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Abstract <p> We prove that a unitary representation <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11503_2025_3241_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\rho\)</EquationSource> </InlineEquation> of an amenable locally compact group <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11503_2025_3241_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(G\)</EquationSource> </InlineEquation> such that <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11503_2025_3241_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="180" /> </InlineMediaObject> <EquationSource Format="TEX">\(\|\rho(g)-\pi(g)\|\le q&lt;1/2\)</EquationSource> </InlineEquation> for all <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11503_2025_3241_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">\(g\in G\)</EquationSource> </InlineEquation> and for some continuous unitary representation <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11503_2025_3241_Article_IEq5.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\pi\)</EquationSource> </InlineEquation> of <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11503_2025_3241_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(G\)</EquationSource> </InlineEquation> in the same Hilbert space is unitary equivalent to <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11503_2025_3241_Article_IEq5.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\pi\)</EquationSource> </InlineEquation>. </p> <p> <b> DOI</b> 10.1134/S1061920825020153 </p>

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The Topology of Uniform Convergence on a Group Is Discrete for Unitary Representations of Amenable Groups

  • 期刊论文

  • 发表日期：

    2025年6月1日

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  • A.I. Shtern

摘要

Abstract

We prove that a unitary representation \(\rho\) of an amenable locally compact group \(G\) such that \(\|\rho(g)-\pi(g)\|\le q<1/2\) for all \(g\in G\) and for some continuous unitary representation \(\pi\) of \(G\) in the same Hilbert space is unitary equivalent to \(\pi\) .

DOI 10.1134/S1061920825020153