Summary: NONCOMMUTATIVE INTERPOLATION
AND POISSON TRANSFORMS II
Alvaro Arias and Gelu Popescu
Abstract. We associate to certain weights ! > 0 some weighted left creation
operators W1 ::: Wn on the full Fock space. The weighted noncommutative an-
alytic Toeplitz algebra F1(! ) is the WOT-closure of the algebra generated by
W1 ::: Wn and the identity. Noncommutative Poisson transforms on F1(! ) are
used to provide a WOT-continuous, F1(! )-functional calculus for sequences of
operators satisfying certain positivity conditions.
This leads to completely isometric representations of the quotient algebra
F1(! )=J on Hilbertspaces, where J is any w -closed,2-sided idealin F1(! ). We
obtain noncommutative interpolation problems of Caratheodory and of Nevanlinna-
Pick in F1(! ) and in the space of multipliers of some weighted Hardy spaces in
the unit ball of Cn.
In this paper we use a technique from APo2] to obtain simple and explicit
completely isometric representations of quotients of some weighted Fock spaces
F1(! ) (see Section 1 for notation) and of the space of multipliers M(!k) of some
weighted Hardy spaces on the unit ball of C n (see Section 3 for notation). The
selection of the weights is motivated by a paper of Quiggin Q].
We use these representations to obtain noncommutative Caratheodory's and