1-Grothendieck $C(K)$ spaces

TL;DR

This paper proves that certain totally disconnected compact spaces with specific algebraic properties have their associated $C(K)$ spaces being 1-Grothendieck, extending the class of known spaces with this property.

Contribution

It establishes that $C(K)$ spaces are 1-Grothendieck when $K$ is totally disconnected with the Subsequential completeness property.

Findings

$C(K)$ spaces are 1-Grothendieck under specified conditions.

Extension of the class of spaces with the Grothendieck property.

Connection between algebraic properties of $K$ and Banach space properties.

Abstract

A Banach space is said to be Grothendieck if weak and weak$^*$ convergent sequences in the dual space coincide. This notion has been quantificated by H. Bendov\'{a}. She has proved that $\ell_\infty$ has the quantitative Grothendieck property, namely, it is 1-Grothendieck. Our aim is to show that Banach spaces from a certain wider class are 1-Grothendieck, precisely, $C(K)$ is 1-Grothendieck provided $K$ is a totally disconnected compact space such that its algebra of clopen subsets has the so called Subsequential completeness property.