Groupoid actions on C*-correspondences

الأصل · EN

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 G and the fixed point algebra Oₕᵍ and obtain similar results as in D, where G was a group. Under certain conditions, we prove that Oₕ G OH G, where H G is the crossed product C*-correspondence and that OₕᵍOρ, 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) G is stably isomorphic to a graph algebra. We illustrate with some examples.