Wickstead's conjecture on positive projections and non-representable Banach lattice algebras

TL;DR

This paper proves Wickstead's conjecture for positive projections on Dedekind complete Banach lattices and demonstrates the existence of non-representable 2-dimensional Banach lattice algebras.

Contribution

It confirms Wickstead's conjecture in general and addresses the representation problem for Banach lattice algebras.

Findings

Wickstead's conjecture holds for all Dedekind complete Banach lattices.

Existence of 2-dimensional Banach lattice algebras without faithful regular operator representations.

Abstract

Let $X$ be a Dedekind complete Banach lattice, and let $P\colon X\to X$ be a positive projection for which the largest central operator below $P$ is $\alpha \operatorname{id}_X$, for some $\alpha \ge 0$. Wickstead conjectured that $\alpha $ must either be $0$ or $1/n$, for some $n \in \mathbb{N}$, and proved it for finite-dimensional $X$. In this paper, we show that the conjecture holds in general. As a consequence, we settle the representation problem for Banach lattice algebras: we show that there exist Banach lattice algebras of dimension $2$ that do not admit a faithful representation as regular operators on any Dedekind complete Banach lattice.