Frames in Hilbert C*-modules and C*-algebras

Michael Frank; David R. Larson

doi:10.48550/arXiv.math/0010189

Frames in Hilbert C-modules and C-algebras

Operator Algebras 2025-05-08 v2 Mathematical Physics Functional Analysis math.MP

Authors: Michael Frank , David R. Larson

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Abstract

We present a general approach to a modular frame theory in C*-algebras and Hilbert C*-modules. The investigations rely on the idea of geometric dilation to standard Hilbert C*-modules over unital C*-algebras that possess orthonormal Hilbert bases, and of reconstruction of the frames by projections and by other bounded modular operators with suitable ranges. We obtain frame representations and decomposition theorems, as well as similarity and equivalence results for frames. Hilbert space frames and quasi-bases for conditional expectations of finite index on C*-algebras appear as special cases. Using a canonical categorical equivalence of Hilbert C*-modules over commutative C*-algebras and (F)Hilbert bundles the results find a reintepretation for frames in vector and (F)Hilbert bundles. Fields of applications are investigations on Cuntz-Krieger-Pimsner algebras, on conditional expectations of finite index, on various ranks of C*-algebras, on classical frame theory of Hilbert spaces (wavelet and Gabor frames), and others. 2001: In the introduction we refer to related publications in detail.

Keywords

hilbert modules frame theory c*-algebras

Cite

@article{arxiv.math/0010189,
  title  = {Frames in Hilbert C*-modules and C*-algebras},
  author = {Michael Frank and David R. Larson},
  journal= {arXiv preprint arXiv:math/0010189},
  year   = {2025}
}

Comments

40 pp., about 50 references. Report 1998, University of Houston, Houston, TX and Texas A&M, College Station, TX, U.S.A / submitted, revised version: introduction extended, some proofs with more details, improved explanations, updated literature list