Reconstructing directed graphs from generalised gauge actions on their Toeplitz algebras

TL;DR

This paper demonstrates how to reconstruct finite directed graphs from their Toeplitz algebras and specific gauge actions, highlighting the importance of additional structures for accurate reconstruction.

Contribution

It introduces methods to recover directed graphs from Toeplitz algebras using generalized gauge actions and identifies limitations when certain conditions are not met.

Findings

Graph reconstruction from Toeplitz algebras is possible with gauge actions and vertex projections.

Without additional structures, reconstruction from gauge actions alone is not feasible.

Recovery is guaranteed for graphs without sinks using generalized gauge actions.

Abstract

We show how to reconstruct a finite directed graph E from its Toeplitz algebra, its gauge action, and the canonical finite-dimensional abelian subalgebra generated by the vertex projections. We also show that if E has no sinks, then we can recover E from its Toeplitz algebra and the generalised gauge action that has, for each vertex, an independent copy of the circle acting on the generators corresponding to edges emanating from that vertex. We show by example that it is not possible to recover E from its Toeplitz algebra and gauge action alone.