Year: 2026
Type: article
Abstract: Let X be an Archimedean vector lattice. We investigate subalgebras of $$\mathscr {L}(X)$$ L ( X ) consisting of regular operators that contain all rank-one operators of the form $$a \otimes \varphi _b$$ a ⊗ φ b , where a and b are atoms of X and $$\varphi _b$$ φ b denotes the coordinate functional associated with b . Our main result shows that every positive automorphism of such a subalgebra contained in $$\mathscr {L}(c_{00}(\Lambda ))$$ L ( c 00 ( Λ ) ) , is necessarily spatial, meaning that it is implemented by a transformation of the form $$ T \mapsto P D\, T\, D^{-1} P^{-1}, $$ T ↦ 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.
Source: Positivity
Authors: G. Cigler, M. Kandić
Language: English
Topic: Advanced Banach Space Theory
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W7153514049