Revista Matemática Iberoamericana

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Volume 27, Issue 1, 2011, pp. 123–179
DOI: 10.4171/RMI/632

Published online: 2011-04-30

Universal objects in categories of reproducing kernels

Daniel Beltiţă[1] and José E. Galé[2]

(1) Romanian Academy, Bucharest, Romania
(2) Universidad de Zaragoza, Spain

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 $\ell^2(\mathbb{N})$.

Keywords: Reproducing kernel, category theory, vector bundle, tautological bundle, Grassmann manifold, completely positive map, universal object.

Beltiţă Daniel, Galé José: Universal objects in categories of reproducing kernels. Rev. Mat. Iberoam. 27 (2011), 123-179. doi: 10.4171/RMI/632