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

Let $X$ be a Dedekind complete Banach lattice, and let $P\text{\hspace{0.17em}⁣}:X\to X$ be a positive projection for which the largest central operator below $P$ is $\alpha {\mathrm{id}}_X$, for some $\alpha \ge 0$. Wickstead conjectured that $\alpha$ must either be $0$ or $1\mathrm{/}n$, 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.