Approximate identities for Ideals in L(Lᵖ))

The main result is that the only non trivial closed ideal in the Banach algebra L(Lᵖ) of bounded linear operators on Lᵖ(0,1), 1≤ p < ∞, that has a left approximate identity is the ideal of compact operators. The algebra L(L¹) 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.