Approximate identities for ideals of Segal algebras on a compact group (Q1614771)

A left approximate identity \((e_\alpha)\) of a normed algebra \(A\) satisfies condition \((U)\) if for every compact set \(K\subset A\) \(\|e_\alpha- a\|\to 0\) uniformly for \(a\in K\). It is proved that for a compact group \(G\) every closed two-sided ideal of \(L^1(G)\) has an approximate identity that lies in the center of \(L^1(G)\) and satisfies condition \((U)\). Moreover it is proved that every closed left (right) ideal \(E\) of \(L^1(G)\) has a right (left) approximate identity satisfying condition \((U)\) if and only if it is an approximately complemented subspace in \(L^1(G)\). The last property means that there is a net of continuous operators \(P_\alpha: X\to E\) such that \(\lim P_\alpha x=x\) uniformly on compact subsets of \(E\). The same results are proved for a Segal algebra \(S^1(G)\) of a compact group \(G\).