We shall show a property of decomposability of the generalized Fredholm element in a general context of Banach algebras. This is an extension of [<CitationRef CitationID="CR22">22</CitationRef>, Theorem 1.1]. More precisely, for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$t \in \mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>∈</mo> <mi mathvariant="script">A</mi> </mrow> </math></EquationSource> </InlineEquation> which is a complex semisimple Banach algebra with identity, we shall prove the following main result: t is a generalized Fredholm element if and only if there exist an idempotent p in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation>, a Fredholm element <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$t_{1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>t</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$p\mathcal {A}p$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mi mathvariant="script">A</mi> <mi>p</mi> </mrow> </math></EquationSource> </InlineEquation> and a nilpotent element <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$t_{2}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>t</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$soc((1-p)\mathcal {A}(1-p))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mi>o</mi> <mi>c</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mi mathvariant="script">A</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$t=t_{1} + t_{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>=</mo> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>t</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>.

On generalized Fredholm elements in the context of Banach algebras

期刊论文
发表日期：

2026年7月22日

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Youness Hadder,
Abdelkhalek El Amrani

摘要

We shall show a property of decomposability of the generalized Fredholm element in a general context of Banach algebras. This is an extension of [22, Theorem 1.1]. More precisely, for $t \in \mathcal {A}$ $t \in A$ which is a complex semisimple Banach algebra with identity, we shall prove the following main result: t is a generalized Fredholm element if and only if there exist an idempotent p in $\mathcal {A}$ $A$ , a Fredholm element $t_{1}$ $t_{1}$ in $p\mathcal {A}p$ $p A p$ and a nilpotent element $t_{2}$ $t_{2}$ in $soc((1-p)\mathcal {A}(1-p))$ $s o c ((1 - p) A (1 - p))$ such that $t=t_{1} + t_{2}$ $t = t_{1} + t_{2}$ .