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

Abstract

In this article we generalize the main results of [7] and [12]. More specifically, we show that there are branching systems (which induce representations of the graph $C^{\ast }\left(E\right)$) associated to each row-countable graph $E$. For row-countable graphs, we characterize the Condition $\left(L\right)$ via branching systems. Moreover, we show that each permutative representation by operators in Hilbert spaces 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 $C^{\ast }$-algebra is permutative.