Dilation Theory for Rank 2 Graph Algebras

An analysis is given of $\ast$-representations of rank 2 single vertex graphs. We develop dilation theory for the non-selfadjoint algebras ${\text{A}}_{\theta }$ and ${\text{A}}_u$ which are associated with the commutation relation permutation $\theta$ of a 2 graph and, more generally, with commutation relations determined by a unitary matrix $u$ in $M_m(\text{bC})\otimes M_n(\text{bC})$. We show that a defect free row contractive representation has a unique minimal dilation to a $\ast$-representation and we provide a new simpler proof of Solel’s row isometric dilation of two $u$-commuting row contractions. Furthermore it is shown that the C*-envelope of ${\text{A}}_u$ is the generalised Cuntz algebra ${\text{O}}_{X_u}$ for the product system $X_u$ of $u$; that for $m\ge 2$ and $n\ge 2$ contractive representations of $\text{Ath}$ need not be completely contractive; and that the universal tensor algebra ${\text{T}}_{+}(X_u)$ need not be isometrically isomorphic to ${\text{A}}_u$.