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Abstract <p>This article investigates the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8237_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(C^{*}\)</EquationSource> <!--LobJMat2560555Grigoryan-m1--> </InlineEquation>-algebras generated by regular representations of free products of abelian semigroups. It provides a comprehensive analysis of the structure of these <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8237_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(C^{*}\)</EquationSource> <!--LobJMat2560555Grigoryan-m2--> </InlineEquation>-algebras, employing tools from semigroup theory, operator algebras, and harmonic analysis. A criterion for the simplicity of these <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8237_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(C^{*}\)</EquationSource> <!--LobJMat2560555Grigoryan-m3--> </InlineEquation>-algebras is established, connecting the algebraic properties of the underlying semigroup to the ideal structure of the generated <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8237_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(C^{*}\)</EquationSource> <!--LobJMat2560555Grigoryan-m4--> </InlineEquation>-algebra. Specifically, the existence of a finite basis for the semigroup is shown to be equivalent to the inclusion of the compact operators as a subalgebra, while the absence of a finite basis is equivalent to the <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8237_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(C^{*}\)</EquationSource> <!--LobJMat2560555Grigoryan-m5--> </InlineEquation>-algebra being simple. The introduction of monomial indices and the construction of Fell bundles are key techniques used to obtain these results.</p>

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Inverse Semigroups Generated by a Free Product of Abelian Semigroups

  • 期刊论文

  • 发表日期：

    2025年3月1日

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  • S. A. Grigoryan,
  • T. A. Grigoryan

摘要

Abstract

This article investigates the \(C^{*}\) -algebras generated by regular representations of free products of abelian semigroups. It provides a comprehensive analysis of the structure of these \(C^{*}\) -algebras, employing tools from semigroup theory, operator algebras, and harmonic analysis. A criterion for the simplicity of these \(C^{*}\) -algebras is established, connecting the algebraic properties of the underlying semigroup to the ideal structure of the generated \(C^{*}\) -algebra. Specifically, the existence of a finite basis for the semigroup is shown to be equivalent to the inclusion of the compact operators as a subalgebra, while the absence of a finite basis is equivalent to the \(C^{*}\) -algebra being simple. The introduction of monomial indices and the construction of Fell bundles are key techniques used to obtain these results.