Noncommutative interpolation and Poisson transforms

General results of interpolation (e.g., Nevanlinna-Pick) by elements in the noncommutative analytic Toeplitz algebra F-infinity (resp., noncommutative disc algebra A(n)) with consequences to the interpolation by bounded opera tor-valued analytic functions in the unit ball of C-n are obtained. Noncomm utative Poisson transforms are used to provide new von Neumann type inequal ities. Completely isometric representations of the quotient algebra F-infin ity/J on Hilbert spaces, where J is any w*-closed, 2-sided ideal of F-infin ity, are obtained and used to construct a w*-continuous, F-infinity/J-funct ional calculus associated to row contractions T = [T-1,...,T-n] when f(T-1, ...,T-n) = 0 for any f epsilon J. Other properties of the dual algebra F-in finity/J are considered.