G-Hilbert bundles

Abstract

A notion of Hilbert bundle is proposed which leads to the construction of a ’’big’’ Hilbert space ℋ starting from a family of Hilbert spaces. For this, such a family is equipped with a suitable structure, called Borel field structure. A meaningful relationship is established between the Borel structures which can be defined on the union of the Hilbert spaces of the family and the Borel field structures with which the family can be equipped. For a topological group G, the structure of G-Hilbert bundle is defined linking in a suitable way a Hilbert bundle with actions of G. In the framework of a G-Hilbert bundle, a continuous unitary representation of G in ℋ can be constructed. The transitive G-Hilbert bundles which are often used in the theory of induced representations of groups are shown to be a subclass of the class of the G-Hilbert bundles which are proposed in this paper.