Embedding of a Banach algebra $A$ into $L(A')$

Given a Banach space $X$, we have shown in 1994 that a product can be defined on it in such a way that the resulting Banach algebra is isomorphic to a compact subalgebra of the algebra $L(X')$ of all bounded linear operators on the topological dual $X'$ of $X$. Our purpose here is to prove that, more generally, any Banach algebra $A$ admitting a left approximate identity, is isomorphic to a subalgebra of $L(A')$, the isomorphism being isometric, provided the approximate identity is bounded by 1. As a consequence, we get a factorization through $L(A')$, of the elements in $A'$.