On positive automorphisms of algebras of operators on atomic Archimedean vector lattices

Cigler, Gregor (author), Kandić, Marko (author)

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 $a \otimes \varphi_b$, where $a$ and $b$ are atoms of $X$ and $\varphi_b$ denotes the coordinate functional associated with $b$. Our main result shows that every positive automorphism of such a subalgebra contained in ${\mathscr L}(c_{00}(\Lambda))$, 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. We also use the Kakutani representation theorem to establish that every finite-dimensional vector subspace of $X$ is order closed.

• Language: English
• Keywords: vector lattices, order algebra automorphisms, inner automorphisms, atom, order continuous operators
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.04.2026
• Year: 2026
• Number of pages: 26 str.
• Numbering: Vol. 30, iss. 2, article no. 30
• PID: 20.500.12556/RUL-181712
• UDC: 517.9
• ISSN on article: 1385-1292
• DOI: 10.1007/s11117-026-01190-y
• COBISS.SI-ID: 275059203
• Publication date in RUL: 14.04.2026

Record is a part of a journal

• Title: Positivity
• Shortened title: Positivity
• Publisher: Springer Nature
• ISSN: 1385-1292
• COBISS.SI-ID: 512122649

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Projects

• Funder: ARIS - Slovenian Research and Innovation Agency
• Project number: P1-0222
• Name: Algebra, teorija operatorjev in finančna matematika

• Funder: ARIS - Slovenian Research and Innovation Agency
• Project number: J1-50002
• Name: Realna algebraična geometrija v matričnih spremenljivkah