Representations of Banach algebras as algebras of completely bounded maps

Abstract

We use the notion of $\Pi_2$-hyperreflexivity to construct, for a wide variety of Banach algebras $\mathcal A$, an operator space $X$ and a representation $\pi : {\mathcal A} \to CB(X)$, such that $CB(X)$ consists of $2$-summing perturbations of $\pi({\mathcal A})$. This gives rise to some examples of operator spaces with interesting properties.