  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.
