Narrow Results

In his seminal paper [On the XY-model on two-sided infinite chain, Publ. RIMS Kyoto Univ. 20 (1984) 277–296], Araki introduced an elegant extension of the Jordan–Wigner transformation which establishes a precise connection between quantum spin systems and Fermi lattice gases in one dimension in the so-called infinite system idealization of quantum statistical mechanics. His extension allows in particular for the rigorous study of numerous aspects of the prominent XY chain over the two-sided infinite discrete line without having to resort to a thermodynamic limit procedure at an intermediate or at the final stage. We rigorously review and elaborate this extension from scratch which makes the paper rather self-contained. In the course of the construction, we also present a simple and concrete realization of Araki’s crossed product extension.

We establish three Hahn–Banach type extension criteria for (sets of) bounded C*-linear maps of Hilbert C*-modules to the underlying C*-algebras of coefficients. One criterion gives an alternative description of the property of C*-algebras to be monotone complete or additively complete.

The notion of frame has some generalizations such as frames of subspaces, fusion frames and g-frames. In this paper, we introduce fusion frames and g-frames in Hilbert C*-modules and we show that they share many useful properties with their corresponding notions in Hilbert space. We also generalize a perturbation result in frame theory to g-frames in Hilbert spaces. We also show that tensor product of fusion frames (g-frames) is a fusion frame (g-frame) and tensor product of resolution of identity is a resolution of identity.

In this paper we introduce modular Riesz basis, modular g-Riesz basis in Hilbert C*-modules in a very natural way and we show that they share many properties with Riesz basis and g-Riesz basis in Hilbert spaces.

We also found that by using the fact that every finitely or countably generated Hilbert C*-module over a unital C*-algebra has a standard Parseval frame, we characterize g-frames, modular Riesz bases and modular g-Riesz bases. Finally we obtain a perturbation result for modular g-Riesz bases.

In this paper, we develop the basic stability theory for g-frames and g-Riesz bases in Hilbert C*-module. Also, we study stability of the dual g-frames in Hilbert C*-module. We extend the Casazza–Christensen perturbation theorem to g-frames in Hilbert C*-module but this is not valid for g-Riesz bases. We prove some characterizations of g-frames and g-Riesz bases in Hilbert C*-module.

Recently, Rashidi-Kouchi et al. devoted their efforts to giving analogs of the Casazza–Christensen general perturbation theorem of Hilbert space frames for the case of g-frames in Hilbert C^*-modules. We found, however, that in the proof they did not obtain the lower bound inequality of g-frames in Hilbert C^*-modules. In this paper, we establish an equivalent definition of g-frames in Hilbert C^*-modules, using it we present a complete proof for the stability result given by them.

The main purpose of this paper is to introduce the notion of *-T-controlled frames in Hilbert $C^{*}$-modules. We present some results of frames in the view of *-T-controlled frames in Hilbert $C^{*}$-modules. Also, we define *-w-frames and give their relations to *-T-controlled frames and multiplier operators.

In this paper, we introduce the notion of (θ₁, θ₂, θ₃, ϕ)-derivations on Hilbert C*-modules. Moreover, we investigate the generalized Hyers–Ulam–Rassias stability, Isac–Rassias type stability and superstability of (θ₁, θ₂, θ₃, ϕ)-derivations on Hilbert C*-modules.