Sphere bundle over the set of inner products in a Hilbert space

Abstract

Let (H,〈,〉) be a complex Hilbert space and B(H) the space of bounded linear operators in H. Any other equivalent inner product in H is of the form 〈f,g〉A=〈Af,g〉 (f,g∈H) for some positive invertible operator A∈B(H). In this paper we study the bundle M which consist of the unit sphere {f∈H:〈f,f〉A=1} over each (equivalent) inner product 〈,〉A, which due to the observation above can be definedM={(A,f)∈B(H)×H:A is positive and invertible and 〈Af,f〉=1}. We prove that M is a complemented submanifold of the Banach space B(H)×H and a homogeneous space of the Banach-Lie group G(H)⊂B(H) of invertible operators. We introduce a reductive structure in M, and study properties of the geodesics of the linear connection induced by this reductive structure. We consider certain submanifolds of M, for instance, the one obtained when the positive elements A describing the inner products lie in a prescribed C⁎-algebra A⊂B(H).