Representations of C*-algebras of row-countable graphs and unitary equivalence

TL;DR

This paper explores how branching systems induce representations of C*-algebras from row-countable graphs, characterizes certain conditions, and shows unitary equivalence of permutative representations, including non-separable cases.

Contribution

It introduces branching systems for row-countable graphs, characterizes the condition (L), and proves unitary equivalence of permutative representations, extending previous results to non-separable spaces.

Findings

Branching systems induce representations for each row-countable graph.

Condition (L) is characterized via branching systems.

Permutative representations are unitarily equivalent to those induced by branching systems.

Abstract

In this article we show that there are branching systems (which induce representations of the graph algebra $C^*(E)$) associated to each row-countable graph $E$. For row-countable graphs, we characterize the condition $(L)$ via branching systems. Moreover, we show that each permutative representation in Hilbert spaces operators is unitarily equivalent to one induced by a branching system, even the spaces being not separable. Furthermore, under some hypothesis on the graph, we show that each representation of the graph C*-algebra is permutative.