When Locally Contractive Representations Are Completely Contractive

A complete lattice theoretic characterization as "interpolating digraphs" is given for the class of matrix algebras containing the diagonal for which every locally contractive representation has a unitary ∗-dilation. Combined with [Davidson et al., Bull. London Math. Soc. (3)68 (1994), 178-202.], this also yields a lattice theoretic characterization of those algebras for which the commutant lifting theorem is valid. An appropriate generalization to infinite dimensions is given. It is shown that these algebras have the complete compact approximation property with respect to the class of finite dimensional interpolating algebras, and hence all weak-∗ continuous contractive representations have unitary ∗-dilations.