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

Abstract

Let $(H,〈,〉)$ be a complex Hilbert space and $B\left(H\right)$ the space of bounded linear operators in $H$. Any other equivalent inner product in $H$ is of the form ${〈f,g〉}_A=〈Af,g〉$ ($f,g\in H$) for some positive invertible operator $A\in B\left(H\right)$. In this paper we study the bundle $M$ which consist of the unit sphere $\{f\in H:{〈f,f〉}_A=1\}$ over each (equivalent) inner product ${〈,〉}_A$, which due to the observation above can be defined$M=\left\{\right(A,f)\in B(H)\times H:A\text{ is positive and invertible and }〈Af,f〉=1\}.$ We prove that $M$ is a complemented submanifold of the Banach space $B\left(H\right)\times H$ and a homogeneous space of the Banach-Lie group $G\left(H\right)\subset B\left(H\right)$ 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 $A\subset B\left(H\right)$.