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

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