Dilation Theory for Rank 2 Graph Algebras

2007/05/31 by Davidson, Kenneth R., Power, Stephen C., Yang, Dilian
#47L55 #FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.0705.4496

Abstract

An analysis is given of *-representations of rank 2 single vertex graphs. We develop dilation theory for the non-selfadjoint algebras \A_θ and \A_u 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 M_m(\bC) ⊗ 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 Ø_Xu for the product system X_u of u; that for m≥ 2 and n ≥ 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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