Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation> be a von Neumann algebra without central abelian projections. We show that if <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ \delta : \mathcal {A} \rightarrow \mathcal {A} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo>:</mo> <mi mathvariant="script">A</mi> <mo stretchy="false">→</mo> <mi mathvariant="script">A</mi> </mrow> </math></EquationSource> </InlineEquation> is an additive map satisfying <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\delta ([A, B]_\xi )= [\delta (A), B]_\xi + [A, \delta (B)]_\xi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mrow> <mo stretchy="false">[</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">]</mo> </mrow> <mi>ξ</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mrow> <mo stretchy="false">[</mo> <mi>δ</mi> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mi>B</mi> <mo stretchy="false">]</mo> </mrow> <mi>ξ</mi> </msub> <mo>+</mo> <msub> <mrow> <mo stretchy="false">[</mo> <mi>A</mi> <mo>,</mo> <mi>δ</mi> <mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">]</mo> </mrow> <mi>ξ</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> for any <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ A, B \in \mathcal {A} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>∈</mo> <mi mathvariant="script">A</mi> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ AB=E $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mi>B</mi> <mo>=</mo> <mi>E</mi> </mrow> </math></EquationSource> </InlineEquation>, where E is an arbitrary but fixed operator of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation>, then there exists an additive derivation <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\theta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation> such that: (1) if <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\xi =1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ξ</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, then <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\delta =\theta +\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo>=</mo> <mi>θ</mi> <mo>+</mo> <mi>γ</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation> is an additive map into the center vanishing at every commutator [A, B] when <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$AB=E$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mi>B</mi> <mo>=</mo> <mi>E</mi> </mrow> </math></EquationSource> </InlineEquation>; (2) if <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\xi =0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ξ</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, then <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\delta (A)=\theta (A)+\delta (I)A$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mi>A</mi> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$A\in \mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>∈</mo> <mi mathvariant="script">A</mi> </mrow> </math></EquationSource> </InlineEquation>; (3) if <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\xi \ne 0, \pm 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ξ</mi> <mo>≠</mo> <mn>0</mn> <mo>,</mo> <mo>±</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, then <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$\delta (A)=\theta (A)+\delta (I)A$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mi>A</mi> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$A \in \mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>∈</mo> <mi mathvariant="script">A</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq19"> <EquationSource Format="TEX">$\theta (\xi I)=\xi \delta (I)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>ξ</mi> <mi>I</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ξ</mi> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>; (4) if <InlineEquation ID="IEq20"> <EquationSource Format="TEX">$\xi =-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ξ</mi> <mo>=</mo> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, then <InlineEquation ID="IEq21"> <EquationSource Format="TEX">$\delta =\theta .$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo>=</mo> <mi>θ</mi> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation>

Lie derivable maps on von neumann algebras

期刊论文
发表日期：

2025年12月16日

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Lei Liu,
Junkai Kong

摘要

Let $\mathcal {A}$ $A$ be a von Neumann algebra without central abelian projections. We show that if $ \delta : \mathcal {A} \rightarrow \mathcal {A} $ $δ : A \to A$ is an additive map satisfying $\delta ([A, B]_\xi )= [\delta (A), B]_\xi + [A, \delta (B)]_\xi $ $δ ({[A, B]}_{ξ}) = {[δ (A), B]}_{ξ} + {[A, δ (B)]}_{ξ}$ for any $ A, B \in \mathcal {A} $ $A, B \in A$ with $ AB=E $ $A B = E$ , where E is an arbitrary but fixed operator of $\mathcal {A}$ $A$ , then there exists an additive derivation $\theta $ $θ$ on $\mathcal {A}$ $A$ such that: (1) if $\xi =1$ $ξ = 1$ , then $\delta =\theta +\gamma $ $δ = θ + γ$ , where $\gamma $ $γ$ is an additive map into the center vanishing at every commutator [A, B] when $AB=E$ $A B = E$ ; (2) if $\xi =0$ $ξ = 0$ , then $\delta (A)=\theta (A)+\delta (I)A$ $δ (A) = θ (A) + δ (I) A$ for all $A\in \mathcal {A}$ $A \in A$ ; (3) if $\xi \ne 0, \pm 1$ $ξ \neq 0, \pm 1$ , then $\delta (A)=\theta (A)+\delta (I)A$ $δ (A) = θ (A) + δ (I) A$ for all $A \in \mathcal {A}$ $A \in A$ and $\theta (\xi I)=\xi \delta (I)$ $θ (ξ I) = ξ δ (I)$ ; (4) if $\xi =-1$ $ξ = - 1$ , then $\delta =\theta .$ $δ = θ .$