Universal objects in categories of reproducing kernels

2009/12/01 by Daniel Beltiţă, Daniel Beltita, Beltita, Daniel +3
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Holomorphic and Operator Theory #math.OA #math.RT #msc:18A05 #msc:46E22 #msc:46L05 #msc:47B32 #msc:58B12

paper · pdf · doi:10.48550/arxiv.0912.0091

34 pages; to appear in Rev. Mat. Iberoamericana

arxiv created 2009/12/01 · arxiv updated 2009/12/08

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 ℓ²(\mathbb N).

Related

• Vector valued reproducing kernel Hilbert spaces and universality
• An algebra structure for reproducing kernel Hilbert spaces
• Reproducing kernels and positivity of vector bundles in infinite dimensions
• Holomorphic geometric models for representations of C^*-algebras
• Deformations of Reproducing Kernel Hilbert Spaces on Homogeneous Subvarieties of the Unit Ball