Approximate identities for Ideals in $L(L^p))$

The main result is that the only non trivial closed ideal in the Banach algebra $L(L^p)$ of bounded linear operators on $L^p(0,1)$, $1\le p<\mathrm{\infty }$, that has a left approximate identity is the ideal of compact operators. The algebra $L(L^1)$ 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.