Abstract Let X be an Archimedean vector lattice. We investigate subalgebras of L (X) L (X) consisting of regular operators that contain all rank-one operators of the form a b a ⊗ φ b, where a and b are atoms of X and b φ b denotes the coordinate functional associated with b. Our main result shows that every positive automorphism of such a subalgebra contained in L (c₀₀ () ) L (c 00 (Λ) ), is necessarily spatial, meaning that it is implemented by a transformation of the form T 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.
Cigler et al. (Wed,) studied this question.