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

In a recent paper, we introduced and studied the class of admissible noncommutative domains Dg−1(H) in B(H)n. Each such a domain admits a universal model W:=(W1,…,Wn) of weighted left creation operators acting on the full Fock space. In the present paper, we obtain a Beurling type characterization of the invariant subspaces of the universal model W and develop a dilation theory for the elements of the domain Dg−1(H). We also obtain results concerning the commutant lifting and Toeplitz-corona in our setting as as well as some results on the boundary property for universal models.