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

TL;DR

This paper investigates the structure of stopping-time Banach spaces, focusing on their bases and operator ideals, and develops criteria for the uniqueness of maximal ideals applicable to various classical Banach spaces.

Contribution

It introduces new criteria for the uniqueness of maximal ideals in operator algebras and applies them to stopping-time Banach spaces and classical spaces.

Findings

Established criteria for maximal ideal uniqueness

Analyzed canonical bases in stopping-time spaces

Applied results to classical Banach spaces such as L^p, BMO, and SL^∞

Abstract

Stopping-time Banach spaces is a collective term for the class of spaces of eventually null integrable processes that are defined in terms of the behaviour of the stopping times with respect to some fixed filtration. From the point of view of Banach space theory, these spaces in many regards resemble the classical spaces such as $L^1$ or $C(\Delta)$, but unlike these, they do have unconditional bases. In the present paper we study the canonical bases in the stopping-time spaces in relation to factorising the identity operator thereon. Since we work exclusively with the dyadic-tree filtration, this set-up enables us to work with tree-indexed bases rather than directly with stochastic processes. \emph{En route} to the factorisation results, we develop general criteria that allow one to deduce the uniqueness of the maximal ideal in the algebra of operators on a Banach space. These criteria are applicable to many classical Banach spaces such as (mixed-norm) $L^p$-spaces, BMO, $\mathrm{SL^\infty}$ and others.