We demonstrate how exact structures can be placed on the additive category of right operator modules over an operator algebra in order to discuss global dimension for operator algebras. The properties of the Haagerup tensor product play a decisive role in this.

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.