Unitary dilations of commuting contractions

The Sz.-Nagy dilation theorem is a seminal result in the theory of contractions on Hilbert space. It states that every contraction has a unitary dilation. An elegant generalisation was given by Ando who proved that every pair of commuting contractions has a unitary dilation. It is somewhat surprising that this phenomenon does not generalise further: Parrott gave an example of three commuting contractions that do not have a unitary dilation. <br/> <br/>This raises the question, when does a tuple of commuting operators have a unitary dilation? We give a characterisation in terms of the existence of a positive map with certain properties. As an application we extend a dilation theorem of Sz.-Nagy and Foias concerning regular dilations. We also explore the close connection with generalisations of the commutant lifting theorem to multivariable settings.