Noncommutative domains, universal operator models, and operator algebras, II

In a recent paper, we introduced and studied the class of admissible noncommutative domains D_g⁻¹(H) in B(H)ⁿ associated with admissible free holomorphic functions g in noncommutative indeterminates Z₁,…, Zₙ. Each such a domain admits a universal model W:=(W₁,…, Wₙ) of weighted left creation operators acting on the full Fock space with n generators. In the present paper, we continue the study of these domains and their universal models in connection with the Hardy algebras and the C^*-algebras they generate. We obtain a Beurling type characterization of the invariant subspaces of the universal model W:=(W₁,…, Wₙ) and develop a dilation theory for the elements of the noncommutative domain D_g⁻¹(H). We also obtain results concerning the commutant lifting and Toeplitz-corrona in our setting as as well as some results on the boundary property for universal models.