 
Summary: NONCOMMUTATIVE INTERPOLATION
AND POISSON TRANSFORMS
Alvaro Arias and Gelu Popescu
Abstract. Generalresultsof interpolation(eg. NevanlinnaPick)by elementsin the
noncommutative analytic Toeplitz algebra F1 (resp. noncommutative disc algebra
An) with consequences to the interpolation by bounded operatorvalued analytic
functions in the unit ball of Cn are obtained.
NoncommutativePoisson transformsare used to provide new von Neumann type
inequalities. Completely isometric representations of the quotient algebra F1=J on
Hilbert spaces, where J is any w closed, 2sided ideal of F1, are obtained and used
to construct a w continuous, F1=J{functional calculus associated to row contrac
tions T = T1 : :: Tn] when f(T1 : :: Tn) = 0 for any f 2 J. Other properties of
the dual algebra F1=J are considered.
In Po5], the second author proved the following version of von Neumann's in
equality for row contractions: if T1 ::: Tn 2 B(H) (the algebra of all bounded
linear operators on the Hilbert space H) and T = T1 ::: Tn] is a contraction, i.e.,
Pn
i=1 TiTi IH, then for every polynomial p(X1 ::: Xn) on n noncommuting
indeterminates,
(1) p(T1 ::: Tn) B(H) p(S1 ::: Sn) B(F2)
