Positive definite functions admit integral representationas Fourier transforms of positive measures. This famous result,since its discovery in the early part of the century, has shownitself key to applications in probability, moment theory,interpolation by analytic functions, and other branches ofanalysis.
Parallel applications in other settings require a moregeneral integral representation theorem, that includes as a bonusan extension property. The procedure involves dilating to a newcontaining space with preservation of norm and other properties.
The dilation representation extends to positivedefinite vector-valued functions, and to completely positive maps,leading to non-commutative results.
The dilation construction can be set in scatteringstructures with several (not necessarily commuting) evolutiongroups. The corresponding integral representations provideapplications to function spaces in several variables (like theHardy space in the polydisk and in symplectic spaces).
(Joint work with Mischa Cotlar.)