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 uses the following scheme. To each bounded bilinear form $\beta$ we attach a separable test algebra $A(\beta)$ whose multiplication records $\beta$. Any homomorphic embedding of $A(\beta)$ into a candidate $B$ forces the linearisation of $\beta$ to factor through the fixed separable space $B\widehat{\otimes}_{\pi}B$. Choosing $\beta$ so that the associated operator fails to factor through $B\widehat{\otimes}_{\pi}B$, by the theorem of Johnson--Szankowski, yields a contradiction. In the commutative case, we take $\beta$ symmetric so $A(\beta)$ is commutative.