Trace-Controlled Operator Selection and Discretization of Continuous Frames in Hilbert C*-Modules
Abstract
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 𝒜 finite-dimensional, and assuming that every closed submodule of ℋ is orthogonally complemented. The resulting sampling family is a Hilbert C*-module frame with explicit frame bounds and controlled separation.
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BibTeX
@misc{tarif2026tracecontrolled,
title = {{Trace-Controlled Operator Selection and Discretization of Continuous Frames in Hilbert \(C^*\)-Modules}},
author = {Hicham Tarif},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05073},
eprint = {2610.05073},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05073},
}
RIS
TY - UNPB
TI - Trace-Controlled Operator Selection and Discretization of Continuous Frames in Hilbert C^*-Modules
AU - Hicham Tarif
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05073
UR - https://arxiv.org/abs/2610.05073
AB - 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 𝒜 finite-dimensional, and assuming that every closed submodule of ℋ is orthogonally complemented. The resulting sampling family is a Hilbert C^*-module frame with explicit frame bounds and controlled separation.
ER -