Revista Matemática Iberoamericana
- Rev. Mat. Iberoamericana
- Volume 27, Number 1 (2011), 123-179.
Universal objects in categories of reproducing kernels
Daniel Beltiţă and José E. Galé
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 $\ell^2(\mathbb{N})$.
Article information
Source
Rev. Mat. Iberoamericana, Volume 27, Number 1 (2011), 123-179.
Dates
First available in Project Euclid: 4 February 2011
Permanent link to this document
https://projecteuclid.org/euclid.rmi/1296828831
Mathematical Reviews number (MathSciNet)
MR2815734
Zentralblatt MATH identifier
1234.46026
Subjects
Primary: 46E22: Hilbert spaces with reproducing kernels (= [proper] functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) [See also 47B32]
Secondary: 47B32: Operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) [See also 46E22] 46L05: General theory of $C^*$-algebras 18A05: Definitions, generalizations 58B12: Questions of holomorphy [See also 32-XX, 46G20]
Keywords
reproducing kernel category theory vector bundle tautological bundle Grassmann manifold completely positive map universal object
Citation
Beltiţă, Daniel; Galé, José E. Universal objects in categories of reproducing kernels. Rev. Mat. Iberoamericana 27 (2011), no. 1, 123--179. https://projecteuclid.org/euclid.rmi/1296828831
