Freeness of Actions of Finite Groups on C*-algebras

We describe some of the forms of freeness of group actions on noncommutative C*-algebras that have been used, with emphasis on actions of finite groups. We give some indications of their strengths, weaknesses, applications, and relationships to each other. The properties discussed include the Rokhlin property, K-theoretic freeness, the tracial Rokhlin property, pointwise outerness, saturation, hereditary saturation, and the requirement that the strong Connes spectrum be the entire dual. Recall that an action (g, x) 7→ gx of a group G on a space X is free if whenever g ∈ G \ {1} and x ∈ X, then gx 6= x. That is, every nontrivial group element acts without fixed points. So what is a free action on a C*-algebra? There are several reasons for being interested in free actions on C*-algebras. First, there is the general principle of noncommutative topology: one should find the C*-algebra analogs of useful concepts from topology. Free (or free and proper) actions of locally compact groups on locally compact Hausdorff spaces have a number of good properties, some of which are visible from topological considerations and some of which become apparent only when one looks at crossed product C*algebras. We describe some of these in Section 1. Second, analogs of freeness, particularly pointwise outerness and the Rokhlin property, have proved important in von Neumann algebras, especially for the classification of group actions on von Neumann algebras. Again, this fact suggests that one should see to what extent the concepts and theorems carry over to C*-algebras. Third, the classification of group actions on C*-algebras is intrinsically interesting. Experience both with the commutative case and with von Neumann algebras suggests that free actions are easier to understand and classify than general actions. (A free action of a finite group on a path connected space corresponds to a finite covering space.) Fourth, noncommutative analogs of freeness play an important role in questions about the structure of crossed products. Freeness hypotheses are important for results on both simplicity and classifiability of crossed products. It turns out that there are many versions of noncommutative freeness. They vary enormously in strength, from saturation (or full Arveson spectrum) all the way up to the Rokhlin property. The various conditions have different uses. The main point of this article is to describe some of the forms of noncommutative freeness that have been used, and give some indications of their strengths, weaknesses, applications, and relationships to each other. To keep things simple, and to keep the focus on freeness, we restrict whenever convenient to actions of finite groups. For one thing, Date: 19 February 2009. 2000 Mathematics Subject Classification. Primary 46L55; Secondary 46L35, 46L40. Research partially supported by NSF grant DMS-0701076.