<p>Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_409_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="67" /> </InlineMediaObject> <EquationSource Format="TEX">$({\mathcal {A}},G,\alpha )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">A</mi> <mo>,</mo> <mi>G</mi> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be a separable groupoid <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_409_Article_IEq2.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>-dynamical system and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_409_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">$C_0(G^{(0)},{\mathcal {A}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>G</mi> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> <mi mathvariant="script">A</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> the <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_409_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> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-algebra of continuous sections that vanish at infinity. When <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_409_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="67" /> </InlineMediaObject> <EquationSource Format="TEX">$({\mathcal {A}},G,\alpha )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">A</mi> <mo>,</mo> <mi>G</mi> <mo>,</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> has the approximation property, we prove that the crossed product <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_409_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="70" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {A}}\rtimes _{\alpha ,r}G$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">A</mi> <msub> <mo>⋊</mo> <mrow> <mi>α</mi> <mo>,</mo> <mi>r</mi> </mrow> </msub> <mi>G</mi> </mrow> </math></EquationSource> </InlineEquation> is exact if and only if <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_409_Article_IEq7.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">$C_0(G^{(0)},{\mathcal {A}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>G</mi> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> <mi mathvariant="script">A</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is exact. In particular, if <i>G</i> is topologically amenable and <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_409_Article_IEq8.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">$C_0(G^{(0)},{\mathcal {A}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>G</mi> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> <mi mathvariant="script">A</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is exact, then <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43034_2025_409_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="70" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {A}}\rtimes _{\alpha ,r}G$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">A</mi> <msub> <mo>⋊</mo> <mrow> <mi>α</mi> <mo>,</mo> <mi>r</mi> </mrow> </msub> <mi>G</mi> </mrow> </math></EquationSource> </InlineEquation> is exact.</p>

On the exactness of groupoid crossed products

期刊论文
发表日期：

2025年2月7日

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Changyuan Gao

摘要

Let $({\mathcal {A}},G,\alpha )$ $(A, G, α)$ be a separable groupoid $C^*$ $C^{*}$ -dynamical system and $C_0(G^{(0)},{\mathcal {A}})$ $C_{0} (G^{(0)}, A)$ the $C^*$ $C^{*}$ -algebra of continuous sections that vanish at infinity. When $({\mathcal {A}},G,\alpha )$ $(A, G, α)$ has the approximation property, we prove that the crossed product ${\mathcal {A}}\rtimes _{\alpha ,r}G$ $A ⋊_{α, r} G$ is exact if and only if $C_0(G^{(0)},{\mathcal {A}})$ $C_{0} (G^{(0)}, A)$ is exact. In particular, if G is topologically amenable and $C_0(G^{(0)},{\mathcal {A}})$ $C_{0} (G^{(0)}, A)$ is exact, then ${\mathcal {A}}\rtimes _{\alpha ,r}G$ $A ⋊_{α, r} G$ is exact.