Groupoid actions and Koopman representations

2022/12/27 by Valentin Deaconu, Deaconu, Valentin, Marius Ionescu +1
Mathematics · Medicine · Neuroscience · #46L05 #Advanced Operator Algebra Research #Cerebrospinal fluid and hydrocephalus #FOS: Mathematics #Operator Algebras (math.OA) #Representation Theory (math.RT) #Traumatic Brain Injury and Neurovascular Disturbances

paper · pdf · doi:10.48550/arxiv.2212.13633

openalex publication_date 2022/12/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the C^*-algebra C^*(κ) generated by the Koopman representation κ=κ^μ of a locally compact groupoid G acting on a measure space (X,μ), where μ is quasi-invariant for the action. We interpret κ as an induced representation and we prove that if the groupoid G\ltimes X is amenable, then κ is weakly contained in the regular representation ρ=ρ^μ associated to μ, so we have a surjective homomorphism C^*_r(G)→ C^*(κ). We consider the particular case of Renault-Deaconu groupoids G= G(X,T) acting on their unit space X and show that in some cases C^*(κ)≅ C^*(G).

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