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

Abstract

Ultragraphs give rise to labelled graphs. We realize algebras associated to such labelled graphs as groupoid algebras, generalizing a known groupoid algebra realization of ultragraph C*-algebras to any ultragraph. Then, we characterize the shift space of an ultragraph as the tight spectrum of the inverse semigroup associated with an ultragraph via its labelled graph. 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 a 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.