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Given a Banach space      X 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 ′ ) L(X’)       of all bounded linear operators on the topological dual      X ′ X’       of      X X      . Our purpose here is to prove that, more generally, any Banach algebra      A A       admitting a  left approximate identity , is isomorphic to a subalgebra of      L ( A ′ ) L(A’)      , the isomorphism being isometric, provided the approximate identity is bounded by 1. As a consequence, we get a factorization through      L ( A ′ ) L(A’)      , of the elements in      A ′ A’      .

Embedding of a Banach algebra 𝐴 into 𝐿(𝐴’)