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

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