Intersections of translates of finite-dimensionally valued frame spaces are conditionally slice-full and almost slice-full
Nizar El Idrissi
TL;DR
The paper addresses the geometry and measure-theoretic structure of intersections of translates of frame spaces in finite-dimensional Hilbert $C^*$-modules. The main approach combines real algebraic geometry, showing the degeneracy locus $V=\\{\\Phi:\\det(S_\\Phi)=0\\}$ is an affine algebraic subvariety of $\\mathcal{E}=L^2(X,\\mu;\\mathcal{H})$, with measure-theoretic arguments to deduce that intersections $\\bigcap_k (\\mathcal{F}+a_k)$ are conditionally slice-full and, for almost every translate sequence under a product measure, slice-full. Key contributions include introducing real affine algebraic subvarieties in infinite-dimensional settings and defining (conditionally) slice-full subsets, plus establishing the results uniformly across three algebraic settings: finite-dimensional Hilbert spaces, finite-dimensional real C*-algebras, and finite-dimensional Hilbert C*-modules. The findings offer a novel structural lens on frame degeneracies and translation-invariant intersections, linking algebraic geometry with prevalence-type concepts in infinite-dimensional analysis.
Abstract
In recent work, the topology of frame spaces $\mathcal{F}_{(X,μ),n}$ 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}. These investigations naturally lead to finer questions about the linear geometry 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 is not merely a "small" subset in a topological or measure-theoretic sense but has a precise algebraic description as a real affine algebraic subvariety. In particular, using additional measure-theoretic arguments, we prove that for any finite-dimensional Hilbert $C^*$-module $\mathcal{H}$ and any countable collection of translates of the frame space $\mathcal{F}_{(X,μ),\mathcal{H}}$, the intersection is conditionally slice-full in $L^2(X,μ;\mathcal{H})$ and almost surely slice-full. We inform the reader that the notions of real affine algebraic subvarieties (although closely related to ind-varieties) and (conditionally) slice-full subsets (although closely related to prevalence and Haar-null sets) of a Hausdorff topological vector space are, to our knowledge, both new.