 

Valentin Deaconu
Groupoid actions on C*correspondences
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Published: 
October 31, 2018. 
Keywords: 
C*correspondence; groupoid action; groupoid representation; graph algebra; CuntzPimsner algebra. 
Subject: 
Primary 46L05. 


Abstract
Let the groupoid G with unit space G^{0} act via a representation ρ on a C*correspondence H over the C_{0}(G^{0})algebra A. By the universal property, G acts on the CuntzPimsner algebra O_{H} which becomes a C_{0}(G^{0})algebra. The action of G commutes with the gauge action on O_{H}, therefore G acts also on the core algebra O_{H}^{T}.
We study the crossed product O_{H} x G and the fixed point algebra O_{H}^{G} and obtain similar results as in [5], where G was a group. Under certain conditions, we prove that O_{H} x G ≅
O_{H x G}, where H x G is the crossed product C*correspondence and that O_{H}^{G} ≅ O_{ρ}, where O_{ρ} is the DoplicherRoberts algebra defined using intertwiners.
The motivation of this paper comes from groupoid actions on graphs. Suppose G with compact isotropy acts on a discrete locally finite graph E with no sources. Since C*(G) is strongly Morita equivalent to a commutative C*algebra, we prove that the crossed product C*(E) x G is stably isomorphic to a graph algebra. We illustrate with some examples. 

Acknowledgements
The author would like to thank Alex Kumjian and Leonard Huang for helpful and illuminating discussions.


Author information
Valentin Deaconu:
Department of Mathematics and Statistics
University of Nevada
Reno, NV 895570084, USA
vdeaconu@unr.edu

