1-Grothendieck C(K) spaces

A Banach space is said to be Grothendieck if weak and weak⁎ convergent sequences in the dual space coincide. This notion has been quantified by H. Bendová. She has proved that ℓ∞ℓ∞ 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)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.