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

@article{Horvath2018WhenAF,
title={When are full representations of algebras of operators on Banach spaces automatically faithful?},
author={Bence Horv'ath},
journal={arXiv: Functional Analysis},
year={2018}
}
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 $\psi: \, \mathcal{B}(X) \rightarrow \mathcal{B}(Y)$ is automatically injective. In the second part of the paper we consider the question in the opposite direction: Building… Expand
2 Citations
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#### References

SHOWING 1-10 OF 57 REFERENCES
MAXIMAL IDEALS IN THE ALGEBRA OF OPERATORS ON CERTAIN BANACH SPACES
For a Banach space $\mathfrak{X}$, let $\mathcal{B}(\mathfrak{X})$ denote the Banach algebra of all continuous linear operators on $\mathfrak{X}$. First, we study the lattice of closed ideals inExpand
A hierarchy of Banach spaces with $C(K)$ Calkin Algebras
• Mathematics
• 2014
For every well founded tree $\mathcal{T}$ having a unique root such that every non-maximal node of it has countable infinitely many immediate successors, we construct a $\mathcal{L}_\infty$-spaceExpand
Local and global liftings of analytic families of idempotents in Banach algebras
• Mathematics
• 2014
Generalizing results of our earlier paper, we investigate the following question. Let $\pi(\lambda) : A \to B$ be an analytic family of surjective homomorphisms between two Banach algebras, andExpand
Closed ideals of operators on and complemented subspaces of Banach spaces of functions with countable support
• Mathematics
• 2015
Let $\lambda$ be an infinite cardinal number and let $\ell_\infty^c(\lambda)$ denote the subspace of $\ell_\infty(\lambda)$ consisting of all functions that assume at most countably many non-zeroExpand
ON RING-THEORETIC (IN)FINITENESS OF BANACH ALGEBRAS OF OPERATORS ON BANACH SPACES
Let B(X) denote the Banach algebra of all bounded linear operators on a Banach space X. We show that B(X) is finite if and only if no proper, complemented subspace of X is isomorphic to X, and weExpand
Operators on two Banach spaces of continuous functions on locally compact spaces of ordinals
• Mathematics
• 2015
Denote by $[0,\omega_1)$ the set of countable ordinals, equipped with the order topology, let $L_0$ be the disjoint union of the compact ordinal intervals $[0,\alpha]$ for $\alpha$ countable, andExpand
Banach algebras and automatic continuity
Banach algebras combine algebraic and analytical aspects: it is the interplay of these structures that gives the subject its fascination. This volume expounds the general theory of Banach algebras,Expand
The lattice of closed ideals in the Banach algebra of operators on certain Banach spaces
• Mathematics
• 2004
Very few Banach spaces E are known for which the lattice of closed ideals in the Banach algebra of all (bounded, linear) operators on E is fully understood. Indeed, up to now the only such BanachExpand
Ideal structure of the algebra of bounded operators acting on a Banach space
• Mathematics
• 2015
We construct a Banach space $Z$ such that the lattice of closed two-sided ideals of the Banach algebra $\mathscr{B}(Z)$ of bounded operators on $Z$ is as follows:  \{0\}\subsetExpand
Operators on Banach spaces of Bourgain-Delbaen type
We begin by giving a detailed exposition of the original Bourgain-Delbaen construction and the generalised construction due to Argyros and Haydon. We show how these two constructions are related, andExpand