Noncommutative Interpolation and Poisson transforms

General results of interpolation (eg. Nevanlinna-Pick) by elements in the noncommutative analytic Toeplitz algebra $F^\infty$ (resp. noncommutative disc algebra $A_n$) with consequences to the interpolation by bounded operator-valued analytic functions in the unit ball of ${\bf C}^n$ are obtained. Non-commutative Poisson transforms are used to provide new von Neumann type inequalities. Completely isometric representations of the quotient algebra $F^\infty/J$ on Hilbert spaces, where $J$ is any $w^*$-closed, 2-sided ideal of $F^\infty$, are obtained and used to construct a $w^*$-continuous, $F^\infty/J$--functional calculus associated to row contractions $T=[T_1,\dots, T_n]$ when $f(T_1,\dots,T_n)=0$ for any $f\in J$. Other properties of the dual algebra $F^\infty/J$ are considered.

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