Invariantly complemented and amenability in Banach algebras related to locally compact groups (Q527039)

For a locally compact group \(G\), let \(LUC(G)\) be the space of right uniformly continuous functions on \(G\) and let \(\mathcal{B} ( LUC(G) ) \) be the space of bounded linear operators on \(LUC(G)\). In this paper, the authors relate the amenability of \(G\) with the existence of projections in \(\mathcal{B} ( LUC(G) ) \). The authors also completely determine the weak\(^*\) closed left translation invariant subspace \(X\) of \(LUC(G)\) which is the range of a weak\(^*\)-weak\(^*\) continuous projection \(P\) on \(LUC(G)\) commuting with left translations. The concept of approximately complemented subspaces of Banach algebras associated to \(G\) is also studied.