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

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.

40 pp., about 50 references. Report 1998, University of Houston, Houston, TX and Texas A&M, College Station, TX, U.S.A / submitted, revised version: introduction extended, some proofs with more details, improved explanations, updated literature list

References in corpus (1)

• The Standard Model - the Commutative Case: Spinors, Dirac Operator and de Rham Algebra

Cited by in corpus (49)

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• C*-Algebras of algebraic dynamical systems and right LCM semigroups
• A Hilbert C*-module admitting no frames
• Projection decomposition in multiplier algebras
• Sigma-model solitons on noncommutative spaces
• The gap between unbounded regular operators
• Smooth lattice orbits of nilpotent groups and strict comparison of projections
• Structured Parseval Frames in Hilbert $C^{\ast }$-modules
• On orthogonal systems in Hilbert C*-modules
• Cohomology of the de Rham complex twisted by the oscillatory representation
• Injective and projective Hilbert C*-modules, and C*-algebras of compact operators
• Projective multiresolution analyses for dilations in higher dimensions
• Pimsner algebras and circle bundles
• The Higson-Roe exact sequence and ${\ell }^2$ eta invariants
• A Hilbert C*-module with extremal properties
• Operator valued frames on C*-modules
• Quantum Tomography and the Quantum Radon Transform
• On frames in Hilbert modules over pro-C*-algebras
• Hodge decomposition for elliptic complexes over unital $C^{\ast }$-algebras
• Weak frames in Hilbert C*-modules with application in Gabor analysis
• Noncommutative Shift-Invariant Spaces
• Left multipliers of reproducing kernel Hilbert $C^{\ast }$-modules and the Papadakis theorem
• Controlled $K$-frames in Hilbert $C^{\ast }$-modules
• Operator valued frames
• A Cuntz-Pimsner Model for the $C^{\ast }$-algebra of a Graph of Groups
• Splittings for C*-correspondences and strong shift equivalence
• Power-norms based on Hilbert $C^{\ast }$-modules
• The Balian-Low theorem for locally compact abelian groups and vector bundles
• Hilbertian versus Hilbert W*-modules, and applications to $L^2$- and other invariants
• Index theory on the Miščenko bundle
• Strict essential extensions of C*-algebras and Hilbert C*-modules
• Wavelets and Hilbert modules
• Integral $K$-Operator Frames for $En{d}_{\mathcal{A}}^{\ast }(\mathcal{H})$
• A geometric Elliott invariant and noncommutative rigidity of mapping tori
• Intersections of translates of finite-dimensionally valued frame spaces are conditionally slice-full and almost slice-full
• Frames generated by the functional calculus and function frames of a normal operator
• Controlled $g$-frames in Hilbert $C^{\ast }$-modules
• Standard $\lambda$-lattices, rigid ${\mathrm{C}}^{\ast }$ tensor categories, and (bi)modules
• Perturbation and Stability of Continuous Operator Frames in Hilbert $C^{\ast }$-Modules
• Controlled continuous g-Frames in Hilbert $C^{\ast }$-Modules
• Fourier transform, Schrödinger representation, and Heisenberg modules
• A remark on the gauge action and noncommutative solitons
• *-K-g-Frames and their duals for Hilbert A-modules
• Operator algebras generated by left invertibles
• Finite Operator-Valued Frames
• Operator-valued free Fisher information of random matrices
• *-Frames for operators on Hilbert modules
• Frames generated by actions of countable discrete groups