Surjective Homomorphisms from Algebras of Operators on Long Sequence Spaces are Automatically Injective

Abstract

We study automatic injectivity of surjective algebra homomorphisms from $\mathscr{B}(X)$, the algebra of (bounded, linear) operators on X, to $\mathscr{B}(Y)$, where X is one of the following long sequence spaces: c0(λ), $\ell_{\infty}^c(\lambda)$, and $\ell_p(\lambda)$ ($1 \leqslant p \lt \infty$) and Y is arbitrary. En route to the proof that these spaces do indeed enjoy such a property, we classify two-sided ideals of the algebra of operators of any of the aforementioned Banach spaces that are closed with respect to the ‘sequential strong operator topology’.

Smooth norms in dense subspaces of $\ell_p(Γ)$ and operator ranges

“…This ceases to be true for uncountable Γ, as the next result shows. It is essentially due to Rosenthal [18] and, in a slightly weaker form, it can also be found in [13,Lemma 3.8]. Yet, a direct proof is so short that we give it here for the sake of completeness.…”

Factorisation in stopping-time Banach spaces: identifying unique maximal ideals

“…It follows from the above-mentioned results by Apatsidis that M X coincides with the set S S 1 (S 1 ) comprising all S 1 -singular operators, that is operators that do not fix any copies of S 1 . As S 1 is complementably homogeneous S S 1 (S 1 ) (hence M S 1 ) is the unique maximal ideal of B(S 1 ) (see [14,Corollary 2.3]).…”

Classifying the closed ideals of bounded operators on two families of non-separable classical Banach spaces

“…An alternative description of the closed ideals of B(X) is given in [12,Theorem 1.5]; see also [10,Theorem 3.7]. Daws' theorem generalises and unifies previous results of Calkin [3] for X = 2 , Gohberg, Markus and Feldman [8] for X = c 0 or X = p , 1 p < ∞, and Gramsch [9] and Luft [21] independently for X = 2 (Γ), where Γ is an arbitrary infinite cardinal.…”