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

Michael Frank, David R. Larson

TL;DR

The paper extends frame theory from Hilbert spaces to Hilbert C*-modules, addressing how frames can replace bases in the more general setting where Riesz bases may fail to exist. It develops a geometric dilation approach to embed modules into standard Hilbert C*-modules over unital C*-algebras, defines a frame transform, and proves a reconstruction formula for standard normalized tight frames; it also develops dual frames, frame operator, and notions of complementarity and equivalence of frames. The results have applications to Cuntz-Krieger-Pimsner algebras, finite-index conditional expectations, and connections to vector bundles and noncommutative geometry, with potential impact on wavelet and Gabor-type analyses in C*-algebraic contexts. Overall, the work provides a robust framework for module frames, including canonical duals and classification, enabling structural and geometric analysis of frames in a broad operator-algebraic setting.

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.

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

TL;DR

The paper extends frame theory from Hilbert spaces to Hilbert C*-modules, addressing how frames can replace bases in the more general setting where Riesz bases may fail to exist. It develops a geometric dilation approach to embed modules into standard Hilbert C*-modules over unital C*-algebras, defines a frame transform, and proves a reconstruction formula for standard normalized tight frames; it also develops dual frames, frame operator, and notions of complementarity and equivalence of frames. The results have applications to Cuntz-Krieger-Pimsner algebras, finite-index conditional expectations, and connections to vector bundles and noncommutative geometry, with potential impact on wavelet and Gabor-type analyses in C*-algebraic contexts. Overall, the work provides a robust framework for module frames, including canonical duals and classification, enabling structural and geometric analysis of frames in a broad operator-algebraic setting.

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.

Paper Structure

This paper contains 9 sections, 30 theorems, 63 equations.

Table of Contents

  1. Preliminaries
  2. Basic definitions
  3. Examples of frames
  4. Frame transform and reconstruction formula
  5. Complementary frames, unitary equivalence and similarity
  6. The canonical dual frame and alternate dual frames
  7. A classification result
  8. Final remarks
  9. Appendix

Key Result

Proposition 2.2

Let $A$ be a C*-algebra and $\mathcal{H}$ be a finitely or countably generated Hilbert $A$-module. In general, the inequality $\langle x_j,x_j \rangle \leq 1_A$ holds for every element $x_j$ of normalized tight frames $\{ x_j : J \in {\mathbb J} \}$ of $\mathcal{H}$.

Theorems & Definitions (66)

  • Example 1.1
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • Example 2.4
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Example 3.1
  • ...and 56 more