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<p>We show that the unitary group of any SOT-separable <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\textrm{II}_1\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>II</mtext> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> factor <i>M</i>, with the strong operator topology, is contractible. Combined with several old results, this implies that the same is true for any SOT-separable von Neumann algebra with no type <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\textrm{I}_n\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>I</mtext> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> direct summands (<InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(n &lt; \infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>&lt;</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>). The proof for the <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\textrm{II}_1\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>II</mtext> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>-factor case uses regularization via free convolution and Popa’s theorem on the existence of approximately free Haar unitaries in <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\textrm{II}_1\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>II</mtext> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> factors.</p>

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The unitary group of a \(\textrm{II}_1\) factor is SOT-contractible

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  • 发表日期：

    2025年9月27日

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  • David Jekel

摘要

We show that the unitary group of any SOT-separable \(\textrm{II}_1\) II 1 factor M, with the strong operator topology, is contractible. Combined with several old results, this implies that the same is true for any SOT-separable von Neumann algebra with no type \(\textrm{I}_n\) I n direct summands ( \(n < \infty \) n < ∞ ). The proof for the \(\textrm{II}_1\) II 1 -factor case uses regularization via free convolution and Popa’s theorem on the existence of approximately free Haar unitaries in \(\textrm{II}_1\) II 1 factors.