Projections onto invariant subspaces of some Banach algebras (Q943548)

The author studies the weak\(^*\)-closed left translation invariant complemented subspace of semigroup algebras and group algebras. The author proved, among other things, that if \(G\) is a locally compact group and \(X\) is the range of a weak\(^*\)-weak\(^*\) continuous projection on \(L^\infty(G)\) commuting with left translation, then \(X\) admits a left translation invariant weak\(^*\)-closed complement. It is also proved that if \(G\) is an amenable locally compact group and \(A\) is a right Banach \(G\)-module, then any weak\(^*\)-closed translation invariant complemented subspace \(X\) of \(A^*\) is a range of a projection on \(A^*\) commuting with left translation.