Universal objects in categories of reproducing kernels

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 2 (N).

Reproducing Kernels and Positivity of Vector Bundles in Infinite Dimensions

“…Reproducing kernels and covariant derivatives often occur simultaneously on the vector bundles involved in various problems in areas such as the geometric quantization, the geometric representation theory of Lie groups, the theory of Cowen-Douglas operators etc. This simple remark additionally motivated the present note, as we tried to provide an explanation for the aforementioned occurrence by using the universality properties of the reproducing kernels established in our previous paper [BG11]. A related issue is the infinite-dimensional extension of the Chern correspondence between the connections and the (almost) complex structures on the total space of a bundle, that we plan to consider in a forthcoming work; see Problem 3.17 below for some more details and references.…”

“…Maybe we should also mention at this point that the interaction between complex geometry and operator theory has already surfaced in the literature, as for instance in the famous theory of the Cowen-Douglas operators ( [CD78]), without to emphasize particularly the idea of positivity. As regards the specific topic of the present paper, we will discuss the relationship between the Griffiths positivity of holomorphic vector bundles ( [Gr69], [GH78]) and the reproducing kernels on infinite-dimensional vector bundles that we have studied recently ( [BG08], [BG09], [BG11], [BG13]).…”

“…The mapping Q H is the universal reproducing kernel associated with the Hilbert space H (see [BG11] and Example 2.5 below).…”

“…We will briefly survey some facts from our earlier papers [BG08], [BG09], [BG11], and [BG13], and will also raise some problems and announce a few partial results with sketchy proofs. Full details of these proofs will be published elsewhere.…”

Geometric Perspectives on Reproducing Kernels

“…In conclusion, for a given CP map ‰, the Stinespring dilation theorem (7.12) can be regarded as an instance of the universality theorem for reproducing kernels of vector bundles, in the sense of [5], and the CP map itself can be interpreted as (admissible) reproducing kernel, connection, and covariant derivative. Thus CP maps appear themselves as geometric objects.…”