Free biholomorphic functions and operator model theory, II

In a companion to this paper, we introduced the class of n  -tuples f=(f1,…,fn)f=(f1,…,fn) of formal power series in noncommutative indeterminates Z1,…,ZnZ1,…,Zn with the model property and developed an operator model theory for pure n  -tuples of operators in noncommutative domains Bf(H)⊂B(H)nBf(H)⊂B(H)n, where the associated universal model is an n  -tuple (MZ1,…,MZn)(MZ1,…,MZn) of multiplication operators on a Hilbert space H2(f)H2(f) of formal power series. In the present paper, we continue this work by considering the completely non-coisometric   (c.n.c) case. In the second part of the paper, several results concerning the noncommutative multivariable operator theory on the unit ball [B(H)n]1− are extended to noncommutative varieties Vf,J(H)⊆Bf(H)Vf,J(H)⊆Bf(H) defined byVf,J(H):={(T1,…,Tn)∈Bf(H):ψ(T1,…,Tn)=0 for any ψ∈J}Vf,J(H):={(T1,…,Tn)∈Bf(H):ψ(T1,…,Tn)=0 for any ψ∈J} for an appropriate evaluation ψ(T1,…,Tn)ψ(T1,…,Tn) and associated with WOT-closed two-sided ideals J   of the Hardy algebra H∞(Bf)H∞(Bf). In particular, we obtain commutative versions for all the results.