Ultragraph algebras via labelled graph groupoids, with applications to generalized uniqueness theorems

TL;DR

This paper extends the groupoid algebra realization of ultragraph C*-algebras to all ultragraphs, including those with sinks, and applies this to characterize shift spaces and prove generalized uniqueness theorems.

Contribution

It provides a new groupoid algebra framework for ultragraph algebras that includes cases with sinks and establishes connections with Steinberg algebras and partial actions.

Findings

Extended groupoid algebra realization to all ultragraphs.

Characterized shift spaces as tight spectra of inverse semigroups.

Proved generalized uniqueness theorems for ultragraph algebras.

Abstract

An ultragraph gives rise to a labelled graph with some particular properties. In this paper we describe the algebras associated to such labelled graphs as groupoid algebras. More precisely, we show that the known groupoid algebra realization of ultragraph C*-algebras is only valid for ultragraphs for which the range of each edge is finite, and we extend this realization to any ultragraph (including ultragraphs with sinks). Using our machinery, we characterize the shift space associated to an ultragraph as the tight spectrum of the inverse semigroup associated to the ultragraph (viewed as a labelled graph). Furthermore, in the purely algebraic setting, we show that the algebraic partial action used to describe an ultragraph Leavitt path algebra as a partial skew group ring is equivalent to the dual of a topological partial action, and we use this to describe ultragraph Leavitt path algebras as Steinberg algebras. Finally, we prove generalized uniqueness theorems for both ultragraph C*-algebras and ultragraph Leavitt path algebras and characterize their abelian core subalgebras.