Proper Actions on Graph Algebras

The c*-algebra C*(E) of a discrete graph E is generated by a family of orthogonal projections and partial isometries. If a discrete group G acts on E, this action induces an action of G on C*(E). In [5], Kumijan and Pask showed that if E is locally finite and the action G on C*E is free, then the C*-algebra C*(GE) of the quotient graph is Morita equivalent to the crossed product C*(E) Xα G. This result has a striking similarity with a theorem of Green [3, Theorem 14], which implies that, if X is a locally compact space and G is a locally compact group which acts freely and properly on X, then C₀(GX) (the C*-algebra of continuous functions ∫:GX → ℂ such that for all ∈>0, the set {z ∈ GX : |∫(z)|≥∈} is compact) is Morita equivalent to the crossed product C₀(X) Xα G.