Representations of relative Cohn path algebras

TL;DR

This paper explores the structure and representations of relative Cohn path algebras, establishing their realization as partial skew group rings and providing conditions for faithful representations via graph-based systems.

Contribution

It introduces a new realization of relative Cohn path algebras as partial skew group rings and characterizes faithfulness of their representations through graph-based branching systems.

Findings

Realization of relative Cohn path algebras as partial skew group rings.

Necessary and sufficient conditions for faithfulness of representations.

Extension of reduction theorems to relative Cohn path algebras.

Abstract

We study relative Cohn path algebras, also known as Leavitt-Cohn path algebras, and we realize them as partial skew group rings (to do this we prove uniqueness theorems for relative Cohn path algebras). Furthermore, given any graph $E$ we define $E$-relative branching systems and prove how they induce representations of the associated relative Cohn path algebra. We give necessary and sufficient conditions for faithfulness of the representations associated to $E$-relative branching systems (this improves previous results known to Leavitt path algebras of row-finite graphs with no sinks). To prove this last result we show first a version, for relative Cohn-path algebras, of the reduction theorem for Leavitt path algebras.