We continue our earlier investigation on generalized reproducing kernels, in connection with the complex geometry of - 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 .
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Daniel Beltiţă, José E. Galé, Universal objects in categories of reproducing kernels. Rev. Mat. Iberoam. 27 (2011), no. 1, pp. 123–179DOI 10.4171/RMI/632