<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\big ((A^{(i)}, G, \alpha ^{(i)}), \phi _i\big )_{i \in \mathbb {N}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <msup> <mi>A</mi> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> <mi>G</mi> <mo>,</mo> <msup> <mi>α</mi> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <msub> <mi>ϕ</mi> <mi>i</mi> </msub> <msub> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mrow> <mi>i</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> be an inductive sequence of partial dynamical systems. We prove the existence of an induced partial action <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$G$</EquationSource> <EquationSource Format="MATHML"><math> <mi>G</mi> </math></EquationSource> </InlineEquation> on the inductive limit <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$A=\varinjlim A^{(i)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>=</mo> <munder accentunder="true"> <mo movablelimits="false">lim</mo> <mo stretchy="false">→</mo> </munder> <msup> <mi>A</mi> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>. We call <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> the inductive limit partial action. Furthermore, we show the corresponding partial crossed product <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$A\rtimes _{\alpha }G$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <msub> <mo>⋊</mo> <mi>α</mi> </msub> <mi>G</mi> </mrow> </math></EquationSource> </InlineEquation> is canonically isomorphic to <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\varinjlim A^{(i)}\rtimes _{\alpha ^{(i)}}G$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <munder accentunder="true"> <mo movablelimits="false">lim</mo> <mo stretchy="false">→</mo> </munder> <msup> <mi>A</mi> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> </msup> <msub> <mo>⋊</mo> <msup> <mi>α</mi> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> </msup> </msub> <mi>G</mi> </mrow> </math></EquationSource> </InlineEquation>. We also study the globalization of the inductive limit partial action <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>, its finite Rokhlin dimension and tracial states on <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$A\rtimes _{\alpha }G$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <msub> <mo>⋊</mo> <mi>α</mi> </msub> <mi>G</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

Inductive limits of partial crossed products

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2026年1月5日

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Md Amir Hossain

摘要

Let $\big ((A^{(i)}, G, \alpha ^{(i)}), \phi _i\big )_{i \in \mathbb {N}}$ $((A^{(i)}, G, α^{(i)}), ϕ_{i})_{i \in N}$ be an inductive sequence of partial dynamical systems. We prove the existence of an induced partial action $\alpha $ $α$ of $G$ $G$ on the inductive limit $A=\varinjlim A^{(i)}$ $A = lim_{\to} A^{(i)}$ . We call $\alpha $ $α$ the inductive limit partial action. Furthermore, we show the corresponding partial crossed product $A\rtimes _{\alpha }G$ $A ⋊_{α} G$ is canonically isomorphic to $\varinjlim A^{(i)}\rtimes _{\alpha ^{(i)}}G$ $lim_{\to} A^{(i)} ⋊_{α^{(i)}} G$ . We also study the globalization of the inductive limit partial action $\alpha $ $α$ , its finite Rokhlin dimension and tracial states on $A\rtimes _{\alpha }G$ $A ⋊_{α} G$ .