Coherent state map quantization in a Hermitian-like setting

For vector bundles having an involution on the base space, Hermitian-like structures are defined in terms of such an involution. We prove a universality theorem for suitable self-involutive reproducing kernels on Hermitian-like vector bundles. This result relies on pullback operations involving the tautological bundle on the Grassmann manifold of a Hilbert space and exhibits the aforementioned reproducing kernels as pullbacks of universal reproducing kernels that live on the Hermitian-like tautological bundle. To this end we use a certain type of classifying morphisms, which are geometric versions of the coherent state maps from quantum theory. As a consequence of that theorem, we obtain some differential geometric properties of these reproducing kernels in this setting.