Boundary representations from constrained interpolation

In this paper, we study C^⁎-envelopes of finite-dimensional operator algebras arising from constrained interpolation problems on the unit disc. In particular, we consider interpolation problems for the algebra H_node^∞ that consists of bounded analytic functions on the unit disk that satisfy f(0)=f(λ) for some 0≠λ∈D. We show that there exist choices of four interpolation nodes that exclude both 0 and λ, such that if I is the ideal of functions that vanish at the interpolation nodes, then C_e^⁎(H_node^∞/I) is infinite-dimensional. This differs markedly from the behavior of the algebra corresponding to interpolation nodes that contain the constrained points studied in the literature. Additionally, we use the distance formula to provide a completely isometric embedding of C_e^⁎(H_node^∞/I) for any choice of n interpolation nodes that do not contain the constrained points into M_n(G_nc²), where G_nc² is Brown's noncommutative Grassmannian.