Branching Systems and General Cuntz-Krieger Uniqueness Theorem for Ultragraph C*-algebras

TL;DR

This paper introduces branching systems for ultragraphs, constructs concrete representations of their C*-algebras, and generalizes the Cuntz-Krieger uniqueness theorem to ensure faithfulness of these representations.

Contribution

It develops a framework of branching systems for ultragraphs, linking them to representations of ultragraph C*-algebras, and extends the Cuntz-Krieger uniqueness theorem.

Findings

Every permutative representation is unitarily equivalent to one from a branching system.

Provides a sufficient condition for representations to be faithful.

Generalizes Szymański's Cuntz-Krieger uniqueness theorem for ultragraphs.

Abstract

We give a notion of branching systems on ultragraphs. From this we build concrete representations of ultragraph C*-algebras on the bounded linear operators of Hilbert spaces. To each branching system of an ultragraph we describe the associated Perron-Frobenius operator in terms of the induced representation. We show that every permutative representation of an ultragraph C*-algebra is unitary equivalent to a representation arising from a branching system. We give a sufficient condition on ultragraphs such that a large class of representations of the C*-algebras of these ultragraphs is permutative. To give a sufficient condition on branching systems so that their induced representations are faithful we generalize Szyma{\'n}ski's version of the Cuntz-Krieger uniqueness theorem for ultragraph C*-algebras.