Abstract

Abstract. We present a general approach to a module 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 other bounded module operators with suitable ranges. We obtain frame representation and decomposition theorems, 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 notion ’frame ’ might generalize the notion ’Hilbert basis ’ for Hilbert C*-modules in a very efficient way avoiding the doubtful 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 reasonable 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 of the work by Deguang Han and David R. Larson [26] in the main. They include the standard

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