Universal objects in categories of reproducing kernels

Abstract

We continue our earlier investigation on generalized reproducing kernels, in connection with the complex geometry of $C^{\ast }$- algebra representations, by looking at them as the objects of an appropriate category. Thus the correspondence between reproducing $(-\ast )$-kernels and the associated Hilbert spaces of sections of vector bundles is made into a functor. We construct reproducing $(-\ast )$-kernels with universality properties with respect to the operation of pull-back. We show how completely positive maps can be regarded as pull-backs of universal ones linked to the tautological bundle over the Grassmann manifold of the Hilbert space ${\ell }^2(\mathbb{N})$.