Surjective homomorphisms from algebras of operators on long sequence spaces are automatically injective

Abstract

We study automatic injectivity of surjective algebra homomorphisms from B(X), the algebra of (bounded, linear) operators on X, to B(Y ), where X is one of the following long sequence spaces: c0(λ), c ∞ (λ), and p(λ) (1 p < ∞) 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'.

The SHAI property for the operators on L^p

“…The continuity assumption is redundant by an automatic continuity theorem of B. E. Johnson [5,Theorem 5.1.5]. The spaces ℓ p for 1 ≤ p ≤ ∞ are known to have the SHAI property [9, Proposition 1.2], as do some other classical spaces [9], [10], but there are many spaces that do not have the SHAI property [9]. Our research on the SHAI property was motivated by the problem mentioned by Horvath [9] whether L p = L p (0, 1) has the SHAI property.…”

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 generalizes and unifies previous results of Calkin [3] for X = ℓ 2 , Gohberg, Markus and Feldman for X = c 0 or X = ℓ p , 1 p < ∞, and Gramsch [9] and Luft [21] independently for X = ℓ 2 (Γ), where Γ is an arbitrary infinite cardinal.…”