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as a spectral set is called a Γ n -contraction. We construct a minimal Γ n -isometric dilation for a particular class of c.n.u. Γ n -contractions and hence express a c.n.u.for some operator C i . We also produce an abstract model for a particular class of c.n.u. Γ n -contractions. We provide an example to show that such non-commutative model does not exists for any c.n.u. Γ n -contraction. By an application of this model we obtain a complete unitary invariant for such c.n.u. Γ ncontraction. The entire program is based on Nagy-Foias model theory for a c.n.u. contraction.

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