Unitarily invariant valuations on convex functions

Abstract: Continuous, dually epi-translation invariant valuations on the space of finite-valued convex functions on $\mathbb{C}<sup>n$ that are invariant under the unitary group are investigated. It is shown that elements belonging to the dense subspace of smooth valuations admit a unique integral representation in terms of two families of Monge-Amp`ere-type operators. In addition, it is proved that homogeneous valuations are uniquely determined by restrictions to subspaces of appropriate dimension and that this information is encoded in the Fourier-Laplace transform of the associated Goodey-Weil distributions. These results are then used to show that a continuous unitarily invariant valuation is uniquely determined by its restriction to a certain finite family of subspaces of $\mathbb{C}<sup>n$.