A note on Banach spaces E for which $E_{w}$ is homeomorphic to $C_p(X)$

Abstract

$C_p(X)$ denotes the space of continuous real-valued functions on a Tychonoff space X endowed with the topology of pointwise convergence. A Banach space E equipped with the weak topology is denoted by $E_{w}$. It is unknown whether $C_p(K)$ and $C(L)_{w}$ can be homeomorphic for infinite compact spaces K and L (Krupski, Rev R Acad Cienc Exact Fis Nat Ser A Mat (RACSAM) 110:557–563, 2016; Krupski and Marciszewski, J Math Anal Appl 452:646–658, 2017 ). In this paper we deal with a more general question: does there exist a Banach space E such that $E_{w}$ is homeomorphic to the space $C_p(X)$ for some infinite Tychonoff space X? We show that if such homeomorphism exists, then (a) X is a countable union of compact sets $X_n, n \in \omega$, where at least one component $X_n$ is non-scattered; (b) the Banach space E necessarily contains an isomorphic copy of the Banach space $\ell _{1}$.