Universal objects in categories of reproducing kernels

TL;DR

This paper explores the categorical framework of generalized reproducing kernels, establishing a functorial correspondence with Hilbert spaces, and introduces universal kernels with pull-back properties linked to $C^*$-algebra representations.

Contribution

It develops a categorical approach to reproducing kernels, constructs universal kernels with pull-back properties, and relates completely positive maps to universal kernels via Grassmannian bundles.

Findings

Reproducing $(-*)$-kernels form a functor with Hilbert spaces.

Universal kernels with pull-back properties are constructed.

Connections between completely positive maps and universal kernels are established.

Abstract

We continue our earlier investigation on generalized reproducing kernels, in connection with the complex geometry of $C^*$- algebra representations, by looking at them as the objects of an appropriate category. Thus the correspondence between reproducing $(-*)$-kernels and the associated Hilbert spaces of sections of vector bundles is made into a functor. We construct reproducing $(-*)$-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 $\ell^2({\mathbb N})$.