Unitary equivalence of representations of graph algebras and branching systems

Abstract

It is shown that, for many countable graphs, every representation of the associated graph algebra in a separable Hilbert space is unitarily equivalent to a representation obtained via branching systems.

On the representations of Leavitt path algebras

“…Most of this section is inspired by corresponding results and ideas for graph C * -algebras, as done in [6] and [7]. For the reader's convenience, we adapt the necessary definitions and results below.…”

“…Proceeding inductively we define the level n vertices set, X n , and the level n edges set, Y n , if such vertices and edges exist. For more details see [7]. Our aim in this section is to describe a sufficient condition, over the graph E, which guarantees that a representation of L K (E) is equivalent to a representation induced by an E-algebraic branching system.…”

“…Our aim is to prove analogue versions of the representation theorems (for graph C * -algebras) in [5][6][7], that is, to show how to obtain representations of Leavitt path algebras from E-algebraic branching systems, to study these representations and to…”

“…(See[7].) A vertex v ∈ E 0 is an extreme vertex of E if #{r −1 (v) ∪ s −1 (v)} = 1 and if there does not exist an edge e ∈ E 1 such that r(e) = v = s(e).…”

“…2In the following proposition, we obtain a sufficient condition over the graph E to conclude that r E 1 ∪ s E 1 = Proposition 6.4. (See[7].) If r(E 1 ) ∪ s(E 1 ) is finite and connected and if E is P -simple then r(E 1…”