A module frame concept for Hilbert C*-modules

TL;DR

The paper extends frame theory from Hilbert spaces to Hilbert $A$-modules, introducing module frames and their core operators within the $A$-valued inner-product framework. It develops the frame transform $\theta$ and frame operator $S$, establishes reconstruction formulas and dual frames for standard frames, and demonstrates existence results (via partial isometries) that guarantee standard frames in finitely or countably generated modules. A comprehensive theory of inner sums, complements, and various disjointness notions is developed, with precise criteria expressed through the ranges of frame transforms. The work also connects module-frame representations to operator decompositions and to representations of Cuntz algebras ${\mathcal O}_n$, generalizing Casazza’s Hilbert space results to the Hilbert $A$-module setting and highlighting how Hilbert-space intuition extends to this broader context.

Abstract

The goal of the present paper is a short introduction to a general module frame theory in C*-algebras and Hilbert C*-modules. The reported investigations rely on the idea of geometric dilation to standard Hilbert C*-modules over unital C*-algebras that possess orthonormal bases, and of reconstruction of the frames by projections and other bounded module operators with suitable ranges. We obtain frame representation and decomposition theorems, as well as similarity and equivalence results. The relative position of two and more frames in terms of being complementary or disjoint is investigated in detail. In the last section some recent results by P. G. Casazza are generalized to our setting. The Hilbert space situation appears as a special case. For detailled proofs we refer to another paper also contained in the ArXiv.

Theorems & Definitions (50)

• Definition 2.1
• Example 2.2
• Theorem 2.3
• Example 2.4
• Theorem 2.5
• Definition 3.1
• Theorem 3.2
• Example 3.3
• Theorem 4.1
• Example 4.2