We apply Arveson's non-commutative boundary theory to dilate every TCK family of a directed graph G to a full CK family for G. We do this by describing all representations of the Toeplitz algebra T (G) that have a unique extension when restricted to the tensor algebra T+(G). This yields an alternative proof to a result of Katsoulis and Kribs that the C* -envelope of T+(G) is the CK algebra 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.
Bibliographical noteFunding Information:
Received 14 February 2017; revised 2 April 2018; published online 23 May 2018. 2010 Mathematics Subject Classification 47L55, 47A20, 47L75, 47L80 (primary). The first author was partially supported by an Ontario Trillium scholarship. The second author was partially supported by the Clore Foundation.
© 2018 London Mathematical Society
- 47L80 (primary)
ASJC Scopus subject areas
- Mathematics (all)