Amenable actions of locally compact groups on von Neumann algebras are investigated by exploiting the natural module structure of the crossed product over the Fourier algebra of the acting group. The resulting characterization of injectivity for crossed products generalizes a result of Anantharaman-Delaroche on discrete groups. Amenable actions of locally compact groups on $C^*$ -algebras are investigated in the same way, and amenability of the action is related to nuclearity of the corresponding crossed product. A survey is given to show that this notion of amenable action for $C^*$ -algebras satisfies a number of expected properties. A notion of inner amenability for actions of locally compact groups is introduced, and a number of applications are given in the form of averaging arguments, relating approximation properties of crossed product von Neumann algebras to properties of the components of the underlying $w^*$ -dynamical system. We use these results to answer a recent question of Buss, Echterhoff, and Willett.

通过在作用群的傅里叶代数上利用叉积的自然模结构来研究局部紧群在冯诺依曼代数上的顺从作用。由此产生的交叉产物的注入性表征概括了 Anantharaman-Delaroche 在离散组上的结果。以同样的方式研究了局部紧群在 $C^*$ - 代数上的顺从性，并且该作用的顺从性与相应叉积的核性有关。进行了一项调查以表明这种对 $C^*$ 的顺从行动的概念-代数满足许多预期的性质。引入了局部紧群作用的内在顺从性概念，并以平均参数的形式给出了许多应用，将叉积冯诺依曼代数的近似性质与基础 $w^*$ 的分量的性质相关联-动力系统。我们使用这些结果来回答 Buss、Echterhoff 和 Willett 最近提出的问题。