Claire Anantharaman-Delaroche first defined amenability for an action of a discrete group on a C*-algebra A in the 1980s. If an action is amenable, then there are desirable structural consequences, such as nuclearity passing from A to the crossed product, and equality of the maximal and reduced crossed products. Actions of amenable groups are always amenable, but there are also many interesting examples of amenable actions of non-amenable groups. I’ll survey some of this, give some new results that we hope clarify the theory a little, and discuss connections to other notions like exactness and equivariant injectivity.

The talk will be based on joint work with Siegfried Echterhoff and Alcides Buss.