Completely Contractive Extensions of Hilbert Modules Over Tensor Algebras

This paper studies completely contractive extensions of Hilbert modules over tensor algebras over C*-correspondences. Using a result of Sz-Nagy and Foias on triangular contractions, extensions are parametrized in terms of contractive intertwining maps between certain defect spaces. These maps have a simple description when initial data consists of partial isometries. Sufficient conditions for the vanishing and nonvanishing of completely contractive Hilbert module Ext(1) are given that parallel results for the classical disc algebra.