(ℤ/2ℤ)≀𝔽_d的字长谱三元组不是度量
Word-Length Spectral Triples of $(\mathbb{Z}/2\mathbb{Z})\wr\mathbb{F}_{d}$ Are Not Metric
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中文总结 AI 辅助
本文针对非交换度量几何中紧量子度量空间的判定问题，以灯夫群(ℤ/2ℤ)≀𝔽_d为研究对象，证明其典范谱三元组无法成为谱度量空间，给出该领域首个明确的反例族。
中文摘要 AI 辅助
给定一个配备真长度函数的可数离散群，可在其约化群C*-代数上构造一个自然的谱三元组。非交换度量几何中一个被广泛研究的问题是，此类三元组对应的Connes伪度量是否能恢复态空间上的弱*拓扑，从而得到Rieffel意义下的紧量子度量空间。尽管已知该度量性质对若干类群成立，包括多项式增长群和字双曲群，但学界曾普遍认为并非每个字长函数都能诱导紧量子度量空间，不过迄今尚未找到明确的反例。本文中，我们通过证明对每个整数d≥2，配备有限对称生成集对应的字长函数的灯夫群(ℤ/2ℤ)≀𝔽_d的典范谱三元组，无法成为谱度量空间，给出了首个反例族。
英文摘要
Given a countable discrete group equipped with a proper length function, one can construct a natural spectral triple on its reduced group C$^{\ast}$-algebra. A well-studied question in non-commutative metric geometry is whether the Connes pseudo-metric associated with such a triple recovers the weak$^{\ast}$-topology on the state space, thereby yielding a compact quantum metric space in the sense of Rieffel. While this metric property is known to hold for several classes of groups - including those of polynomial growth and word-hyperbolic groups - it was widely expected that not every word-length function induces a compact quantum metric space. Despite this, no explicit counterexample has been identified to date. In this note, we provide the first family of counterexamples by proving that for every integer $d \geq 2$ the canonical spectral triple of the Lamplighter group $(\mathbb{Z}/2\mathbb{Z})\wr\mathbb{F}_{d}$, equipped with the word-length function associated with a finite symmetric generating set, fails to be a spectral metric space.
