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arXiv · 2604.14697

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

David Muñoz-Lahoz
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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. All the main results in this paper have been formalized in Lean 4 using the Banach lattice Lean library.

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David Muñoz-Lahoz. 2026-04-16. Wickstead's conjecture on positive projections and non-representable Banach lattice algebras. https://arxiv.org/abs/2604.14697

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