论文标题
$ C^*$ - 代数的本地紧凑型组的行动的舒适性和较弱的遏制
Amenability and weak containment for actions of locally compact groups on $C^*$-algebras
论文作者
论文摘要
在这项工作中，我们介绍并研究了在$ C^*$ - 代数上对本地紧凑型组的行动的新概念。我们的定义扩展了因Claire Anantharaman-Delararoche而导致离散群体行为的舒适性的定义。我们表明，我们的定义具有类似于离散情况下已知的属性的几种特征和持久性属性。例如，对于交换性$ c^*$ - 代数的行动，我们表明我们的不舒适性概念等于衡量敏感性。加上Alex Bearden和Jason Crann的最新结果，这也解决了关于拓扑舒适性和第二个可数$ G $ -G $ -SPACE $ x $的等效性的长期开放问题。当最大和减少的交叉产品同意时，我们使用新的修正性概念来研究。我们的主要结果之一概括了Matsumura的定理：我们表明，对于局部紧凑型$ g $的动作，在本地紧凑的空间上$ x $ $ x $ $ c_0（x）\ rtimes_ \ rtimes_ \ max g $和$ c_0（x）\ rtimes _ $ $ $ com} $ x $是可正常的。我们还表明，该定理的模拟不适合在非交通$ c^*$ - 代数上采取行动。最后，我们研究的是与更详细的结构有关的情况，如果$ c^*$ - 代数为适当的$ g $ -space $ x $的代数，以及舒适性与各种规律性属性的相互作用，例如核，精确性，（l）LP，以及（L）LP，以及Injextitive and Injextitive and Injextitive and wep的等效版本。
In this work we introduce and study a new notion of amenability for actions of locally compact groups on $C^*$-algebras. Our definition extends the definition of amenability for actions of discrete groups due to Claire Anantharaman-Delaroche. We show that our definition has several characterizations and permanence properties analogous to those known in the discrete case. For example, for actions on commutative $C^*$-algebras, we show that our notion of amenability is equivalent to measurewise amenability. Combined with a recent result of Alex Bearden and Jason Crann, this also settles a long standing open problem about the equivalence of topological amenability and measurewise amenability for a second countable $G$-space $X$. We use our new notion of amenability to study when the maximal and reduced crossed products agree. One of our main results generalizes a theorem of Matsumura: we show that for an action of an exact locally compact group $G$ on a locally compact space $X$ the full and reduced crossed products $C_0(X)\rtimes_\max G$ and $C_0(X)\rtimes_{\operatorname{red}} G$ coincide if and only if the action of $G$ on $X$ is amenable. We also show that the analogue of this theorem does not hold for actions on noncommutative $C^*$-algebras. Finally, we study amenability as it relates to more detailed structure in the case of $C^*$-algebras that fibre over an appropriate $G$-space $X$, and the interaction of amenability with various regularity properties such as nuclearity, exactness, and the (L)LP, and the equivariant versions of injectivity and the WEP.
