On positive automorphisms of algebras of operators on atomic Archimedean vector lattices
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| Format: | Preprint |
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2026
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| _version_ | 1866915761203707904 |
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| author | Cigler, Gregor Kandić, Marko |
| author_facet | Cigler, Gregor Kandić, Marko |
| contents | Let $X$ be an Archimedean vector lattice. We investigate subalgebras of $\mathscr{L}(X)$ consisting of regular operators that contain all rank-one operators of the form $a \otimes φ_b$, where $a$ and $b$ are atoms of $X$ and $φ_b$ denotes the coordinate functional associated with $b$. Our main result shows that every positive automorphism of such a subalgebra contained in $\mathscr{L}(c_{00}(Λ))$, is necessarily spatial, meaning that it is implemented by a transformation of the form $$ T \mapsto P D\, T\, D^{-1} P^{-1}, $$ where $P$ is a permutation operator and $D$ is a positive diagonal operator. An important tool for this analysis-one that is also of independent interest-is the Kakutani representation theorem, which we use to establish that every finite-dimensional vector subspace of $X$ is order closed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_21811 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On positive automorphisms of algebras of operators on atomic Archimedean vector lattices Cigler, Gregor Kandić, Marko Functional Analysis 46A40, 47B65, 16W20, 47L05 Let $X$ be an Archimedean vector lattice. We investigate subalgebras of $\mathscr{L}(X)$ consisting of regular operators that contain all rank-one operators of the form $a \otimes φ_b$, where $a$ and $b$ are atoms of $X$ and $φ_b$ denotes the coordinate functional associated with $b$. Our main result shows that every positive automorphism of such a subalgebra contained in $\mathscr{L}(c_{00}(Λ))$, is necessarily spatial, meaning that it is implemented by a transformation of the form $$ T \mapsto P D\, T\, D^{-1} P^{-1}, $$ where $P$ is a permutation operator and $D$ is a positive diagonal operator. An important tool for this analysis-one that is also of independent interest-is the Kakutani representation theorem, which we use to establish that every finite-dimensional vector subspace of $X$ is order closed. |
| title | On positive automorphisms of algebras of operators on atomic Archimedean vector lattices |
| topic | Functional Analysis 46A40, 47B65, 16W20, 47L05 |
| url | https://arxiv.org/abs/2601.21811 |