Cuntz-Krieger dilations of Toeplitz-Cuntz-Krieger families via Choquet theory

Perhaps the simplest dilation result in operator theory is the dilation of an isometry to a unitary. However, when one generalizes an isometry to a Toeplitz-Cuntz-Krieger family of a directed graph, things become much more complicated.

The analogue of a unitary operator in this case is a (full) Cuntz-Krieger family, and a result of Skalski and Zacharias on C-correspondences supplies us with a such a dilation when the graph is row-finite and sourceless. We apply Arveson’s non-commutative Choquet theory to answer this question for arbitrary graphs. We compute the non-commutative Choquet boundary of graph tensor algebras and are able to recover a result of Katsoulis and Kribs on the computation of the C-envelope of these algebras.

However, as the non-commutative Choquet boundary of the operator algebra is a more delicate information than the C*-envelope, we are able to dilate any TCK family to a (full) CK family. In fact, we are able to make progress on a decade old problem of Skalski and Zacharias, that asks for the multivariable analogue, generalizing a result of Ito’s dilation theorem. More precisely, we are able to show that TCK families of graphs $G_1,…,G_d$ that commute according to a higher rank row-finite sourceless directed graph have CK dilations that still commute according to the higher rank graph structure.