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 $Z_1,\ldots, Z_n$. Each such a domain admits a universal model ${\bf W}:=(W_1,\ldots, W_n)$ 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 ${\bf W}:=(W_1,\ldots, W_n)$ 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.