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Abstract <p>Inspired by the condition <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8341_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="186" /> </InlineMediaObject> <EquationSource Format="TEX">\(\sup_{\lambda\in\mathbb{C}}||e^{\lambda Y}Xe^{-\lambda Y}||&lt;\infty\)</EquationSource> <!--LobJMat2560781Bikchentaev-m1--> </InlineEquation>, which is equivalent to the commutativity of two operators <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8341_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="97" /> </InlineMediaObject> <EquationSource Format="TEX">\(X,Y\in\mathcal{B}(\mathcal{H})\)</EquationSource> <!--LobJMat2560781Bikchentaev-m2--> </InlineEquation>, we establish that for a state <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8341_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varphi\)</EquationSource> <!--LobJMat2560781Bikchentaev-m3--> </InlineEquation> on a von Neumann algebra <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8341_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal{M}\)</EquationSource> <!--LobJMat2560781Bikchentaev-m4--> </InlineEquation> the following conditions are equivalent: (i) <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8341_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varphi\)</EquationSource> <!--LobJMat2560781Bikchentaev-m5--> </InlineEquation> is tracial; (ii) <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8341_Article_IEq6.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="200" /> </InlineMediaObject> <EquationSource Format="TEX">\(\sup_{\lambda\in\mathbb{C}}|\varphi(e^{\lambda Y}Xe^{-\lambda Y})|&lt;\infty\)</EquationSource> <!--LobJMat2560781Bikchentaev-m6--> </InlineEquation> for all positive operators <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8341_Article_IEq7.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">\(X,Y\in\mathcal{M}\)</EquationSource> <!--LobJMat2560781Bikchentaev-m7--> </InlineEquation>; (iii) <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8341_Article_IEq8.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="166" /> </InlineMediaObject> <EquationSource Format="TEX">\(|\varphi(\textrm{Re}(X^{2})|\leq\varphi(X^{*}X)\)</EquationSource> <!--LobJMat2560781Bikchentaev-m8--> </InlineEquation> for all <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8341_Article_IEq9.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="62" /> </InlineMediaObject> <EquationSource Format="TEX">\(X\in\mathcal{M}\)</EquationSource> <!--LobJMat2560781Bikchentaev-m9--> </InlineEquation>; (iv) <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8341_Article_IEq10.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="253" /> </InlineMediaObject> <EquationSource Format="TEX">\(\varphi(X^{*}Y+Y^{*}X)=\varphi(XY^{*}+YX^{*})\)</EquationSource> <!--LobJMat2560781Bikchentaev-m10--> </InlineEquation> for all unitary operators <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8341_Article_IEq7.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">\(X,Y\in\mathcal{M}\)</EquationSource> <!--LobJMat2560781Bikchentaev-m11--> </InlineEquation>. We also provide new criteria for the commutativity of von Neumann algebras.</p>

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Characterization of Tracial Functionals on von Neumann Algebras

  • 期刊论文

  • 发表日期：

    2025年6月1日

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  • A. M. Bikchentaev,
  • M. S. Moslehian,
  • V. Zh. Sakbaev

摘要

Abstract

Inspired by the condition \(\sup_{\lambda\in\mathbb{C}}||e^{\lambda Y}Xe^{-\lambda Y}||<\infty\) , which is equivalent to the commutativity of two operators \(X,Y\in\mathcal{B}(\mathcal{H})\) , we establish that for a state \(\varphi\) on a von Neumann algebra \(\mathcal{M}\) the following conditions are equivalent: (i) \(\varphi\) is tracial; (ii) \(\sup_{\lambda\in\mathbb{C}}|\varphi(e^{\lambda Y}Xe^{-\lambda Y})|<\infty\) for all positive operators \(X,Y\in\mathcal{M}\) ; (iii) \(|\varphi(\textrm{Re}(X^{2})|\leq\varphi(X^{*}X)\) for all \(X\in\mathcal{M}\) ; (iv) \(\varphi(X^{*}Y+Y^{*}X)=\varphi(XY^{*}+YX^{*})\) for all unitary operators \(X,Y\in\mathcal{M}\) . We also provide new criteria for the commutativity of von Neumann algebras.