<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${\mathbf {\mathcal {A}}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation> be a unital infinite dimensional semisimple Banach algebra and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\mathbf {\Phi ({\mathcal {A}})}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">A</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be the set of Fredholm elements in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ {\mathbf {\mathcal {A}}} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation>. An element <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$a \in \mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>∈</mo> <mi mathvariant="script">A</mi> </mrow> </math></EquationSource> </InlineEquation> is called <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ {\mathbf {\Phi }}({\mathbf {\mathcal {A}}}) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">A</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-consistent provided that <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$ab\in \Phi (\mathcal {A})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mi>b</mi> <mo>∈</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">A</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> if and only if <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$ba\in \Phi (\mathcal {A})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mi>a</mi> <mo>∈</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">A</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for every <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$b\in \mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mo>∈</mo> <mi mathvariant="script">A</mi> </mrow> </math></EquationSource> </InlineEquation>. We first characterize the <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\Phi (\mathcal {A})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">A</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-consistent elements and show that the set of such elements forms an upper semiregularity. Building on this, we introduce the consistent Fredholm spectrum, establish its spectral mapping theorem, and obtain a characterization for algebraic elements in <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation>. As an application, we characterize the stability of <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\Phi (\mathcal {A})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">A</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-consistent elements in terms of nullity and defect in primitive <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$c^*$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>c</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation> - algebras.</p>

Consistency of Fredholmness Under Products in Semisimple Banach Algebras

期刊论文
发表日期：

2026年7月27日

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Nan Li,
Lining Jiang,
Tengjie Zhang

摘要

Let ${\mathbf {\mathcal {A}}}$ $A$ be a unital infinite dimensional semisimple Banach algebra and ${\mathbf {\Phi ({\mathcal {A}})}}$ $Φ (A)$ be the set of Fredholm elements in $ {\mathbf {\mathcal {A}}} $ $A$ . An element $a \in \mathcal {A}$ $a \in A$ is called $ {\mathbf {\Phi }}({\mathbf {\mathcal {A}}}) $ $Φ (A)$ -consistent provided that $ab\in \Phi (\mathcal {A})$ $a b \in Φ (A)$ if and only if $ba\in \Phi (\mathcal {A})$ $b a \in Φ (A)$ for every $b\in \mathcal {A}$ $b \in A$ . We first characterize the $\Phi (\mathcal {A})$ $Φ (A)$ -consistent elements and show that the set of such elements forms an upper semiregularity. Building on this, we introduce the consistent Fredholm spectrum, establish its spectral mapping theorem, and obtain a characterization for algebraic elements in $\mathcal {A}$ $A$ . As an application, we characterize the stability of $\Phi (\mathcal {A})$ $Φ (A)$ -consistent elements in terms of nullity and defect in primitive $c^*$ $c^{*}$ - algebras.