When are full representations of algebras of operators on Banach spaces automatically faithful?
Volume 253 / 2020
Abstract
We examine when surjective algebra homomorphisms between algebras of operators on Banach spaces are automatically injective. In the first part of the paper we show that for certain Banach spaces $X$ the following property holds: For every non-zero Banach space $Y$ every surjective algebra homomorphism $\psi : \, \mathcal {B}(X) \rightarrow \mathcal {B}(Y)$ is automatically injective. In the second part we consider the question in the opposite direction: Building on the work of Kania, Koszmider, and Laustsen [Trans. London Math. Soc., 2014] we show that for every separable, reflexive Banach space $X$ there is a Banach space $Y_X$ and a surjective but not injective algebra homomorphism $\psi : \, \mathcal {B}(Y_X) \rightarrow \mathcal {B}(X)$.
