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We proved in a previous article a path-connectedness property of theintersection of translates of the space of finite-dimensional Hilbert spaceframes, which we formulated in the language of Stiefel manifolds $St(n,H)$.Yet, the notion of a Hilbert space frame has already been successfully extendedto $C^*$-algebras and Hilbert modules. In this article, we prove a new propertyof the intersection of translates of the space of frames both in thefinite-dimensional Hilbert space and finite-dimensional $C^*$-algebraic cases.This property expresses that there are many almost-vector spaces in theseintersections, and it relies crucially on the hypothesis of finite dimension.When the translating $l$-tuple is made of frames, we see that the intersectionalmost contains the vector space spanned by these frames. We finally conjecturethat the property and its consequence are true in a finite-dimensionalleft-Hilbert module because of the simple and related structure of the latter.

Almost-vector spaces in the intersection of translates of the frame space in finite dimension