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

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

In this paper, we introduce the concept of Continuous $\ast$-K-g-Frame in Hilbert $C^{\ast}$-Modules and we give some properties.

Controlled continuous g-Frames in Hilbert $C^{\ast}$-Modules

Frame Theory has a great revolution in recent years, this Theory have been extended from Hilbert spaces to Hilbert $C^{\ast}$-modules. The purpose of this paper is the introduction and the study of the concept of Controlled Continuous g-Frames in Hilbert $C^{\ast}$-Modules. Also we give some properties.

有限维C*代数上互为强Birkhoff-James正交性的加法保持器

Additive preservers of mutual strong Birkhoff-James orthogonality on finite-dimensional $C^\ast$-algebras

本文研究有限维C*代数上加法映射保持单向奇异性的性质，并分类保持互为强Birkhoff-James正交性的加法映射。

Some generalizations of K-g-frames in Hilbert $C^{\ast}$- module

In this papers we investigate the g-frame and Bessel g-sequence related to a linear bounded operator $K$ in Hilbert $C^{\ast}$-module and we establish some results.

Finite generation in $C^\ast$-algebras and Hilbert $C^\ast$-modules

We characterize $C^*$-algebras and $C^*$-modules such that every maximal right ideal (resp. right submodule) is algebraically finitely generated. In particular, $C^*$-algebras satisfy the Dales--Żelazko conjecture.

Multipliers of Hilbert pro-C*-bimodules and crossed products by Hilbert pro-C*-bimodules

In this paper we introduce the notion of multiplier of a Hilbert pro-$C^{\ast }$-bimodule and we investigate the structure of the multiplier bimodule of a Hilbert pro-$C^{\ast}$-bimodule. We also investigate the relationship between the crossed product $A\times _{X}\mathbb{Z}$ of a pro-$% C^{\ast }$-algebra $A$ by a Hilbert pro-$C^{\ast }$-bimodule $X$ over $A$, the crossed product $M(A)\times _{M(X)}\mathbb{Z}$ of the multiplier algebra $M(A)$ of $A$ by the multiplier bimodule $M(X)$ of $X$ and the multiplier algebra $M(A\times _{X}\mathbb{Z})$ of $A\times _{X}\mathbb{Z}$.

Additivity of maps preserving Jordan $η_{\ast}$-products on $C^{*}$-algebras

Let $\mathcal{A}$ and $\mathcal{B}$ be two $C^{*}$-algebras such that $\mathcal{B}$ is prime. In this paper, we investigate the additivity of map $Φ$ from $\mathcal{A}$ onto $\mathcal{B}$ that are bijective unital and satisfies $$Φ(AP+ηPA^{*})=Φ(A)Φ(P)+ηΦ(P)Φ(A)^{*},$$ for all $A\in\mathcal{A}$ and $P\in\{P_{1},I_{\mathcal{A}}-P_{1}\}$ where $P_{1}$ is a nontrivial projection in $\mathcal{A}$. Let $η$ be a non-zero complex number such that $|η|\neq1$, then $Φ$ is additive. Moreover, if $η$ is rational then $Φ$ is $\ast$-additive.

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…

Additive mappings in algebras of unbounded operators preserving operators of rank one

Let $ \mathcal D$ be a dense linear manifold in a Hilbert space $\mathcal H$ and let $L^+(\mathcal D)$ be the *-algebra of all linear operators $A$ such that $A \mathcal D \subset \mathcal D, A^* \mathcal D \subset \mathcal D$. Denote by $F(\mathcal D)$ the *-ideal of $L^+(\mathcal D)$ consisting of all finite-rank operators. A characterization of the structure of additive mappings of $F(\mathcal D)$ preserving operators of rank one or projections of rank one is given. The corresponding results for algebras of bounded operators on Banach spaces were given by P. uSemrl.

Additive maps preserving idempotency of products or Jordan products of operators

Let H and K be infinite dimensional Hilbert spaces, while B(H) and B(K) denote the algebras of all linear bounded operators on H and K, respectively. We characterize the forms of additive mappings from B(H) into B(K) that preserve the nonzero idempotency of either Jordan products of operators or usual products of operators in both directions.

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.

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.

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。

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。

关于Hilbert C*-模的Hahn-Banach型定理

On Hahn-Banach type theorems for Hilbert C*-modules

本文针对Hilbert C*-模上的有界C*-线性映射建立了三个Hahn-Banach型延拓准则，并利用其中一个准则给出了AW*-C*-代数单调完备与加法完备性质的新等价刻画。

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…

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.

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…

$\ast$-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.

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

Let $M_q(H_{\mathbb{R}})$ be the $q$-Gaussian von Neumann algebra associated with a separable infinite dimensional real Hilbert space $H_{\mathbb{R}}$ where $-1 < q < 1$. We show that $M_q(H_{\mathbb{R}}) \not \simeq M_0(H_{\mathbb{R}})$ for $-1 < q \not = 0 < 1$. The C$^\ast$-algebraic counterpart of this result was obtained recently in [BCKW22]. Using ideas of Ozawa we show that this non-isomorphism result also holds on the level of von Neumann algebras.

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.

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.

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].

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 $A=U^\ast AU$.

Maps preserving the local spectral subspace of skew-product of operators

Let $B(H)$ be the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert space $H$. For $T \in B(H)$ and $λ\in \mathbb{C}$, let $H_{T}(\{λ\})$ denotes the local spectral subspace of $T$ associated with $\{λ\}$. We prove that if $φ:B(H)\rightarrow B(H)$ be an additive map such that its range contains all operators of rank at most two and satisfies $$H_{φ(T)φ(S)^{\ast}}(\{λ\})= H_{TS^{\ast}}(\{λ\})$$ for all $T, S \in B(H)$ and $λ\in \mathbb{C}$, then there exist a unitary operator $V$ in $B(H)$ and a nonzero scalar $μ$ such that $φ(T) = μTV^{\ast}$ for all $T \in B(H…

$C^\ast$-extreme maps and nests

The generalized state space $ S_{\mathcal{H}}(\mathcal{\mathcal{A}})$ of all unital completely positive (UCP) maps on a unital $C^*$-algebra $\mathcal{A}$ taking values in the algebra $\mathcal{B}(\mathcal{H})$ of all bounded operators on a Hilbert space $\mathcal{H}$, is a $C^\ast$-convex set. In this paper, we establish a connection between $C^\ast$-extreme points of $S_{\mathcal{H}}(\mathcal{A})$ and a factorization property of certain algebras associated to the UCP map. In particular, this factorization property of some nest algebras is used to give a complete characterization of those $C^…