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The Koopman representation for self-similar groupoid actions (2112.14341v1)
Published 29 Dec 2021 in math.OA and math.RT
Abstract: We introduce the $C*$-algebra $C*(\kappa)$ generated by the Koopman representation $\kappa$ of an \'etale groupoid $G$ acting on a measure space $(X,\mu)$. We prove that for a level transitive self-similar action $(G,E)$ with $E$ finite and $|uE1|$ constant, there is an invariant measure $\nu$ on $X=E\infty$ and that $C*(\kappa)$ is residually finite-dimensional with a normalized self-similar trace. We also discus $p$-fold similarities of Hilbert spaces in connection to representations of the graph algebra $C*(E)$ and self-similar representations of $G$ in connection to the Cuntz-Pimsner algebra $C*(G,E)$.