标题： 关于长度函数的 Étale 群胚的$C^{\ast}$-代数上的度量 显示英文标题

标题： Metrics on $C^{\ast}$-algebras of Étale groupoids from length functions

摘要： 我们证明，对于具有紧致单位空间的平展群胚，由连续长度函数产生的自然Dirac型算子在相应约化$C^{\ast}$-代数的状态空间上产生一个自然伪度量。对于具有快速衰减的连续、正规长度函数的变换群胚，状态空间分解为均匀有界直径的真正度量空间，这些空间纤维化在紧致单位空间的状态空间上。此外，当变换群胚的单位空间只有有限多个点时，每条纤维上的度量都与弱*拓扑一致。 显示英文摘要

摘要： We show that for an \'etale groupoid with compact unit space, the natural Dirac type operator from a continuous length function produces a natural pseudo-metric on the state space of the corresponding reduced $C^{\ast}$-algebra. For a transformation groupoid with a continuous, proper length function with rapid decay, the state space decomposes into genuine metric spaces with a uniform finite diameter fibred over the state space of the compact unit space. Moreover, when the unit space of the transformation groupoid has finitely many points, the metric on each fibre metrizes the weak*-topology.