迹唯一的简单约化$L^p$算子交叉积
Simple reduced $L^p$ operator crossed products with unique trace
摘要：
在这篇文章中，我们研究简化$L^p$算子交叉产品$F^p_{\mathrm{r}}(G, A, α)$的简单性和痕迹。给定$p \in (1, \infty)$，设$G$是幂群，$α\colon G \to Aut(A)$是$G$在有单位的$L^p$算子代数$A$上的等距作用，使得$A$是$G$ -单的.证明了$A$与$G$的约化$L^p$算子交叉积$F^p_{\mathrm{r}}(G, A, α)$是简单的.此外，我们还证明了$F^p_{\mathrm{r}}(G, A, α)$上的迹与A上的$G$ -不变迹是一致的.我们的结果推广了de la Harpe在1985年关于约化$C^*$交叉积的结果.作为一种特殊情况，设$G$是可数非交换自由群，我们得到了Powers在1975年的一个结果的类似结果.对于$p = 1$情形，证明了（约化）$L^p$算子群代数不是单的.
英文摘要：
In this article we study simplicity and traces of reduced $L^p$ operator crossed products $F^p_{\mathrm{r}}(G, A, α)$. Given $p \in (1, \infty)$, let $G$ be a Powers group, and let $α\colon G \to Aut(A)$ be an isometric action of $G$ on a unital $L^p$ operator algebra $A$ such that $A$ is $G$-simple. We prove that the reduced $L^p$ operator crossed product of $A$ by $G$, $F^p_{\mathrm{r}}(G, A, α)$, is simple. Moreover, we show that traces on $F^p_{\mathrm{r}}(G, A, α)$ are in correspondence with $G$-invariant traces on A. Our results generalize the results obtained by de la Harpe for reduced $C^*$crossed products in 1985. By letting $G$ be a countable nonabelian free group as a special case, we recover an analogue of a result of Powers from 1975. For the case $p = 1$, it turns out that (reduced) $L^p$ operator group algebras are not simple.
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