Certain ideals in L1-algebras -- The property P1, applications -- Some properties equivalent to P1. Relativization -- Segal algebras -- Examples -- Approximate units in Banach algebras: definitions -- Approximate units in Banach algebras: lemmas -- Approximate units in Segal algebras -- Some general results on Segal algebras -- Certain ideals in Segal algebras -- Canonical representation of the ideals -- Segal algebras and normal subgroups -- Segal algebras on quotient groups -- Closed ideals with approximate units in Segal algebras -- Symmetric Segal algebras on compact groups -- Segal algebras on abelian groups -- L1-algebras of compact and of abelian groups