New York Journal of Mathematics
Volume 24 (2018), 1020-1038


Valentin Deaconu

Groupoid actions on C*-correspondences

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Published: October 31, 2018.
Keywords: C*-correspondence; groupoid action; groupoid representation; graph algebra; Cuntz-Pimsner algebra.
Subject: Primary 46L05.

Let the groupoid G with unit space G0 act via a representation ρ on a C*-correspondence H over the C0(G0)-algebra A. By the universal property, G acts on the Cuntz-Pimsner algebra OH which becomes a C0(G0)-algebra. The action of G commutes with the gauge action on OH, therefore G acts also on the core algebra OHT.

We study the crossed product OH x G and the fixed point algebra OHG and obtain similar results as in [5], where G was a group. Under certain conditions, we prove that OH x G ≅ OH x G, where H x G is the crossed product C*-correspondence and that OHG ≅ Oρ, where Oρ is the Doplicher-Roberts 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.


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 89557-0084, USA