<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$K_1 \subset H$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>K</mi> <mn>1</mn> </msub> <mo>⊂</mo> <mi>H</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$K_2 \subset H$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>K</mi> <mn>2</mn> </msub> <mo>⊂</mo> <mi>H</mi> </mrow> </math></EquationSource> </InlineEquation> be half-sided modular inclusions in a common standard subspace <i>H</i>. We prove that the inclusion <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$K_1 \subset K_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>K</mi> <mn>1</mn> </msub> <mo>⊂</mo> <msub> <mi>K</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> holds if and only if we have an inclusion of spectral subspaces of the generators of the positive one-parameter groups associated to the half-sided modular inclusions <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$K_1 \subset H$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>K</mi> <mn>1</mn> </msub> <mo>⊂</mo> <mi>H</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$K_2 \subset H$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>K</mi> <mn>2</mn> </msub> <mo>⊂</mo> <mi>H</mi> </mrow> </math></EquationSource> </InlineEquation>. From this we give a characterization of this situation in terms of (operator-valued) symmetric inner functions. We illustrate these characterizations with some examples of non-trivial phenomena occurring in this setting.</p>

Relative positions of half-sided modular inclusions

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2025年10月29日

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Ian Koot

摘要

Let $K_1 \subset H$ $K_{1} \subset H$ and $K_2 \subset H$ $K_{2} \subset H$ be half-sided modular inclusions in a common standard subspace H. We prove that the inclusion $K_1 \subset K_2$ $K_{1} \subset K_{2}$ holds if and only if we have an inclusion of spectral subspaces of the generators of the positive one-parameter groups associated to the half-sided modular inclusions $K_1 \subset H$ $K_{1} \subset H$ and $K_2 \subset H$ $K_{2} \subset H$ . From this we give a characterization of this situation in terms of (operator-valued) symmetric inner functions. We illustrate these characterizations with some examples of non-trivial phenomena occurring in this setting.