发表机构
TICLab, College of Engineering and Architecture, International University of Rabat（拉巴特国际大学工程与建筑学院 TICLab）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对有限维$C^*$-代数上的Hilbert $C^*$-模，提出迹控制算子选择定理，并应用于连续框架的离散化，得到具有显式框架界和受控分离的采样族。
AI 中文摘要
Hilbert空间中连续框架的离散化已得到充分研究，而Hilbert $C^*$-模中的相应问题因$C^*$-代数取值的内积以及Hilbert模上紧算子的结构而带来额外困难。本文针对有限维$C^*$-代数上的Hilbert $C^*$-模，发展了一个用于连续框架离散化的算子选择框架。我们的主要结果是一个关于作用在有限维$C^*$-代数上可数生成Hilbert $C^*$-模上的正紧算子的迹控制选择定理。特别地，我们建立了正迹控制紧算子的二元选择原理的模版本。证明结合了有限秩逼近、有限维$C^*$-代数的矩阵表示以及适应于$C^*$-模设定的算子估计。然后我们将这些选择结果应用于连续Hilbert $C^*$-模框架的离散化。在底层度量测度空间的适当假设下，系数代数$\mathcal A$为有限维，并假设$\mathcal H$的每个闭子模都是正交补的。所得采样族是一个具有显式框架界和受控分离性质的Hilbert $C^*$-模框架。
英文摘要
The discretization of continuous frames in Hilbert spaces is well established, while the corresponding problem for Hilbert $C^*$-modules presents additional difficulties arising from the $C^*$-algebra-valued inner product and the structure of compact operators on Hilbert modules. In this paper, we develop an operator-selection framework for the discretization of continuous frames in Hilbert $C^*$-modules over finite-dimensional $C^*$-algebras. Our main result is a trace-controlled selection theorem for positive compact operators acting on countably generated Hilbert $C^*$-modules over finite-dimensional $C^*$-algebras. In particular, we establish a module version of the binary selection principle for positive trace-controlled compact operators. The proof combines finite-rank approximation, the matrix representation of finite-dimensional $C^*$-algebras, and operator estimates adapted to the $C^*$-module setting. We then apply these selection results to the discretization of continuous Hilbert $C^*$-module frames. Under suitable assumptions on the underlying metric measure space, with the coefficient algebra $\mathcal A$ finite-dimensional, and assuming that every closed submodule of $\mathcal H$ is orthogonally complemented. The resulting sampling family is a Hilbert $C^*$-module frame with explicit frame bounds and controlled separation.
Comments41 pages, This is a preprint and has not yet undergone peer review. Although the proofs have been carefully checked, unintended errors or inaccuracies may remain
