We generalize the Li–Yang notion of self-similar k-graph <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43037_2024_401_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">$(G,\Lambda )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and its <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43037_2024_401_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$C^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-algebra <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43037_2024_401_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {O}}_{G,\Lambda }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">O</mi> <mrow> <mi>G</mi> <mo>,</mo> <mi mathvariant="normal">Λ</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> to any finitely aligned k-graph <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43037_2024_401_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\Lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation>. We then introduce an inverse semigroup model for <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43037_2024_401_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {O}}_{G,\Lambda }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">O</mi> <mrow> <mi>G</mi> <mo>,</mo> <mi mathvariant="normal">Λ</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> and analyze its tight groupoid and <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43037_2024_401_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$C^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-algebra via inverse semigroup methods.

An inverse semigroup approach to self-similar k-graph $C^*$-algebras and simplicity

期刊论文
发表日期：

2025年1月9日

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Hossein Larki

摘要

We generalize the Li–Yang notion of self-similar k-graph $(G,\Lambda )$ $(G, Λ)$ and its $C^*$ $C^{*}$ -algebra ${\mathcal {O}}_{G,\Lambda }$ $O_{G, Λ}$ to any finitely aligned k-graph $\Lambda $ $Λ$ . We then introduce an inverse semigroup model for ${\mathcal {O}}_{G,\Lambda }$ $O_{G, Λ}$ and analyze its tight groupoid and $C^*$ $C^{*}$ -algebra via inverse semigroup methods.

An inverse semigroup approach to self-similar k-graph \(C^*\)-algebras and simplicity

摘要