Let X be an Archimedean vector lattice. We investigate subalgebras of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathscr {L}(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">L</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> consisting of regular operators that contain all rank-one operators of the form <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$a \otimes \varphi _b$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>⊗</mo> <msub> <mi>φ</mi> <mi>b</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, where a and b are atoms of X and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\varphi _b$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>φ</mi> <mi>b</mi> </msub> </math></EquationSource> </InlineEquation> denotes the coordinate functional associated with b. Our main result shows that every positive automorphism of such a subalgebra contained in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathscr {L}(c_{00}(\Lambda ))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">L</mi> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mn>00</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, is necessarily spatial, meaning that it is implemented by a transformation of the form <Equation ID="Equ5"> <EquationSource Format="TEX">$ T \mapsto P D\, T\, D^{-1} P^{-1}, $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>T</mi> <mo>↦</mo> <mi>P</mi> <mi>D</mi> <mspace width="0.166667em" /> <mi>T</mi> <mspace width="0.166667em" /> <msup> <mi>D</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>P</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> </mrow> </math></EquationSource> </Equation>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.

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

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2026年4月11日

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G. Cigler,
M. Kandić

摘要

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