You Can See the Arrows in a Quiver Operator Algebra

TL;DR

This paper establishes that the structure of quiver operator algebras uniquely determines the underlying directed graphs, and demonstrates how to recover the graph from algebra representations.

Contribution

It proves that isometric isomorphism of quiver operator algebras implies graph isomorphism and provides a method to recover the graph from algebra representations.

Findings

Two quiver operator algebras are isometrically isomorphic only if their quivers are isomorphic.

The underlying directed graph can be reconstructed from certain algebra representations.

The result links algebraic isomorphisms directly to graph isomorphisms.

Abstract

We prove that two quiver operator algebras can be isometrically isomorphic only if the quivers (=directed graphs) are isomorphic. We also show how the graph can be recovered from certain representations of the algebra.