  Let $X$ be a Banach space. Then $X$ is complemented in the bidual $X^{**}$ if
and only if there exists an invariant mean $\ell_\infty(G, X)\to X$ with
respect to a free Abelian group $G$ of rank equal to the cardinality of
$X^{**}$, and this happens if and only if there exists an invariant mean with
respect to the additive group of $X^{**}$. This improves upon previous results
due to Bustos Domecq =and the second-named author, where certain idempotent
semigroups of cardinality equal to the cardinality of $X^{**}$ were considered,
and answers a question of J.M.F. Castillo (private communication). En route to
the proof of the main result, we endow the family of all finite-dimensional
subspaces of an infinite-dimensional vector space with a structure of a free
commutative monoid with the property that the product of two subspaces contains
the respective subspaces, which is possibly of interest in itself.
