用酉群刻画群$C^\ast$ -代数：阿贝尔情形
Characterizing group $C^\ast$-algebras through their unitary groups: the Abelian case
摘要：
研究了群$C^\ast$ -代数在多大程度上被它们的酉群刻画.给出了交换群$C^\ast$ -代数具有同构的酉群的一个完整刻画.我们比较这些结果与其他酉相关的不变量的$C^\ast(Γ)$，如$K$理论的$K_1(C^\ast(Γ))$和发现，$C^\ast$ -代数的非同构无挠阿贝尔群可能有同构的$K_1$ -群，在鲜明的对比，众所周知的事实，即$C^\ast(Γ)$（甚至$Γ$）是由其酉群的拓扑群结构的特点时，$Γ$是无挠和阿贝尔。
英文摘要：
We study to what extent group $C^\ast$-algebras are characterized by their unitary groups. A complete characterization of which Abelian group $C^\ast$-algebras have isomorphic unitary groups is obtained. We compare these results with other unitary-related invariants of $C^\ast(Γ)$, such as the $K$-theoretic $K_1(C^\ast(Γ))$ and find that $C^\ast$-algebras of nonisomorphic torsion-free Abelian groups may have isomorphic $K_1$-groups, in sharp contrast with the well-known fact that $C^\ast(Γ)$ (even $Γ$) is characterized by the topological group structure of its unitary group when $Γ$ is torsion-free and Abelian.
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