Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Dilation theory for rank two graph algebras.
AU - Davidson, Kenneth R.
AU - Power, Stephen C.
AU - Yang, Dilian
PY - 2010
Y1 - 2010
N2 - 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 $\A_u$.
AB - 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 $\A_u$.
M3 - Journal article
VL - 63
SP - 245
EP - 270
JO - Journal of Operator Theory
JF - Journal of Operator Theory
IS - 2
ER -