Realising the Toeplitz algebra of a higher-rank graph as a Cuntz-Krieger algebra

TL;DR

This paper constructs a higher-rank graph T{\Lambda} to represent the Toeplitz algebra of a given higher-rank graph {\Lambda} as a Cuntz-Krieger algebra, proving T{\Lambda} is aperiodic and providing a new proof of a uniqueness theorem.

Contribution

It introduces a method to realize Toeplitz algebras of higher-rank graphs as Cuntz-Krieger algebras via a new graph T{\Lambda}, establishing its aperiodicity.

Findings

Toeplitz algebra of {\Lambda} is isomorphic to Cuntz-Krieger algebra of T{\Lambda}

T{\Lambda} is always aperiodic

Provides a new proof of the uniqueness theorem for Toeplitz algebras

Abstract

For a row-finite higher-rank graph {\Lambda}, we construct a higher-rank graph T{\Lambda} such that the Toeplitz algebra of {\Lambda} is isomorphic to the Cuntz-Krieger algebra of T{\Lambda}. We then prove that the higher-rank graph T{\Lambda} is always aperiodic and use this fact to give another proof of a uniqueness theorem for the Toeplitz algebras of higher-rank graphs.