<p>A <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_104_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda$</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-graph system is a labeled Bratteli diagram with certain additional structure, which presents a subshift. The class of the <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_104_Article_IEq4.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>-algebras <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_104_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal{O}_ \mathfrak{L}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">O</mi> <mi mathvariant="fraktur">L</mi> </msub> </math></EquationSource> </InlineEquation> associated with <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_104_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda$</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-graph systems is a generalized class of the class of Cuntz–Krieger algebras. In this paper, we will compute the strong extension groups<InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_104_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="70" /> </InlineMediaObject> <EquationSource Format="TEX">$ Ext _{\rm s} (\mathcal{O}_ \mathfrak{L})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>E</mi> <mi>x</mi> <msub> <mi>t</mi> <mi mathvariant="normal">s</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="script">O</mi> <mi mathvariant="fraktur">L</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>for the <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_104_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$C^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-algebras associated with <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_104_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda$</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-graph systems <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_104_Article_IEq10.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$ \mathfrak{L} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">L</mi> </math></EquationSource> </InlineEquation> and study their relation with the weak extension group <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_104_Article_IEq11.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">$ Ext _{\rm w} (\mathcal{O}_ \mathfrak{L})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>E</mi> <mi>x</mi> <msub> <mi>t</mi> <mi mathvariant="normal">w</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="script">O</mi> <mi mathvariant="fraktur">L</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

Extension groups for the $C^*$ -algebras associated with $\lambda$-graph systems

期刊论文
发表日期：

2025年8月13日

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K. Matsumoto

摘要

A $\lambda$ $λ$ -graph system is a labeled Bratteli diagram with certain additional structure, which presents a subshift. The class of the $C^{*}$ $C^{*}$ -algebras $\mathcal{O}_ \mathfrak{L}$ $O_{L}$ associated with $\lambda$ $λ$ -graph systems is a generalized class of the class of Cuntz–Krieger algebras. In this paper, we will compute the strong extension groups $ Ext _{\rm s} (\mathcal{O}_ \mathfrak{L})$ $E x t_{s} (O_{L})$ for the $C^*$ $C^{*}$ -algebras associated with $\lambda$ $λ$ -graph systems $ \mathfrak{L} $ $L$ and study their relation with the weak extension group $ Ext _{\rm w} (\mathcal{O}_ \mathfrak{L})$ $E x t_{w} (O_{L})$ .

Extension groups for the \(C^*\) -algebras associated with \(\lambda\)-graph systems

摘要