The Koopman representation for self-similar groupoid actions

Abstract

We introduce the $C^{\ast }$-algebra $C^{\ast }(\kappa )$ generated by the Koopman representation $\kappa$ of an étale 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 $|u{E}^1|$ constant, there is an invariant measure $\nu$ on $X=E^{\infty }$ and that $C^{\ast }(\kappa )$ 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 $C^{\ast }(E)$ and self-similar representations of $G$ in connection to the Cuntz-Pimsner algebra $C^{\ast }(G,E)$.

arXiv admin note: text overlap with arXiv:2108.01178