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

Abstract Let X be an Archimedean vector lattice. We investigate subalgebras of $$\mathscr {L}(X)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> consisting of regular operators that contain all rank-one operators of the form $$a \otimes \varphi _b$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>⊗</mml:mo> <mml:msub> <mml:mi>φ</mml:mi> <mml:mi>b</mml:mi> </mml:msub> </mml:mrow> </mml:math> , where a and b are atoms of X and $$\varphi _b$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>φ</mml:mi> <mml:mi>b</mml:mi> </mml:msub> </mml:math> denotes the coordinate functional associated with b . Our main result shows that every positive automorphism of such a subalgebra contained in $$\mathscr {L}(c_{00}(\Lambda ))$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>c</mml:mi> <mml:mn>00</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Λ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , is necessarily spatial, meaning that it is implemented by a transformation of the form $$ T \mapsto P D\, T\, D^{-1} P^{-1}, $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>↦</mml:mo> <mml:mi>P</mml:mi> <mml:mi>D</mml:mi> <mml:mspace/> <mml:mi>T</mml:mi> <mml:mspace/> <mml:msup> <mml:mi>D</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:msup> <mml:mi>P</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> 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.