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

Abstract

As proved in Ka̧kol and Leiderman (Proc AMS Ser B 8:86–99, 2021), 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 $$T:C_p(X) \rightarrow C_p(Y)$$ and $$C_p(X)$$ is distinguished, then $$C_p(Y)$$ also is distinguished (Ka̧kol and Leiderman Proc AMS Ser B, 2021). Firstly, in this paper we explore the following question: Under which conditions the operator $$T:C_p(X) \rightarrow C_p(Y)$$ 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 $$T:C_p([1,\alpha ]) \rightarrow C_p(Y)$$ 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 Ka̧kol and Leiderman (Proc AMS Ser B, 2021). Using some properties of $$\Delta $$ -spaces we prove that a linear continuous surjection $$T:C_p(X) \rightarrow C_k(X)_w$$ , 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 $$X \subset {\mathbb {R}}^n$$ ).