Abstract: A classical result of Jones, based on previous work by Connes, states that for actions of finite groups on the hyperfinite $\mathrm{II}_1$ factor, pointwise outerness is equivalent to having the Rokhlin property. In this talk I will discuss analogs of this result in the setting of C*-algebras. I will also discuss several properties of actions of compact groups on C*-algebras with finite Rokhlin dimension, particularly in relation to crossed products. I will show how taking crossed products by such actions preserves a number of relevant classes of C*-algebras, including: D-absorbing C*-algebras (where D is strongly self-absorbing), finite stable rank, C*-algebras with finite nuclear dimension or decomposition rank, C*-algebras that are nuclear and satisfy the UCT, among others. I will introduce a representability dimension for actions of discrete groups on C*-algebras and show that in the abelian setting the dual action of an action of a compact group with finite Rokhlin dimension has finite representability dimension.

This a joint work with Eusebio Gardella and Ilan Hirshberg.