Twisted operator algebras of self-similar groupoid actions on arbitrary graphs

TL;DR

This paper develops a comprehensive theory of twisted operator algebras arising from self-similar groupoid actions on graphs, extending classical results to a broad class of $L^p$-operator algebras and providing criteria for simplicity, pure infiniteness, and other properties.

Contribution

It introduces a unified framework for twisted $L^P$-operator algebras of self-similar groupoid actions, generalizing classical graph algebra results to new algebraic settings.

Findings

Characterization of topological properties of associated groupoids.

Generalized Cuntz--Krieger and Coburn--Toeplitz theorems for twisted $L^P$-algebras.

Criteria for simplicity, pure infiniteness, and when universal and reduced algebras coincide.

Abstract

We study self-similar groupoid actions on arbitrary directed graphs together with $\mathbb{T}$-valued twists that exhaust the second cohomology group of the associated Zappa-Sz\'ep product category. We define and analyse the associated universal, reduced, and essential $C^*$-algebras, along with their Toeplitz versions and core subalgebras. In fact, we develop our theory in the more general setting of $L^P$-operator algebras, where $P\subseteq [1,\infty]$ is any non-empty set of parameters. This includes $C^*$-algebras, $L^p$-operator algebras and symmetrised $L^{p,*}$-operator algebras for $p\in [1,\infty]$, as special cases. We use three complementary approaches: twisted inverse semigroups, twisted ample groupoids, and $C^*$-correspondences. We provide, in terms of the self-similar action, general characterisations of topological freeness, minimality, Hausdorffness, finite non-Hausdorffness, effectiveness, and local contractiveness for the associated ample groupoids. We generalise the classical Cuntz--Krieger Uniqueness and Coburn--Toeplitz Uniqueness Theorems for graph $C^*$-algebras to twisted $L^P$-operator algebras of self-similar groupoid actions. We characterise when the natural inclusions are Cartan, give checkable criteria for simplicity and pure infiniteness of the essential algebras, and discuss when the universal and reduced algebras coincide. We also provide conditions that ensure the singular ideals vanish. Using the groupoid model we show that for any $P\subseteq [1,\infty]$, the $L^P$-operator algebra of a contracting self-similar action is simple if and only if the corresponding Steinberg algebra is simple. Using the Toeplitz-Pimsner model, we prove that for the universal groupoid of any self-similar groupoid action on a row-finite graph, the singular $C^*$-algebraic ideal always vanishes.