There is no universal separable Banach algebra

Abstract

We prove that no separable Banach algebra is universal for homomorphic embeddings of all separable Banach algebras, whether embeddings are merely bounded or required to be contractive. The same holds in the commutative category. The proof is short and direct. For each closed subspace $X$ of $c_0$ we build a separable commutative Banach algebra $A_X$ whose multiplication records the canonical pairing between $X^{\ast }$ and $X$. Any injective homomorphism of $A_X$ into a Banach algebra $B$ forces the identity operator on $X^{\ast }$ to factor through $B$. Thus a universal separable Banach algebra would be complementably universal for the family of duals of subspaces of $c_0$, contrary to a theorem of Johnson and Szankowski.

arXiv comment5 pp., accepted for publication in Journal of Functional Analysis