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

Let $X$ be an Archimedean vector lattice. We investigate subalgebras of $\mathcal{L}(X)$ consisting of regular operators that contain all rank-one operators of the form $a\otimes {\phi }_b$, where $a$ and $b$ are atoms of $X$ and ${\phi }_b$ denotes the coordinate functional associated with $b$. Our main result shows that every positive automorphism of such a subalgebra contained in $\mathcal{L}(c_{00}(\mathrm{\Lambda }))$, is necessarily spatial, meaning that it is implemented by a transformation of the form $$T\mapsto PDT{D}^{-1}{P}^{-1},$$ 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.