标题： 理想量子度量从分数拉普拉斯算子而来 显示英文标题

标题： Ideal quantum metrics from fractional Laplacians

摘要： 我们发展了一种新颖的框架，用于利用施密特理想和Ahlfors正则空间上分数阶Laplacian的交换子来定义Monge-Kantorovich度量。值得注意的是，对于这些度量，我们得到了以高阶分数阶Laplacian的谱表示的闭合公式。为了证明这些结果，我们在非交换几何中开发了新的技术，特别是Weyl定律和施密特类交换子，从而在Borel概率测度空间上产生了精细的量子度量。最后，我们的分数分析扩展到了动力系统。我们通过扩展的代数 $\mathbb Z^m$-作用和某些双曲动力系统的同宿 $C^*$-代数展示了这一点。这些发现展示了分数分析在分形几何、动力系统和非交换几何中的多功能性。 显示英文摘要

摘要： We develop a novel framework for Monge--Kantorovi\v{c} metrics using Schatten ideals and commutators of fractional Laplacians on Ahlfors regular spaces. Notably, for those metrics we derive closed formulas in terms of spectra of higher-order fractional Laplacians. For our proofs we develop new techniques in noncommutative geometry, in particular a Weyl law and Schatten-class commutators, yielding refined quantum metrics on the space of Borel probability measures. Lastly, our fractional analysis extends to dynamical systems. We showcase this in the setting of expansive algebraic $\mathbb Z^m$-actions and homoclinic $C^*$-algebras of certain hyperbolic dynamical systems. These findings illustrate the versatility of fractional analysis in fractal geometry, dynamical systems and noncommutative geometry.