This is an announcement for the paper "1-Grothendieck $C(K)$ spaces" by Jindrich Lechner.

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.

Archive classification: math.FA

Submitted from: jindrich.lechner@seznam.cz

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1511.02202

or

http://arXiv.org/abs/1511.02202