In this paper, among other things, we show that there is a close connection between the existence of a bounded projection on some Banach algebras associated to a locally compact group~$G$ and the existence of a left invariant mean on $L^\infty (G)$. A necessary and sufficient condition is found for a locally compact group to possess a left invariant mean.
"Invariantly complemented and amenability in Banach algebras related to locally compact groups." Rocky Mountain J. Math. 47 (2) 445 - 461, 2017. https://doi.org/10.1216/RMJ-2017-47-2-445