  We initiate the study of a class of noncommutative domains of n-tuples of
bounded linear operators on a Hilbert space, which is generated by certain
positivity conditions on polynomials in n noncommutative indeterminates. We
obtain Fatou type results, functional calculi, and a model theory for n-tuples
of operators in these domains. The main role in this study is played by a class
of noncommutative Berezin transforms which is introduced in this paper. Our
results extend to noncommutative varieties.
