Scholarlink

<p>An old and famous problem from the 1950s, popularized by Halmos, is that whether any pair of almost commuting contractive self-adjoint matrices are norm close to a pair of exactly commuting self-adjoint matrices? This question was solved affirmatively by Lin in the 1990's. In this paper, we study the general Halmos problem concerning unitary elements in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(C^*\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-algebras. Specifically, we first introduce the definition of A<InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathbb{T}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">T</mi> </math></EquationSource> </InlineEquation>-relations, and then we give a necessary and sufficient condition for the stability of A<InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(\mathbb{T}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">T</mi> </math></EquationSource> </InlineEquation>-relations in any unital infinitedimensional simple separable <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(C^*\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-algebra with tracial rank at most one. Finally, as applications, we show that many naturally occurring relations are A<InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\mathbb{T}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">T</mi> </math></EquationSource> </InlineEquation>-relations, and thus the stability results of these relations can be obtained by applying the above conclusions.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Stability of A\(\mathbb{T}\)-relations in \(C^*\)-algebras with tracial rank at most one

  • 期刊论文

  • 发表日期：

    2026年2月24日

查看全文
  • J. Hua

摘要

An old and famous problem from the 1950s, popularized by Halmos, is that whether any pair of almost commuting contractive self-adjoint matrices are norm close to a pair of exactly commuting self-adjoint matrices? This question was solved affirmatively by Lin in the 1990's. In this paper, we study the general Halmos problem concerning unitary elements in \(C^*\) C ∗ -algebras. Specifically, we first introduce the definition of A \(\mathbb{T}\) T -relations, and then we give a necessary and sufficient condition for the stability of A \(\mathbb{T}\) T -relations in any unital infinitedimensional simple separable \(C^*\) C ∗ -algebra with tracial rank at most one. Finally, as applications, we show that many naturally occurring relations are A \(\mathbb{T}\) T -relations, and thus the stability results of these relations can be obtained by applying the above conclusions.