arXiv math/02051952002-05math.OAmath.GRmath.MG

作为紧量子度量空间的群C* 代数

Group C*-algebras as compact quantum metric spaces

Marc A. Rieffel

备注：: 53 pages, yet more minor improvements. To appear in Doc. Math
发表信息：: Documenta Mathematica 7 (2002) 605-651

摘要：

设$\ell$是群$G$上的一个长度函数，$M_{\ell}$表示与$\bell^2(G)$上的$\ell$点乘的算子.在Connes之后，$M_{\ell}$可以用作$C_r^*(G)$的"Dirac“运算符。它在$C_r^*(G)$上定义了一个Lipschitz矩阵，该矩阵在$C_r^*(G)$的状态空间上定义了一个度量。我们研究这个度量的拓扑是否与弱-* 拓扑（我们对“紧致量子度量空间”的定义）一致。当$\ell$是一个字长时，我们对$G = {\mathbb Z}^d$给出了肯定的回答，或者对${\mathbb R}^d$的范数对${\mathbb Z}^d$的限制.这适用于被2-上循环扭曲的$C_r^*(G)$，因此也适用于非交换环面。我们的方法涉及Connes的余球面代数，以及与测地线密切相关的有趣的度量空间紧化。

英文摘要：

Let $\ell$ be a length function on a group $G$, and let $M_{\ell}$ denote the operator of pointwise multiplication by $\ell$ on $\bell^2(G)$. Following Connes, $M_{\ell}$ can be used as a ``Dirac'' operator for $C_r^*(G)$. It defines a Lipschitz seminorm on $C_r^*(G)$, which defines a metric on the state space of $C_r^*(G)$. We investigate whether the topology from this metric coincides with the weak-* topology (our definition of a ``compact quantum metric space''). We give an affirmative answer for $G = {\mathbb Z}^d$ when $\ell$ is a word-length, or the restriction to ${\mathbb Z}^d$ of a norm on ${\mathbb R}^d$. This works for $C_r^*(G)$ twisted by a 2-cocycle, and thus for non-commutative tori. Our approach involves Connes' cosphere algebra, and an interesting compactification of metric spaces which is closely related to geodesic rays.

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