Group Actions on Operator Algebras

We give a brief account of group actions on operator algebras mainly focusing on classification results. We first recall rather classical results on the classification of discrete amenable group actions on the injective factors, which may serve as potential goals in the case of C∗-algebras for the future. We also mention Galois correspondence type results and quantum group actions for von Neumann algebras. Then we report on the recent developments of the classification of group actions on C∗-algebras in terms of K-theoretical invariants. We give conjectures on the classification of a class of countable amenable group actions on Kirchberg algebras and strongly self-absorbing C∗-algebras, which involve the classifying spaces of the groups. Mathematics Subject Classification (2010). Primary 46L40; Secondary 46L35.