非交换齐性空间的度量方面
Metric aspects of noncommutative homogeneous spaces
摘要：
对于局部紧群$G$的闭余紧子群$Γ$，给定$G$的紧交换子群$K$和满足一定条件的同态$ρ:\hat{K}\to G$，Landstad和Raeburn构造了齐性空间$G/Γ$的等变非对易变形$C^*(\hat{G}/Γ, ρ)$，推广了Rieffel构造量子海森堡流形的方法。证明了当$G$是李群且$G/Γ$是连通的时，给定$G$的李代数上的任意范数，由正则$G$ -作用的导子映射诱导的$C^*(\hat{G}/Γ, ρ)$上的量子度规定义了一个紧的量子度规.此外，它表明，这个紧凑的量子度规空间依赖于$ρ$连续，关于量子Gromov-Hausdorff距离。
英文摘要：
For a closed cocompact subgroup $Γ$ of a locally compact group $G$, given a compact abelian subgroup $K$ of $G$ and a homomorphism $ρ:\hat{K}\to G$ satisfying certain conditions, Landstad and Raeburn constructed equivariant noncommutative deformations $C^*(\hat{G}/Γ, ρ)$ of the homogeneous space $G/Γ$, generalizing Rieffel's construction of quantum Heisenberg manifolds. We show that when $G$ is a Lie group and $G/Γ$ is connected, given any norm on the Lie algebra of $G$, the seminorm on $C^*(\hat{G}/Γ, ρ)$ induced by the derivation map of the canonical $G$-action defines a compact quantum metric. Furthermore, it is shown that this compact quantum metric space depends on $ρ$ continuously, with respect to quantum Gromov-Hausdorff distances.
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