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

Desquith, Etienne

doi:10.1090/S0002-9939-96-03335-7

Embedding of a Banach algebra $A$ into $L(A’)$
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by Etienne Desquith

Proc. Amer. Math. Soc. 124 (1996), 2773-2778

DOI: https://doi.org/10.1090/S0002-9939-96-03335-7

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Abstract:

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 $A’$.

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