linear operator from $C_p([1, eta])$ to $C_p(Y)$, and answers an open question about subspaces of distinguished spaces.

Findings

Characterization of all $Y$ admitting a surjective operator from $C_p([1,eta])$

All closed subspaces of $C_p([1,eta])$ are distinguished

No surjective operator exists from $C_p(X)$ to $C_k(X)_w$ for certain compact spaces

Abstract

As proved in [16], for a Tychonoff space $X$, a locally convex space $C_{p}(X)$ is distinguished if and only if $X$ is a $\Delta$-space. If there exists a linear continuous surjective mapping $T:C_p(X) \to C_p(Y)$ and $C_p(X)$ is distinguished, then $C_p(Y)$ also is distinguished [17]. Firstly, in this paper we explore the following question: Under which conditions the operator $T:C_p(X) \to C_p(Y)$ above is open? Secondly, we devote a special attention to concrete distinguished spaces $C_p([1,\alpha])$, where $\alpha$ is a countable ordinal number. A complete characterization of all $Y$ which admit a linear continuous surjective mapping $T:C_p([1,\alpha]) \to C_p(Y)$ is given. We also observe that for every countable ordinal $\alpha$ all closed linear subspaces of $C_p([1,\alpha])$ are distinguished, thereby answering an open question posed in [17]. Using some properties of $\Delta$-spaces we prove that a linear continuous surjection $T:C_p(X) \to C_k(X)_w$, where $C_k(X)_w$ denotes the Banach space $C(X)$ endowed with its weak topology, does not exist for every infinite metrizable compact $C$-space $X$ (in particular, for every infinite compact $X \subset \mathbb{R}^n$).