A famous question of Halmos asks whether every operator on a separable infinite-dimensional Hilbert space is a norm limit of reducible operators. Voiculescu gave this problem an affirmative answer by his remarkable non-commutative Weyl-von Neumann theorem. In the paper, we investigate the analogous question for type <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\textrm{II}_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>II</mtext> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> factors. First, we give a characterization of Murray and von Neumann’s property <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> for type <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\textrm{II}_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>II</mtext> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> factors. Also, we develop a spectral gap property for a single operator in non-<InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> factors of type <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\textrm{II}_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>II</mtext> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>. Based on this spectral gap property, we prove that the set of reducible operators is nowhere dense in each non-<InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> factor of type <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\textrm{II}_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>II</mtext> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> in the operator norm topology, where separable and non-separable cases are both considered. In addition, we obtain equivalent formulations of McDuff factors.

Reducible operators in non-$\Gamma $ type $\textrm{II}_1$ factors

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发表日期：

2026年2月13日

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Junhao Shen,
Rui Shi

摘要

A famous question of Halmos asks whether every operator on a separable infinite-dimensional Hilbert space is a norm limit of reducible operators. Voiculescu gave this problem an affirmative answer by his remarkable non-commutative Weyl-von Neumann theorem. In the paper, we investigate the analogous question for type $\textrm{II}_1$ ${II}_{1}$ factors. First, we give a characterization of Murray and von Neumann’s property $\Gamma $ $Γ$ for type $\textrm{II}_1$ ${II}_{1}$ factors. Also, we develop a spectral gap property for a single operator in non- $\Gamma $ $Γ$ factors of type $\textrm{II}_1$ ${II}_{1}$ . Based on this spectral gap property, we prove that the set of reducible operators is nowhere dense in each non- $\Gamma $ $Γ$ factor of type $\textrm{II}_1$ ${II}_{1}$ in the operator norm topology, where separable and non-separable cases are both considered. In addition, we obtain equivalent formulations of McDuff factors.

Reducible operators in non-\(\Gamma \) type \(\textrm{II}_1\) factors

摘要