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- W1604858148 abstract "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 $mgeqslant 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 $A_u$." @default.
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- W1604858148 date "2010-01-01" @default.
- W1604858148 modified "2023-09-27" @default.
- W1604858148 title "Dilation theory for rank two graph algebras." @default.
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