$C^*$-动力系统的Haagerup性质

On the Haagerup Property of $C^*$-Dynamical Systems

中文关键词:  $C^*$-动力系统  Haagerup性质  拟顺从作用

英文关键词:$C^*$-dynamical system  Haagerup property  quasi-amenable action

中文摘要:

      令$A$是一个单位$C^*$-代数, $\tau$是它的一个态, $\alpha$是一个离散群$G$在$A$上保持$\tau$的作用. 首先, 我们通过考虑 $C^*$-代数的态, 推广了动力系统的Haagerup性质, 并且证明了动力系统有 Haagerup性质当且仅当它的约化交叉积有Haagerup性质. 然后, 我们引入了$G$在$A$上关于$\tau$的拟顺从作用. 最后, 利用上面的结果, 我们证明了如果$\alpha$是$G$在$A$上关于$\tau$的拟顺从作用, 那么$(A,\tau)$有Haagerup性质当且仅当$(A\rtimes_{\alpha,r}G,\tau')$有Haagerup性质, 其中$\tau'$是由$\tau$诱导的$A\rtimes_{\alpha,r}G$上的态. 本文的主要结论推广了一些经典情况下的已知结果.

英文摘要:

      Let $A$ be a unital $C^*$-algebra with a state $\tau$ and $G$ be a discrete group that acts on $A$ through a $\tau$-preserving action $\alpha$. We first generalize the Haagerup property of dynamical systems by considering states and prove that the dynamical system has the Haagerup property if and only if the reduced crossed product does. Then we introduce the quasi-amenable action of $G$ on $A$ with respect to $\tau$. Finally, using the above results, we prove that if $\alpha$ is a quasi-amenable action of $G$ on $A$ with respect to $\tau$, then $(A,\tau)$ has the Haagerup property if and only if $(A\rtimes_{\alpha,r}G,\tau')$ does, where $\tau'$ is the induced state on $A\rtimes_{\alpha,r}G$. As a consequence, our main results improve some well known results in the classical situation.