谱三元组、Coulhon-Varopoulos 维数与热核估计
Spectral triples, Coulhon-Varopoulos dimension and heat kernel estimates
AI总结：
本文研究谱三元组中完全有界局部 Coulhon-Varopoulos 维数与谱维数的关系，证明前者超过后者并蕴含 Hodge-Dirac 算子的可求和性，并进一步证明有限该维数的对称次马尔可夫半群迫使冯·诺依曼代数具有内射性。
摘要：
我们研究作用于有限测度空间上的经典（或非交换）$\mathrm{L}^p$-空间的次马尔可夫半群（或狄利克雷型）所关联的谱三元组的（完全有界）局部 Coulhon-Varopoulos 维数与谱维数之间的关系。更确切地说，我们证明完全有界局部 Coulhon-Varopoulos 维数 $d$ 超过谱维数，并且甚至蕴含相应的 Hodge-Dirac 算子是 $d^+$-可求和的。我们探索不同的情形来比较这两个值：紧黎曼流形、紧李群、次拉普拉斯算子、度量测度空间、非交换环面与量子群。具体地，我们证明，虽然在光滑紧致情形中这两个维数常常相等，但它们也可能出现分歧。最后，我们证明，在一个冯·诺依曼代数上存在具有有限完全有界局部 Coulhon-Varopoulos 维数的对称次马尔可夫半群，蕴含该冯·诺依曼代数必然是内射的。
英文摘要：
We investigate the relations between the (completely bounded) local Coulhon-Varopoulos dimension and the spectral dimension of spectral triples associated to sub-Markovian semigroups (or Dirichlet forms) acting on classical (or noncommutative) $\mathrm{L}^p$-spaces associated to finite measure spaces. More precisely, we prove that the completely bounded local Coulhon-Varopoulos dimension $d$ exceeds the spectral dimension and even implies that the associated Hodge-Dirac operator is $d^+$-summable. We explore different settings to compare these two values: compact Riemannian manifolds, compact Lie groups, sublaplacians, metric measure spaces, noncommutative tori and quantum groups. Specifically, we prove that, while very often equal in smooth compact settings, these dimensions can diverge. Finally, we show that the existence of a symmetric sub-Markovian semigroup on a von Neumann algebra with finite completely bounded local Coulhon-Varopoulos dimension implies that the von Neumann algebra is necessarily injective.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
