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

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.