Noncommutative Function Theory and Its Applications

As the project unfolded, its main focus — free noncommutative function theory — became a major emerging topic in operator theory and operator algebras. This was partially due to the fact that the development of the foundations of the subject has been completed by D.K.-V. and V,V. during the first years of the project, and summarized in a monograph that then became the basic reference for all future Work. Another central achievement of the project, in the last years, was a systematic development of the theory of completely positive noncommutative kernels and correspoinding noncommutative reproducing kernel Hilbert spaces, and a very general realization and interpolation theorem for contractive noncommutative functions with its many consequences.

Both of these major achievements were actual goals set in the original research proposal. Other achievements occured naturally during the project our knowledge increased over time, such noncommutative Hardy spaces, noncommutative fixed point theorems, and noncommutative integrability.

There are numerous connections between the free noncommutative setting and the commutative setting. Accordingly, there were several achievements in commutative multivariable realization theory, both for conservative realizations of contractive functions and for finite-dimensional realizations of rational functions. A blend of commutative and noncommutative ideas and techniques also yielded important results on determinantal representations of stable polynomials, that have a strong bearing on central questions of convex algeraic geometry.