Maximal right ideals of the Banach algebra of bounded operators on a Banach space

Abstract

We study finitely generated maximal right ideals of the Banach algebra $\mathcal{B}(E)$ of bounded operators on a complex Banach space $E$. Every maximal right ideal is either fixed by a non-zero functional or contains the ideal of finite-rank operators; when $E$ is infinite-dimensional, each non-fixed maximal right ideal in fact contains the ideal of inessential operators. Using the elementary representation of finitely generated right ideals as lifting ideals $\operatorname{Lift}(T)=\{TU:U\in\mathcal{B}(E,E^n)\}$, where $T\in\mathcal{B}(E^n,E)$ for some $n\in\mathbb{N}$, we identify the exact operator-theoretic obstruction. The ideal $\operatorname{Lift}(T)$ contains the finite-rank operators precisely when $T$ is surjective, and it equals $\mathcal{B}(E)$ precisely when $T$ is right invertible. If $T$ is surjective but not right invertible, then $\operatorname{Lift}(T)$ is maximal exactly when the row operator $[T\ S]$ is right invertible for every $S\in\mathcal{B}(E)\setminus\operatorname{Lift}(T)$. We apply this framework, together with duality, pullback, lattice-theoretic and cardinality arguments, to obtain maximal right ideals which are not finitely generated for large classes of Banach spaces. These include the following infinite-dimensional spaces: reflexive spaces, separable spaces with an unconditional Schauder decomposition into a countably infinite sequence of non-zero subspaces, spaces containing a complemented copy of $\ell_1$, KB-spaces, Lebesgue spaces $L_p(μ)$ for $1\leqslant p<\infty$, full Orlicz spaces with order-continuous norm, and scalar-plus-compact spaces. We obtain the stronger conclusion that every finitely generated maximal right ideal is fixed for Hilbert spaces, $\ell_1(Γ)$-spaces, reflexive spaces with the bounded approximation property, and the mixed spaces $\ell_1(Γ)\oplus H$ with $H$ a separable Hilbert space.