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

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 ⊗ φ b a ⊗ φ b , where a and b are atoms of X and φ 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(Λ )) 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.