Dilation Theory for Rank 2 Graph Algebras
نویسنده
چکیده
An analysis is given of ∗-representations of rank 2 single vertex graphs. We develop dilation theory for the nonselfadjoint algebras Aθ and Au which are associated with the commutation relation permutation θ of a 2 graph and, more generally, with commutation relations determined by a unitary matrix u in Mm(C)⊗Mn(C). 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 Au is the generalised Cuntz algebra OXu for the product system Xu of u; that for m ≥ 2 and n ≥ 2 contractive representations of Aθ need not be completely contractive; and that the universal tensor algebra T+(Xu) need not be isometrically isomorphic to Au.
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