  Banach spaces that are complemented in the second dual are characterised
precisely as those spaces $X$ which enjoy the property that for every amenable
semigroup $S$ there exists an $X$-valued analogue of an invariant mean defined
on the Banach space of all bounded $X$-valued functions on $S$. This was first
observed by Bustos Domecq (J. Math. Anal. Appl., 2002), however the original
proof was slightly flawed as remarked by Lipecki. The primary aim of this note
is to present a corrected version of the proof. We also demonstrate that
universally separably injective spaces always admit invariant means with
respect to countable amenable semigroups, thus such semigroups are not rich
enough to capture complementation in the second dual as spaces falling into
this class need not be complemented in the second dual.
