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

TL;DR

This paper characterizes positive automorphisms of certain operator subalgebras on atomic Archimedean vector lattices, showing they are spatial and implemented by permutations and positive diagonal operators.

Contribution

It proves that positive automorphisms of these subalgebras are necessarily spatial, extending the understanding of automorphism structure in operator algebras on vector lattices.

Findings

Positive automorphisms are spatial and implemented by permutations and positive diagonal operators.

Every finite-dimensional subspace of the vector lattice is order closed.

The Kakutani representation theorem is used to establish order closure of subspaces.

Abstract

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. 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.