论文标题
光谱三元组，库洪 - 瓦罗普洛斯尺寸和热核估计值
Spectral triples, Coulhon-Varopoulos dimension and heat kernel estimates
论文作者
论文摘要
我们研究了（完全有限的）局部库隆 - 瓦罗普洛斯维度与与经典（或非交通）$ \ mathrm {l}^p $ - 与有限量测量空间相关的经典（或非交通）$ \ mathrm {l}^p $ space相关的与亚马尔可夫半群（或dirichlet形式）相关的光谱尺寸之间的关系。更确切地说，我们证明了完全有限的本地coulhon-varopoulos尺寸$ d $超过光谱尺寸，甚至暗示着相关的hodge-dirac运算符为$ d^+$ - 可总结。我们探索不同的设置以比较这两个值：紧凑的riemannian歧管，紧凑的谎言组，sublaplacians，度量测量空间，非交通性Tori和量子组。具体而言，我们证明，尽管在平滑的紧凑设置中通常相等，但这些维度可能会有所不同。最后，我们表明，在von Neumann代数上存在对称次马克维亚次群的存在，具有有限的局部Coulhon-Varopoulos尺寸，这意味着von Neumann代数必然是注入性的。
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.
