# Journal of Operator Theory

Volume 63, Issue 2, Spring 2010 pp. 245-270.

Dilation theory for rank 2 graph algebras**Authors**: Kenneth R. Davidson, (1) Stephen C. Power (2), and Dilian Yang (3)

**Author institution:**(1) Department of Pure Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada

(2) Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, U.K.

(3) Department of Mathematics and Statistics, University of Windsor, Windsor, ON N9B 3P4, Canada

**Summary:**An analysis is given of $*$-representations of rank 2 single vertex graphs. We develop dilation theory for the non-selfadjoint algebras $\A_\theta$ and $\A_u$ 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 $M_m(\bC) \otimes M_n(\bC)$. We show that a defect free row contractive representation has a unique minimal dilation to a $*$-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 $\A_u$ is the generalised Cuntz algebra $\O_{X_u}$ for the product system $X_u$ of $u$; that for $m\geqslant 2 $ and $n \geqslant 2 $ contractive representations of $\Ath$ need not be completely contractive; and that the universal tensor algebra $\T_+(X_u)$ need not be isometrically isomorphic to $\A_u$.

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