# Full Cuntz-Krieger dilations via non-commutative boundaries

## TL;DR

This paper uses non-commutative boundary theory to dilate Toeplitz-Cuntz-Krieger families to full Cuntz-Krieger families, providing new insights and generalizations involving colored graphs and free products of operator algebras.

## Contribution

It offers a novel dilation approach for Toeplitz-Cuntz-Krieger families using non-commutative boundaries and extends results to colored graphs and free product contexts.

## Key findings

- Characterization of representations with unique extension property
- Alternative proof of the C*-envelope of the tensor algebra
- Generalization to colored directed graphs and free products

## Abstract

We apply Arveson's non-commutative boundary theory to dilate every Toeplitz-Cuntz-Krieger family of a directed graph $G$ to a full Cuntz-Krieger family for $G$. We do this by describing all representations of the Toeplitz algebra $\mathcal{T}(G)$ that have unique extension when restricted to the tensor algebra $\mathcal{T}_+(G)$. This yields an alternative proof to a result of Katsoulis and Kribs that the $C^*$-envelope of $\mathcal T_+(G)$ is the Cuntz-Krieger algebra $\mathcal O(G)$. We then generalize our dilation results further, to the context of colored directed graphs, by investigating free products of operator algebras. These generalizations rely on results of independent interest on complete injectivity and a characterization of representations with the unique extension property for free products of operator algebras.