Continuous $\ast$-K-g-Frame in Hilbert $C^{\ast}$-Modules

Some inequalities for $(α, β)$-normal operators in Hilbert spaces

An operator $T$ acting on a Hilbert space is called $(α,β)$-normal ($0\leq α\leq 1\leq β$) if \begin{equation*} α^{2}T^{\ast }T\leq TT^{\ast}\leq β^{2}T^{\ast}T. \end{equation*} In this paper we establish various inequalities between the operator norm and its numerical radius of $(α,β)$-normal operators in Hilbert spaces. For this purpose, we employ some classical inequalities for vectors in inner product spaces.

Additive preserving rank one maps on Hilbert $C^\ast$-modules

In this paper, we characterize a class of additive maps on Hilbert $C^\ast$-modules which maps a "rank one" adjointable operators to another rank one operators.

Hilbert C*-模上同态与*-同态的Hyers-Ulam稳定性

Hyers-Ulam stability of homomorphisms and *-homomorphisms on Hilbert C*-modules

该研究提出Hilbert C*-模上*-同态的概念，并采用不动点方法证明了此类模上同态与*-同态的Hyers-Ulam稳定性。

Hilbert spaces and ${C}^\ast$-algebras are not finitely concrete

We show that no faithful functor from the category of Hilbert spaces with linear isometries into the category of sets preserves directed colimits. Thus Hilbert spaces cannot form an abstract elementary class, even up to change of language. We deduce an analogous result for the category of commutative unital $C^\ast$-algebras with $\ast$-homomorphisms. This implies, in particular, that this category is not axiomatizable by a first-order theory, a strengthening of a conjecture of Bankston.

Hilbert $C^{\ast}$-模空间中两个算子的欧几里得算子半径的若干不等式

Some inequalities for the Euclidean operator radius of two operators in Hilbert $C^{\ast}$-Modules space

本文研究Hilbert $C^*$-模中两个有界线性算子的欧几里得算子半径，给出若干精确界，并将其与已有数值半径上下界结果建立联系。

On the ternary domain of a completely positive map on a Hilbert C*-module

We associate to an operator valued completely positive linear map $φ$ on a $C^{\ast }$-algebra $A$ and a Hilbert $C^{\ast }$-module $X$ over $A$ a subset $X_{φ}$ of $X,$ called '\textit{ternary domain}' of $φ$ on $X,$ which is a Hilbert $C^{\ast }$-module over the multiplicative domain of $φ$ and every $φ$-map (i.e., associated quaternary map with $φ$) acts on it as a ternary map. We also provide several characterizations for this set. The ternary domain \ of $φ$ on $A\ $ is a closed two-sided $\ast $-ideal $T_{φ}$ of the multiplicative domain of $φ$. We show that $XT_{φ}=X_{φ}$ and give sever…

Crossed products by Hilbert pro-C*-bimodules versus tensor products

We show that if $(X.A)$ and $(Y,B)$ are two isomorphic Hilbert pro-$C^{\ast} $-bimodules, then the crossed product $A\times_{X}\mathbb{Z}$ of $A$ by $X$ and the crossed product $B\times_{Y}\mathbb{Z}$ of $B$ by $Y$ are isomorphic as pro-$C^{\ast}$-algebras. We also prove a property of "associativity" between " $\otimes_{\min}$" and "$\times_{X}$" $\ $as well as " $\otimes_{\max}$" and "$\times_{X}$". As an application of these results we show that the crossed product of a nuclear pro-$C^{\ast}$ -algebra $A$ by a full Hilbert pro-$C^{\ast}$-bimodule $X$ is a nuclear pro-$C^{\ast}$-algebra.

Robustness of controlled $K$-Fusion Frame in Hilbert C$^*$-modules under erasures of submodules

Controlled $\ast$-K-fusion frames are generalization of controlled fusion frames in frame theory. In this paper, we propose the notion of controlled $\ast$-k-fusions frames on Hilbert $C^{\ast}$-modules. We give some caraterizations and some of their properties are obtained. Then we study the erasures of submodules of a controlled $k$-fusion frame in Hilbert $C^{\ast}$-modules and we present some sufficient conditions under which a sequence remains a standart controlled k-fusion frame after deletion of some submodules. Finally, we introduce a perturbation for controlled $K$-fusion frames in Hi…

The reverse order law for Moore-Penrose inverses of operators on Hilbert C*-modules

Suppose $T$ and $S$ are bounded adjointable operators between Hilbert C*-modules admitting bounded Moore-Penrose inverse operators. Some necessary and sufficient conditions are given for the reverse order law $(TS)^{ †} =S^{ †} T^{ †}$ to hold. In particular, we show that the equality holds if and only if $Ran(T^{\ast}TS) \subseteq Ran(S)$ and $Ran(SS^{\ast}T^{\ast}) \subseteq Ran(T^{\ast}),$ which was studied first by Greville [{\it SIAM Rev. 8 (1966) 518--521}] for matrices.

Some generalizations of frames in Hilbert modules

Frames play significant role in various areas of science and engineering. In this paper, we introduce the concepts of frames for $End_{\mathcal{A}}^{\ast}(\mathcal{H, K})$ and their generalizations. Moreover, we obtain some new results for generalized frames in Hilbert modules.

$\ast$-operator frame for $End_{\mathcal{A}}^{\ast}(\mathcal{H})$

In this paper, a new notion of frames is introduced: $\ast$-operator frame as generalization of $\ast$-frames in Hilbert $C^{\ast}$-modules introduced by A. Alijani and M. A. Dehghan \cite{Ali} and we establish some results.

Generalized Fixed-Point Algebras for Twisted $ C^{\ast} $-Dynamical Systems

In his seminal paper "Generalized Fixed Point Algebras and Square-Integrable Group Actions", Ralf Meyer showed how to construct generalized fixed-point algebras for $ C^{\ast} $-dynamical systems via their square-integrable representations on Hilbert $ C^{\ast} $-modules. His method extends Marc Rieffel's construction of generalized fixed-point algebras from proper group actions on $ C^{\ast} $-algebras. This dissertation seeks to generalize Meyer's work to construct generalized fixed-point algebras for twisted $ C^{\ast} $-dynamical systems. To accomplish this, we must introduce some new conc…

The duals of $\ast$-operator frames for $End_{\mathcal{A}}^{\ast}(H)$

Frames play significant role in signal and image processing, which leads to many applications in differents fields. In this paper we define the dual of $\ast$-operator frames and we show their propreties obtained in Hilbert $\mathcal{A}$-modules and we establish some results.

The Covariant Stone-von Neumann Theorem for Actions of Abelian Groups on $ C^{\ast} $-Algebras of Compact Operators

In this paper, we formulate and prove a version of the Stone-von Neumann Theorem for every $ C^{\ast} $-dynamical system of the form $ (G,\mathbb{K}(\mathcal{H}),α) $, where $ G $ is a locally compact Hausdorff abelian group and $ \mathcal{H} $ is a Hilbert space. The novelty of our work stems from our representation of the Weyl Commutation Relation on Hilbert $ \mathbb{K}(\mathcal{H}) $-modules instead of just Hilbert spaces, and our introduction of two additional commutation relations, which are necessary to obtain a uniqueness theorem. Along the way, we apply one of our basic results on Hil…

On the binary relation $\leq_u$ on self-adjoint Hilbert space operators

Given self-adjoint operators $A, B\in\mathbb{B}(\mathscr{H})$ it is said $A\leq_uB$ whenever $A\leq U^*BU$ for some unitary operator $U$. We show that $A\leq_u B$ if and only if $f(g(A)^r)\leq_uf(g(B)^r)$ for any increasing operator convex function $f$, any operator monotone function $g$ and any positive number $r$. We present some sufficient conditions under which if $B\leq A\leq U^*BU$, then $B=A=U^*BU$. Finally we prove that if $A^n\leq U^\ast A^nU$ for all $n\in\mathbb{N}$, then $A=U^\ast AU$.

On frames in Hilbert modules over pro-C*-algebras

We introduce the concept of frame of multipliers in Hilbert modules over pro-C*-algebras and show that many properties of frames in Hilbert C*-modules are valid for frames of multipliers in Hilbert modules over pro-C*-algebras.

On Controlled Frames in Hilbert $C^*$-modules

In this paper, we introduce controlled frames in Hilbert $C^*$-modules and we show that they share many useful properties with their corresponding notions in Hilbert space. Next, we give a characterization of controlled frames in Hilbert $C^*$-module. Also multiplier operators for controlled frames in Hilbert $C^*$-modules will be defined and some of its properties will be shown. Finally, we investigate weighted frames in Hilbert $C^*$-modules and verify their relations to controlled frames and multiplier operators.

*-K-g-Frames and their duals for Hilbert A-modules

Frame theory has a great revolution in recent years. This new Theory have been extended from Hilbert spaces to Hilbert C*-modules. In this paper, we introduce the notion of dual *-K-g-frames in Hilbert A-modules. Lastly we study *-K-g-frames in tensor product of Hilbert C*-Modules and we establish some new results.

A Hilbert C*-module admitting no frames

We show that every infinite-dimensional commutative unital C*-algebra has a Hilbert C*-module admitting no frames. In particular, this shows that Kasparov's stabilization theorem for countably generated Hilbert C*-modules can not be extended to arbitrary Hilbert C*-modules.

Frames and outer frames for Hilbert C^*-modules

The goal of the present paper is to extend the theory of frames for countably generated Hilbert $C^*$-modules over arbitrary $C^*$-algebras. In investigating the non-unital case we introduce the concept of outer frame as a sequence in the multiplier module $M(X)$ that has the standard frame property when applied to elements of the ambient module $X$. Given a Hilbert $\A$-module $X$, we prove that there is a bijective correspondence of the set of all adjointable surjections from the generalized Hilbert space $\ell^2(\A)$ to $X$ and the set consisting of all both frames and outer frames for $X$.…

Controlled $K$-Fusion Frame for Hilbert Spaces

$K$-fusion frames are a generalization of fusion frames in frame theory. In this paper, we extend the concept of controlled fusion frames to controlled $K$-fusion frames, and we develop some results on the controlled $K$-fusion frames for Hilbert spaces, which generalized some well known of controlled fusion frames case. also we discuss some characterizations of controlled Bessel $K$-fusion sequences and of controlled Bessel $K$-fusion. Further, we analyse stability conditions of controlled $K$-fusion frames under perturbation.

Controlled K-g-fusion frames in Hilbert C*-modules

Controlled frames have been the subject of interest because of its ability to improve the numerical efficiency of iterative algorithms for inverting the frame operator. In this paper, we introduce the concepts of controlled g-fusion frame and controlled K-g-fusion frame in Hilbert C*-modules and we give some properties. Also, we study the pertubation problem of controlled K-g-fusion frame. Moreover, an illustrative examples is presented to support the obtained results.

$K$-$b$-frames for Hilbert spaces and the $b$-adjoint operator

In this paper, we will generelize $b$-frames; a new concept of frames for Hilbert spaces, by $K$-$b$-frames. The idea is to take a sequence from a Banach space and see how it can be a frame for a Hilbert space. Instead of the scalar product we will use a new product called the $b$-dual product and it is constructed via a bilinear mapping. We will introduce new results about this product, about $b$-frames, and about $K$-$b$-frames, and we will also give some examples of both $b$-frames and $K$-$b$-frames that have never been given before. We will give the expression of the reconstruction formul…

Structured Parseval Frames in Hilbert $C^*$-modules

We investigate the structured frames for Hilbert $C^{*}$-modules. In the case that the underlying $C^{*}$-algebra is a commutative $W^*$-algebra, we prove that the set of the Parseval frame generators for a unitary operator group can be parameterized by the set of all the unitary operators in the double commutant of the group. Similar result holds for the set of all the general frame generators where the unitary operators are replaced by invertible and adjointable operators. Consequently, the set of all the Parseval frame generators is path-connected. We also obtain the existence and uniquenes…

Hilbert-Schauder Frame Operators

We introduce a new concept of frame operators for Banach spaces we call a Hilbert-Schauder frame operator. This is a hybird between standard frame theory for Hilbert spaces and Schauder frame theory for Banach spaces. Most of our results involve basic structure properties of the Hilbert-Schauder frame operator. Examples of Hilbert-Schauder frames include standard Hilbert frames and classical bases of $\ell_p$ and $L^p$-spaces with $1< p \le 2$. Finally, we give a new isomorphic characterization of Hilbert spaces.

Hilbert C*-模的模框架与框架的对称逼近

Modular frames for Hilbert C*-modules and symmetric approximation of frames

本文系统研究了Hilbert C*-模中的模框架构造及相关算子，通过几何膨胀方法建立与经典框架理论的联系，并利用框架算子范数距离刻画框架接近程度，特别探讨了最接近紧框架的存在唯一性及其与Löwdin正交化的关系。

Hilbert C*-模的模框架概念

A module frame concept for Hilbert C*-modules

本文提出Hilbert C*-模中基于几何膨胀与有界模算子重构的一般模框架理论，建立表示、分解、相似与等价定理，并推广Casazza的近期结果。

Frames and bases in tensor products of Hilbert spaces and Hilbert C*-modules

In this article, we study tensor product of Hilbert $C^*$-modules and Hilbert spaces. We show that if $E$ is a Hilbert $A$-module and $F$ is a Hilbert $B$-module, then tensor product of frames (orthonormal bases) for $E$ and $F$ produce frames (orthonormal bases) for Hilbert $A\otimes B$-module $E\otimes F$, and we get more results. For Hilbert spaces $H$ and $K$, we study tensor product of frames of subspaces for $H$ and $K$, tensor product of resolutions of the identities of $H$ and $K$, and tensor product of frame representations for $H$ and $K$.

Holomorphic Hilbert Bundles and the Frame Existence Problem

We show that if $A=K(l^2)+ \mathbb{C}I_{l^2}$, then there exists a Hilbert $A$-module that possess no frames.

Frames and operators in Hilbert C*-modules

In this paper we introduce the concepts of atomic systems for operators and K-frames in Hilbert C*-modules and we establish some results.

Rank-preserving module maps

In this paper, we characterize rank one preserving module maps on a Hilbert $C^\ast-$module and study its applications on free probability theory.

K-frames in Hilbert $C^*$-modules

In this paper, firstly we investigate conditions under which the action of an operator on a $K$-frame, remain again a $K$-frame for Hilbert module E. We also give a generalization of Douglas Theorem and we shall use it to prove the sum of two $K$-frame under certain condition is again a $K$-frame. Finally, we characterize the $K$-frame generators for a unitary system in terms of operators.

*-Frames for operators on Hilbert modules

K-frames were introduced by L. Gavruta to study atomic systems on Hilbert spaces. Recently some generalizations of this concept are introduced and some of its difference with ordinary frames are studied. In this paper *-K-frames are introduced and some properties of this generalization of K-frames are studied. After proving some characterizations of *-K-frames, direct sum and tensor product of *-K-frames are considered and finally some perturbation results are established.

Hilbert C*-模与C*-代数中的框架

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

本文利用几何膨胀与投影重构方法建立C*-代数及Hilbert C*-模的模框架理论，得到表示、分解、相似性与等价性结果，并应用于Cuntz-Krieger-Pimsner代数、条件期望、C*-代数秩及小波与Gabor框架等领域。

On the isomorphism class of $q$-Gaussian C$^\ast$-algebras for infinite variables

For a real Hilbert space $H_{\mathbb{R}}$ and $-1 < q < 1$ Bozejko and Speicher introduced the C$^\ast$-algebra $A_q(H_{\mathbb{R}})$ and von Neumann algebra $M_q(H_{\mathbb{R}})$ of $q$-Gaussian variables. We prove that if $\dim(H_{\mathbb{R}}) = \infty$ and $-1 < q < 1, q \not = 0$ then $M_q(H_{\mathbb{R}})$ does not have the Akemann-Ostrand property with respect to $A_q(H_{\mathbb{R}})$. It follows that $A_q(H_{\mathbb{R}})$ is not isomorphic to $A_0(H_{\mathbb{R}})$. This gives an answer to the C$^\ast$-algebraic part of Question 1.1 and Question 1.2 in [NeZe18].

Noncommutative Spherical Tight Frames in finitely generated Hilbert C*-modules

In the paper we describe the C*-algebras of noncommutative spherical tight frames over some C*-algebras and then apply to study the noncommutative version of the universal classifying space.

Continuous frames in tensor product Hilbert spaces, localization operators and density operators

Continuous frames and tensor products are important topics in theoretical physics. This paper combines those concepts. We derive fundamental properties of continuous frames for tensor product of Hilbert spaces. This includes, for example, the consistency property, i.e. preservation of the frame property under the tensor product, and the description of the canonical dual tensors by those on the Hilbert space level. We show the full characterization of all dual systems for a given continuous frame, a result interesting by itself, and apply this to dual tensor frames. Furthermore, we discuss the …

经典与量子参考系的希尔伯特模块方法

A Hilbert Module Approach to Classical and Quantum Reference Frames

本文提出基于希尔伯特模块与C*-动力系统等变对应的C*-代数经典与量子参考系模型，给出经典模型为量子模型提供交换基线等结论，并包含相关显式示例。