Unifying interval maps and branching systems with applications to relative graph C*-algebras

TL;DR

This paper unifies the theory of interval maps and branching systems, introducing relative branching systems to analyze representations of relative graph C*-algebras, with applications to injectivity and faithfulness of these representations.

Contribution

It develops the theory of relative branching systems and characterizes when associated representations of relative graph C*-algebras are faithful, especially for maps with escape sets.

Findings

Characterization of faithfulness of representations via branching systems

Improved criteria for injectivity of relative graph algebra representations

Application to Markov interval maps with escape sets

Abstract

We describe Markov interval maps via branching systems and develop the theory of relative branching systems, characterizing when the associated representations of relative graph C*-algebras are faithful. When the Markov interval maps $f$ have escape sets, we use our results to characterize injectivity of the associated relative graph algebra representations, improving on previous work by the first, third, and fourth authors.