Weakly web-compact Banach spaces C(X), and $Lip_0(M)$, $\mathcal {F}(M)$ over metric spaces M

Abstract

The class of web-compact spaces (in sense of Orihuela), which encompasses a number of spaces, like Lindelöf $\Sigma$-spaces (called also countably determined), Quasi-Suslin spaces, separable spaces, etc., applies to distinguish a class of weakly web-compact Banach spaces E whose dual unit ball is weak$^{*}$-sequentially compact, consequently Banach spaces without quotients isomorphic to $\ell _{\infty }.$ We prove however that for a Banach space E the space $E_w$ (i.e. E with the weak topology) is web-compact if and only if $E_w$ is a Lindelöf $\Sigma$-space if and only if $E_w$ contains a web-compact total subset. Consequently, for compact X the space $C(X)_w$ is web-compact if and only if X is Gul’ko compact if and only if $C(X)_w$ is a Lindelöf $\Sigma$-space if and only if $C_p(X)$ contains a web-compact total subset. If X is compact and $C(X)_w$ is web-compact, then $C_p(X)$ contains a complemented copy of the space $(c_{0})_p=\{(x_{n})\in \mathbb R^{\omega }: x_{n}\rightarrow 0\}$ with the topology of $\mathbb {R}^{\omega }$ but does not admit quotients isomorphic to $(\ell _{\infty })_{p}=\{(x_{n})\in \mathbb R^{\omega }: \sup _n|x_{n}|<\infty \}$. We characterize weakly web-compact Banach spaces $Lip_0(M)$ of Lipschitz functions on metric spaces M and their predual $\mathcal {F}(M)$. In fact, $Lip_0(M)_w$ is web-compact if and only if M is separable. Illustrating examples are provided.