Symmetric Imprimitivity Theorems for Graph C∗-algebras

The C∗-algebra C∗(E) of a directed graph E is generated by partial isometries satisfying relations which reflect the path structure of the graph. In [10], Kumjian and Pask considered the action of a group G on C∗(E) induced by an action of G on E. They proved that if G acts freely and E is locally finite, then the crossed product C∗(E) × G is Morita equivalent to the C∗-algebra of the quotient graph E/G [10, Corollary 3.10]. This theorem bears a striking resemblance to a famous theorem of Green, which says that the crossed product C0(X)×G associated to a free and proper action of G on a locally compact space X is Morita equivalent to C0(X/G) [6]. So one naturally asks whether this resemblance can be pushed further: are there analogues for free actions on graphs of the other Morita equivalences associated to free and proper actions on spaces? Here we contribute to this circle of ideas by proving an analogue of the symmetric imprimitivity theorem of [15] and [12] concerning commuting free and proper actions of two different groups. The proofs of the Kumjian-Pask theorem in [10] and [9] rely on a result of Gross and Tucker which realises free actions as translations on skew products, and exhibit a stable isomorphism rather than a Morita equivalence. Here choosing a skew-product realisation for one of the actions would destroy the symmetry of the situation, so instead we construct directly an imprimitivity bimodule implementing the equivalence. This construction is of some interest even in the case of one action; it shows, for example, that the actions on C∗(E) induced by free actions on E provide a new family of actions which are proper and saturated in the sense of Rieffel ([16], see also [5, 8, 17]). Since our methods do not require any local-finiteness hypothesis on the graph E, our one-action result is more general than the original Kumjian-Pask theorem. In fact we shall prove two symmetric imprimitivity theorems, one for reduced crossed products (Theorem 1.9) and one for full crossed products (Theorem 2.1). We prove the theorem about reduced crossed products in §1, following the strategy of [4]. We show that every action on C∗(E) induced by a free action on E is proper in Rieffel’s sense, so that the machine of [16] gives a bimodule for each action. We then take crossed products of these bimodules by actions of the other groups, and tensor them to get a bimodule implementing the desired equivalence. In §2, we prove