Journal of Operator Theory

Volume 48, Issue 2, Fall 2002 pp. 273-314.

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

Authors: Michael Frank (1), and David R. Larson (2)
Author institution: (1) Universitaet Leipzig, Math. Institut, D--04109 Leipzig, Germany
(2) Department of Mathematics, Texas A&M University, College Station, TX 77843, USA

Summary: 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 dilatio n to standard Hilbert $C^*$-modules over unital $C^*$-algebras that possess orthon ormal Hilbert bases, of reconstruction of the frames by projections and by 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 are reinterpretated for frames in vector and (F)Hilbert bundles.

Keywords: Frame, frame transform, frame operator, dilation, frame representation, Riesz basis, Hilbert basis, $C^*$-algebra, Hilbert $C^*$-module

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