Dilation theory for rank two graph algebras.

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 $m\geqslant 2$ 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