Universal objects in categories of reproducing kernels

Rev. Mat. Iberoamericana, Tome 27 (2011) no. 1, p. 123-179 / Harvested from Project Euclid

Résumé

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

Détail

Publié le : 2011-01-15
Classification:  reproducing kernel,  category theory,  vector bundle,  tautological bundle,  Grassmann manifold,  completely positive map,  universal object,  46E22,  47B32,  46L05,  18A05,  58B12