Identification of maximal $C^{\ast }$-covers of some operator algebras

Abstract

We use results on inclusions of free products and extensions of completely positive maps to determine the maximal $C^{\ast }$-envelope for upper triangular $3\times 3$ matrices. We consider these same results in the context of larger upper triangular matrices and graph algebras associated to cycle graphs.

12 pages; corrected proof of Proposition 2.5 with additional hypotheses, clarified the pushout diagrams in Theorem 2.10