Group actions on topological graphs.pdf

Ergod. Th. Dynam. Sys. (First published online 2011), page 1 of 40∗ c doi:10.1017/S014338571100040X Cambridge University Press, 2011 ∗Provisional—final page numbers to be inserted when paper edition is published Group actions on topological graphs VALENTIN DEACONU†, ALEX KUMJIAN† and JOHN QUIGG‡ † Department of Mathematics, University of Nevada, Reno, NV 89557-0084, USA (e-mail: vdeaconu@, alex@) ‡ School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804, USA (e-mail: quigg@) (Received 15 July 2010 and accepted in revised form 10 May 2011) Abstract . We define the action of a locally compact group G on a topological graph E . This action induces a natural action of G on the C∗-correspondence H(E ) and on the graph C∗-algebra C∗ (E ). If the action is free and proper, we prove that C∗ (E ) G is r strongly Morita equivalent to C∗ (E / G). We define the skew product of a locally compact group G by a topological graph E via a cocycle c : E 1 → G. The group acts freely and properly on this new topological graph E × G. If G is abelian, there is a dual action on c ∗ ∗ ∼ ∗ ˆ C (E ) such that C (E ) G = C (E × G). We also define the fundamental group and