Approximate identities for ideals in L(Lp)$L(L^p)$

Abstract The main result is that the only non‐trivial closed ideal in the Banach algebra of bounded linear operators on , , that has a left approximate identity is the ideal of compact operators. The algebra has at least one non‐trivial closed ideal that has a contractive right approximate identity as well as many, including the unique maximal ideal, that do not have a right approximate identity.