Do non-unital Banach subalgebras of $B(L^p)$ have contractive approximate identities?

Question

As stated in the title, I would like to know whether for $p\in[1,\infty)$, a Banach subalgebra of $B(L^p(X,\mu))$ (or a Banach algebra isometrically isomorphic to such a subalgebra) has a contractive approximate identity if it does not already contain a unit.

Answer 1

No. Even for $p=2$: consider the subalgebra $A$ of $M_2$ consisting of strictly upper triangular matrices. (Equivalently, consider $\mathbb{C}$ with its usual norm and the trivial product given by $xy=0$ for all $x,y\in \mathbb{C}$.) Then $A$ is a subalgebra of $\mathcal{B}(\ell^2(\{0,1\}))$ (or a subalgebra of $\mathcal{B}(\ell^p(\{0,1\}))$ for any $p\in [1,\infty)$), but it has no approximate identity, contractive or not.