Wickstead's conjecture on positive projections and non-representable Banach lattice algebras
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917419924062208 |
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| author | Muñoz-Lahoz, David |
| author_facet | Muñoz-Lahoz, David |
| contents | 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 $α\operatorname{id}_X$, for some $α\ge 0$. Wickstead conjectured that $α$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_14697 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Wickstead's conjecture on positive projections and non-representable Banach lattice algebras Muñoz-Lahoz, David Functional Analysis Operator Algebras 47B65, 46B42, 46A40, 06F25 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 $α\operatorname{id}_X$, for some $α\ge 0$. Wickstead conjectured that $α$ 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. |
| title | Wickstead's conjecture on positive projections and non-representable Banach lattice algebras |
| topic | Functional Analysis Operator Algebras 47B65, 46B42, 46A40, 06F25 |
| url | https://arxiv.org/abs/2604.14697 |