On positive automorphisms of algebras of operators on atomic Archimedean vector lattices
Gregor Cigler, Marko Kandić
TL;DR
This paper analyzes positive automorphisms of algebras of regular operators on atomic Archimedean vector lattices. It shows that for subalgebras containing all rank-one atoms, any positive automorphism on a subalgebra of $\mathscr L(c_{00}(\Lambda))$ is spatial and has the form $T \mapsto P D\,T\,D^{-1} P^{-1}$, where $P$ is a permutation operator and $D$ a positive diagonal operator; the Kakutani representation is employed to prove order-closedness of finite-dimensional subspaces, a key structural tool. Specializing to $X=c_{00}(\Lambda)$, the authors prove that all order-continuous and bounded operator algebras satisfy $\mathscr L(c_{00}(\Lambda))=\mathscr L_n(c_{00}(\Lambda))=\mathscr L_b(c_{00}(\Lambda))$, and every positive automorphism of $\mathscr L(c_{00}(\Lambda))$ or $\mathscr B(c_{00}(\Lambda))$ is inner, with explicit $P$ and $D$ and bounded inverses. The paper also clarifies the role of $\mathscr A_0$ as the finite-rank-operator building block, provides criteria for when automorphisms preserve this subalgebra, and discusses order-bounded functionals and lexicographic products to situate the results in a broader lattice-theoretic context.
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 $\mathscr{L}(c_{00}(Λ))$, 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.