<p>We show that for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$1\le p, q<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$p/q\notin \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo stretchy="false">/</mo> <mi>q</mi> <mo>∉</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>, the doubly atomless separable <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L_pL_q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>L</mi> <mi>p</mi> </msub> <msub> <mi>L</mi> <mi>q</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> Banach lattice <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$L_p([0,1]; L_q([0,1]))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>L</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> <mo>;</mo> <msub> <mi>L</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is approximately ultrahomogeneous (AUH) over the class of its finitely generated sublattices with lattice embeddings as corresponding maps. The above is not true when <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$p/q \in \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo stretchy="false">/</mo> <mi>q</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$p\ne q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≠</mo> <mi>q</mi> </mrow> </math></EquationSource> </InlineEquation>. However, for any <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$p\ne q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≠</mo> <mi>q</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$L_p(L_q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>L</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>L</mi> <mi>q</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is AUH over the finitely generated lattices in the class <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$BL_pL_q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <msub> <mi>L</mi> <mi>p</mi> </msub> <msub> <mi>L</mi> <mi>q</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> of bands of <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$L_pL_q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>L</mi> <mi>p</mi> </msub> <msub> <mi>L</mi> <mi>q</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> lattices. In both scenarios, we exploit certain equimeasurability properties which hold in <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$L_p([0,1];L_q([0,1]))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>L</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> <mo>;</mo> <msub> <mi>L</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, and some of which fail to hold when <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$p/q \in \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo stretchy="false">/</mo> <mi>q</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>, to achieve both of the desired results.</p>

Approximate ultrahomogeneity in $L_pL_q$ lattices

期刊论文
发表日期：

2026年1月23日

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Mary Angelica Tursi

摘要

We show that for $1\le p, q<\infty $ $1 \leq p, q < \infty$ with $p/q\notin \mathbb {N}$ $p / q \notin N$ , the doubly atomless separable $L_pL_q$ $L_{p} L_{q}$ Banach lattice $L_p([0,1]; L_q([0,1]))$ $L_{p} ([0, 1]; L_{q} ([0, 1]))$ is approximately ultrahomogeneous (AUH) over the class of its finitely generated sublattices with lattice embeddings as corresponding maps. The above is not true when $p/q \in \mathbb {N}$ $p / q \in N$ and $p\ne q$ $p \neq q$ . However, for any $p\ne q$ $p \neq q$ , $L_p(L_q)$ $L_{p} (L_{q})$ is AUH over the finitely generated lattices in the class $BL_pL_q$ $B L_{p} L_{q}$ of bands of $L_pL_q$ $L_{p} L_{q}$ lattices. In both scenarios, we exploit certain equimeasurability properties which hold in $L_p([0,1];L_q([0,1]))$ $L_{p} ([0, 1]; L_{q} ([0, 1]))$ , and some of which fail to hold when $p/q \in \mathbb {N}$ $p / q \in N$ , to achieve both of the desired results.

Approximate ultrahomogeneity in \(L_pL_q\) lattices

摘要