Intersections of translates of finite-dimensionally valued frame spaces are conditionally slice-full and almost slice-full

TL;DR

This paper investigates the geometric and measure-theoretic structure of intersections of translated frame spaces in finite-dimensional Hilbert C*-modules, revealing they are almost slice-full and possess a slice-wise algebraic structure.

Contribution

It uncovers a new almost-linear structure within intersections of translated frame spaces and introduces novel concepts like slice-fullness and slice-wise algebraic subvarieties.

Findings

The set of non-frames forms a slice-wise real affine algebraic subvariety.

Intersections of translated frame spaces are conditionally slice-full.

Such intersections are almost surely slice-full in the measure-theoretic sense.

Abstract

In recent work, the topology of frame spaces $\mathcal{F}_{(X,\mu),n}$ has been studied via Stiefel manifolds, revealing in particular a connectedness property for intersections of their translates when $\operatorname{span}(\{a_j\}_{j \in J}$ is not too large, in fact when $\operatorname{codim}(\operatorname{span}\{a_j^l\}_{(j,l) \in J \times [\![1,n]\!]}) \geq 3n$, where $\{a_j\}_{j \in J}$ is the translating family \cite{ElIdrissiKabbajMoalige2023}. The investigation of the connectedness of the intersections of translates of the frame space can be extended to questions about the algebro-geometric and measure-theoretic structure of such intersections. The present article addresses these questions by uncovering an almost-linear structure within intersections of translated frame spaces. We show that the set of non-frames in finite-dimensional Hilbert $C^*$-modules inherits the structure of a slice-wise real affine algebraic subvariety. As a consequence, it is a small subset in a precise measure-theoretic sense. In particular, we prove that for any finite-dimensional Hilbert $C^*$-module $\mathcal{H}$ and any countable collection of translates of the frame space $\mathcal{F}_{(X,\mu),\mathcal{H}}$, the intersection is conditionally slice-full in $L^2(X,\mu;\mathcal{H})$ and almost surely slice-full. We inform the reader that the notions of slice-wise real affine algebraic subvarieties (although related to ind-varieties), conditionally slice-full subsets and slice-full subsets (although related to shy sets) of a Hausdorff topological vector space are, to our knowledge, both new.