Full Cuntz-Krieger dilations via non-commutative boundaries

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 $C^{\ast }$-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.