Invariant vector means and complementability of Banach spaces in their second duals

Abstract

Let X be a Banach space. Fix a torsion-free commutative and cancellative semigroup S whose torsion-free rank is the same as the density of X^{**}. We then show that X is complemented in X^{**} if and only if there exists an invariant mean M:ell _infty (S,X)rightarrow X. This improves upon previous results due to Bustos Domecq (J Math Anal Appl 275(2):512–520, 2002), Kania (J Math Anal Appl 445:797–802, 2017), Goucher and Kania (Studia Math 260:91–101, 2021).