Metrizable quotients of $C_p$-spaces

Abstract

The famous Rosenthal-Lacey theorem asserts that for each infinite compact set $K$ the Banach space $C(K)$ admits a quotient which is either a copy of $c$ or ${\ell }_2$. What is the case when the uniform topology of $C(K)$ is replaced by the pointwise topology? Is it true that $C_p(X)$ always has an infinite-dimensional separable (or better metrizable) quotient? In this paper we prove that for a Tychonoff space $X$ the function space $C_p(X)$ has an infinite-dimensional metrizable quotient if $X$ either contains an infinite discrete $C^{\ast }$-embedded subspace or else $X$ has a sequence $(K_n{)}_{n\in \mathbb{N}}$ of compact subsets such that for every $n$ the space $K_n$ contains two disjoint topological copies of $K_{n+1}$. Applying the latter result, we show that under $◊$ there exists a zero-dimensional Efimov space $K$ whose function space $C_p(K)$ has an infinite-dimensional metrizable quotient. These two theorems essentially improve earlier results of Kąkol and Śliwa on infinite-dimensional separable quotients of $C_p$-spaces.

8 pages