Unitary equivalence of representations of graph algebras and branching systems

TL;DR

This paper demonstrates that for certain countable graphs, all representations of the associated graph algebra in separable Hilbert spaces are unitarily equivalent to those derived from branching systems, linking algebraic and combinatorial structures.

Contribution

It establishes a unitarily equivalence between algebraic representations and branching system constructions for a class of countable graphs.

Findings

All representations are unitarily equivalent to branching system representations.

The result applies to a specific class of countable graphs.

Bridges algebraic and combinatorial approaches in graph algebras.

Abstract

In this paper we show that, for a class of countable graphs, every representation of the associated graph algebra in a separable Hilbert space is unitarily equivalent to a representation obtained via branching systems.