Minimal Cuntz-Krieger Dilations and Representations of Cuntz-Krieger Algebras

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Given a contractive tuple of Hilbert space operators satisfying certain A-relations we show that there exists a unique minimal dilation to generators of Cuntz-Krieger algebras or its extension by compact operators. This Cuntz-Krieger dilation can be obtained from the classical minimal isometric dilation as a certain maximal A-relation piece. We define a maximal piece more generally for a finite set of polynomials in n noncommuting variables. We classify all representations of Cuntz-Krieger algebras Oₐ obtained from dilations of commuting tuples satisfying A-relations. The universal properties of the minimal Cuntz-Krieger dilation and the WOT-closed algebra generated by it is studied in terms of invariant subspaces.