FRAMES IN HILBERT C*-MODULES AND C*-ALGEBRAS
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We present a general approach to a module frame theory in C*-algebras and Hilbert C*-modules. The investigations rely on the ideas of geometric dilation to standard Hilbert C*-modules over unital C*-algebras that possess orthonormal Hilbert bases, of reconstruction of the frames by projections and by other bounded module operators with suitable ranges. We obtain frame representation and decomposition the- orems, as well as similarity and equivalence results. 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 reinterpretation for frames in vector and (F)Hilbert bundles. The purpose of this paper is to extend the theory of frames known for (separable) Hilbert spaces to similar sets in C*-algebras and (finitely and countably generated) Hilbert C*-modules. The concept 'frame' may generalize the concept 'Hilbert basis' for Hilbert C*-modules in a very efficient way circumventing the ambiguous condition of 'C*-linear independence' and emphasizing geometrical dilation results and operator properties. This idea is natural in this context because, while such a module may fail to have any reason- able type of basis, it turns out that countably generated Hilbert C*-modules over unital C*-algebras always have an abundance of frames of the strongest (and simplest) type. The considerations follow the line of the geometrical and operator-theoretical approach worked out by Deguang Han and David R. Larson (30) in the main. They include the standard Hilbert space case in full as a special case, see also (12, 13, 29, 31, 34, 57). However, proofs that generalize from the Hilbert space case, when attainable, are usually considerably more difficult for the module case for reasons that do not occur in the sim- pler Hilbert space case. For example, Riesz bases of Hilbert spaces with frame bounds equal to one are automatically orthonormal bases, a straight consequence of the frame definition. A similar statement for standard Riesz bases of certain Hilbert C*-modules still holds, but the proof of the statement requires incomparably more efforts to be estab- lished, see Corollary 4.2. Generally speaking, the known results and obstacles of Hilbert C*-module theory in comparison to Hilbert space and ideal theory would rather suggest to expect a number of counterexamples and diversifications of situations that could appear investigating classes of Hilbert C*-modules and of C*-algebras of coefficients beyond the Hilbert space situation. Surprisingly, almost the entire theory can be shown to survive these significant changes. For complementary results to those explained in the present paper we refer to (24, 28).
Subfields: Applied Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/7481896dc7c29a3bfae8e53ca008e5adf97c2827
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Citations in the corpus
- Operator-valued frames (2007, background): “Also, several papers in the literature deal with frames in Hilbert C∗-modules, including one by the same authors of this paper [11, 12, 19].”
- Frame representations and Parseval duals with applications to Gabor frames (2008, background): “…of attention in recent years, partially because of its applications in signal processing as well as its close connections with other mathematical fields such as wavelet theory, time-frequency analysis, operator and operator algebra theory (cf. [8, 14, 15, 12, 13, 16, 17, 18, 20, 21, 27, 25, 29]).”
- A module frame concept for Hilbert C*-modules (2000, unknown): “Definition 2.1.”
- Projective modules over noncommutative tori are multi-window Gabor frames for modulation spaces (2008, background): “Later Frank and Larson introduced module frames for arbitrary finitely and countably generated Hilbert C-modules in [23].”