2014/10/14 by Valentin Deaconu, Deaconu, Valentin · 1 citation
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Noncommutative and Quantum Gravity Theories #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.1410.3846
openalex publication_date 2014/10/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
If G acts on a C^*-correspondence \mathcal H, then by the universal property G acts on the Cuntz-Pimsner algebra \mathcal O\mathcal H and we study the crossed product \mathcal O\mathcal H\rtimes G and the fixed point algebra \mathcal O\mathcal HG. Using intertwiners, we define the Doplicher-Roberts algebra \mathcal Oρ of a representation ρ of a compact group G on \mathcal H and prove that \mathcal O\mathcal HG is isomorphic to \mathcal Oρ. When the action of G commutes with the gauge action on \mathcal O\mathcal H, then G acts also on the core algebras \mathcal O\mathcal H\mathbb T, where \mathbb T denotes the unit circle. We give applications for the action of a group G on the C^*-correspondence \mathcal HE associated to a directed graph E. If G is finite and E is discrete and locally finite, we prove that the crossed product C^*(E)\rtimes G is isomorphic to the C^*-algebra of a graph of C^*-correspondences and stably isomorphic to a locally finite graph algebra. If C^*(E) is simple and purely infinite and the action of G is outer, then C^*(E)G and C^*(E)\rtimes G are also simple and purely infinite with the same K-theory groups. We illustrate with several examples.