Ergodic Theory and Dynamical Systems

$C^*$-algebras of directed graphs and group actions

a1 Department of Mathematics (084), University of Nevada, Reno NV 89555-0045, USA (e-mail:
a2 Department of Mathematics, The University of Newcastle, NSW 2308, Australia (e-mail:


Given a free action of a group $G$ on a directed graph $E$ we show that the crossed product of $C^* (E)$, the universal $C^*$-algebra of $E$, by the induced action is strongly Morita equivalent to $C^* (E/G)$. Since every connected graph $E$ may be expressed as the quotient of a tree $T$ by an action of a free group $G$ we may use our results to show that $C^* (E)$ is strongly Morita equivalent to the crossed product $C_0 ( \partial T ) \times G$, where $\partial T$ is a certain zero-dimensional space canonically associated to the tree.

(Received October 16 1997)
(Revised November 4 1997)

Dedicated to Marc A. Rieffel on the occasion of his 60th birthday