<p>A compact quantum metric space is a unital <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$C^{}$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mi>C</mi> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> </math></EquationSource> </InlineEquation>-algebra equipped with a Lip-norm. Let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\{(A_n, L_n)\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mi>n</mi> </msub> <mo>,</mo> <msub> <mi>L</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> be a sequence of compact quantum metric spaces, and let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\phi _n:A_n\rightarrow A_{n+1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ϕ</mi> <mi>n</mi> </msub> <mo>:</mo> <msub> <mi>A</mi> <mi>n</mi> </msub> <mo stretchy="false">→</mo> <msub> <mi>A</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> be a unital <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$^{}$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow /> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> </math></EquationSource> </InlineEquation>-homomorphism preserving Lipschitz elements for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$n\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. We show that there exists a compact quantum metric space structure on the inductive limit <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\varinjlim (A_n,\phi _n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <munder accentunder="true"> <mo movablelimits="false">lim</mo> <mo stretchy="false">→</mo> </munder> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mi>n</mi> </msub> <mo>,</mo> <msub> <mi>ϕ</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> by means of the inverse limit of the state spaces <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\{\mathcal {S}(A_n)\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mi mathvariant="script">S</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>. We also give some sufficient conditions that two inductive limits of compact quantum metric spaces are Lipschitz isomorphic.</p>

Inductive Limits of Compact Quantum Metric Spaces

期刊论文
发表日期：

2025年11月18日

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Botao Long,
Ghadir Sadeghi

摘要

A compact quantum metric space is a unital $C^{*}$ $C^{*}$ -algebra equipped with a Lip-norm. Let $\{(A_n, L_n)\}$ ${(A_{n}, L_{n})}$ be a sequence of compact quantum metric spaces, and let $\phi _n:A_n\rightarrow A_{n+1}$ $ϕ_{n} : A_{n} \to A_{n + 1}$ be a unital $^{*}$ $^{*}$ -homomorphism preserving Lipschitz elements for $n\ge 1$ $n \geq 1$ . We show that there exists a compact quantum metric space structure on the inductive limit $\varinjlim (A_n,\phi _n)$ $lim_{\to} (A_{n}, ϕ_{n})$ by means of the inverse limit of the state spaces $\{\mathcal {S}(A_n)\}$ ${S (A_{n})}$ . We also give some sufficient conditions that two inductive limits of compact quantum metric spaces are Lipschitz isomorphic.