Solid sequence $F$-spaces of $L_0$-type over submeasures on $\mathbb{N}$

Abstract

We study solid sequence $F$-spaces $\lambda_{0}(\eta)$, nonseparable in general, and their closed separable subspaces $\lambda_{00}(\eta)$. The space $\lambda_{0}(\eta)$ is associated with a strictly positive submeasure $\eta$ on $\mathbb{N}$ and equipped with the topology of convergence in submeasure. While $\lambda_{0}(\eta)$'s may be viewed as analogs of usual $L_0$-spaces, the relation between $\lambda_{00}(\eta)$ and $\lambda_{0}(\eta )$ often resembles that between $c_0$ and $l_\infty$. For many $\eta$'s, the weak topology of these spaces coincides with that of coordinate-wise convergence, they are not locally pseudoconvex and yet have the Bounded Multiplier Property. Further, in agreement with the analogy to $L_0=L_0[0,1]$, they possess copies of $l_p$ for $0 \lt p \leq 2$, and yet in contrast to $L_0$ they contain a lot of well-located copies of $c_0$ and $l_\infty$; also, the quotient $\lambda_{0}(\eta)/\lambda_{00}(\eta)$ contains a copy of $L_0$. All of this happens already for the spaces $\lambda_{0}=\lambda _{0}(\bar d)$ and $\lambda_{00}=\lambda_{00}(\bar d)$ with $\bar d$ being a submeasure closely related to the standard density $d$, in which case, moreover: (1) There is a series in $\lambda_{00}$ all of whose subseries of density zero are convergent, and yet its partial sums are unbounded. (2) The Orlicz–Pettis theorem fails in $\lambda_{0}$. (3) $\lambda_{00}$ can be used to show that some earlier constructed normed barrelled spaces are not ultrabarrelled.