<p>Let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\mathcal {A}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation> be a <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$C^\star$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>⋆</mo> </msup> </math></EquationSource> </InlineEquation>-algebra. Consider two continuous linear maps, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\delta$</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\tau$</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation>, from <InlineEquation ID="IEq7"> <EquationSource Format="TEX">${\mathcal {A}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation> to its double dual <InlineEquation ID="IEq8"> <EquationSource Format="TEX">${\mathcal {A}}^{**}$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow> <mi mathvariant="script">A</mi> </mrow> <mrow /> <mrow> <mrow /> <mo>∗</mo> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> </math></EquationSource> </InlineEquation>. We assume that <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\delta$</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\tau$</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation> behave like generalized derivations on orthogonal elements in <InlineEquation ID="IEq11"> <EquationSource Format="TEX">${\mathcal {A}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation>, satisfying conditions like <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$ab = 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mi>b</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$ab^\star = 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <msup> <mi>b</mi> <mo>⋆</mo> </msup> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$a^\star b = 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>a</mi> <mo>⋆</mo> </msup> <mi>b</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. We prove the existence of standard solutions for these maps and apply this result to characterize various types of mappings, such as (right or left) centralizers and (generalized) derivations vanishing on zero products, and their local variations. Some of our findings generalize previously established results in this area.</p>

Characterization of Generalized Derivations on $C^\star$-Algebras via Orthogonal Actions and Their Applications

期刊论文
发表日期：

2025年7月21日

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Behrooz Fadaee,
Eghbal Ghaderi,
Saber Naseri

摘要

Let ${\mathcal {A}}$ $A$ be a $C^\star$ $C^{⋆}$ -algebra. Consider two continuous linear maps, $\delta$ $δ$ and $\tau$ $τ$ , from ${\mathcal {A}}$ $A$ to its double dual ${\mathcal {A}}^{**}$ $A^{* *}$ . We assume that $\delta$ $δ$ and $\tau$ $τ$ behave like generalized derivations on orthogonal elements in ${\mathcal {A}}$ $A$ , satisfying conditions like $ab = 0$ $a b = 0$ , $ab^\star = 0$ $a b^{⋆} = 0$ , and $a^\star b = 0$ $a^{⋆} b = 0$ . We prove the existence of standard solutions for these maps and apply this result to characterize various types of mappings, such as (right or left) centralizers and (generalized) derivations vanishing on zero products, and their local variations. Some of our findings generalize previously established results in this area.

Characterization of Generalized Derivations on \(C^\star\)-Algebras via Orthogonal Actions and Their Applications

摘要