When are full representations of algebras of operators on Banach spaces automatically faithful?

Abstract

We examine the phenomenon when surjective algebra homomorphisms between algebras of operators on Banach spaces are automatically injective. In the first part of the paper we shall show that for certain Banach spaces $X$ the following property holds: For every non-zero Banach space $Y$ every surjective algebra homomorphism $\psi :\mathcal{B}(X)\to \mathcal{B}(Y)$ is automatically injective. In the second part of the paper we consider the question in the opposite direction: Building on the work of Kania, Koszmider and Laustsen \textit{(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 $\psi :\mathcal{B}(Y_X)\to \mathcal{B}(X)$.

28 pp. The SHAI property of both real- and complex Hilbert spaces discussed. To appear in Studia Mathematica

References in corpus (5)

• Operators on Banach Spaces of Bourgain-Delbaen Type
• Closed ideals of operators on and complemented subspaces of Banach spaces of functions with countable support
• Closed Ideals of Operators between the Classical Sequence Spaces
• Local and global liftings of analytic families of idempotents in Banach algebras
• Ideals in $L(L_1)$

Cited by in corpus (2)

• Surjective homomorphisms from algebras of operators on long sequence spaces are automatically injective
• The SHAI property for the operators on L^p