Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1860_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation> be a linear mapping from a von Neumann algebra <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1860_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation> into itself. For <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1860_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">$a, b\in \mathcal {A},$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi mathvariant="script">A</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> the following statements are equivalent: <OrderedList> <ListItem> <ItemNumber>(1)</ItemNumber> <ItemContent> <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1860_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$ab=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mi>b</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> implies <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1860_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="124" /> </InlineMediaObject> <EquationSource Format="TEX">$b\delta (a)+\delta (b)a=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. </ItemContent> </ListItem> <ListItem> <ItemNumber>(2)</ItemNumber> <ItemContent> <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1860_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$ab^{}=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mmultiscripts> <mi>b</mi> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> implies <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1860_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="137" /> </InlineMediaObject> <EquationSource Format="TEX">$b^{}\delta (a)+\delta (b)^{}a=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mi>b</mi> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mi>δ</mi> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>δ</mi> <mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1860_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="70" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta (P)=0,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1860_Article_IEq9.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">$P\in \mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>P</mi> <mo>∈</mo> <mi mathvariant="script">A</mi> </mrow> </math></EquationSource> </InlineEquation> is the central projection such that <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1860_Article_IEq10.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">$P\mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>P</mi> <mi mathvariant="script">A</mi> </mrow> </math></EquationSource> </InlineEquation> contains no commutative direct summands. </ItemContent> </ListItem> <ListItem> <ItemNumber>(3)</ItemNumber> <ItemContent> <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1860_Article_IEq11.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta (a)=\xi a$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ξ</mi> <mi>a</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1860_Article_IEq12.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\xi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ξ</mi> </math></EquationSource> </InlineEquation> belongs to the type <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1860_Article_IEq13.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{I}_{1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mtext>I</mtext> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> direct summand of <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1860_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation>. </ItemContent> </ListItem> </OrderedList> If <InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1860_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>δ</mi> </math></EquationSource> </InlineEquation> is continuous, a similar conclusion holds for a <InlineEquation ID="IEq16"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1860_Article_IEq16.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$C^{}$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mi>C</mi> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> </math></EquationSource> </InlineEquation>-algebra <InlineEquation ID="IEq17"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1860_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation> mapping into <InlineEquation ID="IEq18"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1860_Article_IEq18.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {A}^{**}$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow> <mi mathvariant="script">A</mi> </mrow> <mrow /> <mrow> <mrow /> <mo>∗</mo> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> </math></EquationSource> </InlineEquation>, and for a locally compact group <InlineEquation ID="IEq19"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1860_Article_IEq19.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {G}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">G</mi> </math></EquationSource> </InlineEquation>, it remains valid for the group algebra <InlineEquation ID="IEq20"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1860_Article_IEq20.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$L^{1}(\mathcal {G})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> mapping into the measure convolution algebra <InlineEquation ID="IEq21"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1860_Article_IEq21.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$ M(\mathcal {G})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.

Multipliers of Operator Algebras: A Characterization via Antiderivable Mappings at Zero

期刊论文
发表日期：

2025年11月7日

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Wenbo Huang,
Fengjuan Meng

摘要

Let $\delta $ $δ$ be a linear mapping from a von Neumann algebra $\mathcal {A}$ $A$ into itself. For $a, b\in \mathcal {A},$ $a, b \in A,$ the following statements are equivalent: (1)

$ab=0$ $a b = 0$ implies $b\delta (a)+\delta (b)a=0$ $b δ (a) + δ (b) a = 0$ .

(2)

$ab^{*}=0$ $a b^{*} = 0$ implies $b^{*}\delta (a)+\delta (b)^{*}a=0$ $b^{*} δ (a) + δ {(b)}^{*} a = 0$ , and $\delta (P)=0,$ $δ (P) = 0,$ where $P\in \mathcal {A}$ $P \in A$ is the central projection such that $P\mathcal {A}$ $P A$ contains no commutative direct summands.

(3)

$\delta (a)=\xi a$ $δ (a) = ξ a$ , where $\xi $ $ξ$ belongs to the type $\textrm{I}_{1}$ $I_{1}$ direct summand of $\mathcal {A}$ $A$ .

If $\delta $

δ

is continuous, a similar conclusion holds for a $C^{*}$

C^{*}

-algebra $\mathcal {A}$

A

mapping into $\mathcal {A}^{**}$

A^{* *}

, and for a locally compact group $\mathcal {G}$

G

, it remains valid for the group algebra $L^{1}(\mathcal {G})$

L^{1} (G)

mapping into the measure convolution algebra $ M(\mathcal {G})$

M (G)