Covariant version of the Stinespring type theorem for Hilbert C*-modules

Stinespring's Theorem for Unbounded Operator valued Local completely positive maps and Its Applications

Anar A. Dosiev in [Local operator spaces, unbounded operators and multinormed $C^*$-algebras, J. Funct. Anal. 255 (2008), 1724-1760], obtained a Stinespring's theorem for local completely positive maps (in short: local CP-maps) on locally $C^{\ast}$-algebras. In this article a suitable notion of minimality for this construction has been identified so as to ensure uniqueness up to unitary equivalence for the associated representation. Using this a Radon-Nikodym type theorem for local completely positive maps has been proved. Further, a Stinespring's theorem for unbounded operator valued local c…

Minimal Stinespring Representations of Operator Valued Multilinear Maps

A completely positive linear map $φ$ from a C*-algebra $A$ into $B(H)$ has a Stinespring representation as $φ(a) = X^*π(a)X,$ where $π$ is a *-representation of $A$ on a Hilbert space $K$ and $X$ is a bounded operator from $H$ to $K. $ Completely bounded multilinear operators on C*-algebras as well as some densely defined multilinear operators in Connes' non commutative geometry also have Stinespring representations of the form $$ Φ(a_1, \dots, a_k ) = X_0π_1(a_1)X_1 \dots π_k(a_k)X_k$$ such that each $a_i$ is in a *-algebra $A_i$ and $X_0, \dots X_k $ are densely defined closed operators betw…

Stinespring 构造作为一种伴随

Stinespring's construction as an adjunction

本文将 Stinespring 构造刻画为从 $C^*$-代数表示到完全正映射的限制函子的左伴随，并借此给出极小膨胀的泛性质，最终证明所有有限维 $C^*$-代数满足纯化公设。

Matrix KSGNS construction and a Radon--Nikodym type theorem

In this paper, we introduce the concept of completely positive matrix of linear maps on Hilbert $A$-modules over locally $C^{*}$-algebras and prove an analogue of Stinespring theorem for it. We show that any two minimal Stinespring representations for such matrices are unitarily equivalent. Finally, we prove an analogue of the Radon--Nikodym theorem for this type of completely positive $n\times n$ matrices.

Completely positive maps on Hilbert modules over pro-C*-algebras

We derive Paschke's GNS construction for completely positive maps on unital pro-C*-algebras from the KSGNS construction, presented by M. Joita [J. London Math. Soc. {\bf 66} (2002), 421--432], and then we deduce an analogue of Stinespring theorem for Hilbert modules over pro-C*-algebras. Also, we obtain a Radon-Nikodym type theorem for operator valued completely positive maps on Hilbert modules over pro-C*-algebras.

KSGNS type construction for $α$-completely positive maps on Krein $C^{\ast}$-modules

In this paper, we investigate $Φ$-maps associated to a certain type of $α$-completely positive maps. We then prove a KSGNS (Kasparov--Stinespring--Gel'fand--Naimark--Segal) type theorem for $α$-completely positive maps on Krein $C^*$-modules and show that the minimal KSGNS construction is unique up to unitary equivalence. We also establish a covariant version of the KSGNS type theorem for a covariant $α$-completely positive map and study the structure of minimal covariant KSGNS constructions.

Hilbert pro-C*-模上CP映射的Radon-Nikodym定理

A Radon-Nikodym theorem for CP-maps on Hilbert pro-C*-modules

该研究针对pro-C*-代数上Hilbert模间的完全正映射，引入等价关系并分析其Stinespring构造，定义预序关系，最终得到一类Radon-Nikodym型定理，拓展了相关算子代数理论的研究范畴。

Hilbert von Neumann modules

We introduce a way of regarding Hilbert von Neumann modules as spaces of operators between Hilbert space, not unlike [Skei], but in an apparently much simpler manner and involving far less machinery. We verify that our definition is equivalent to that of [Skei], by verifying the `Riesz lemma' or what is called `self-duality' in [Skei]. An advantage with our approach is that we can totally side-step the need to go through $C^*$-modules and avoid the two stages of completion - first in norm, then in the strong operator topology - involved in the former approach. We establish the analogue of the …

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…

Hilbert 局部 $C^*$-模的算子模型

Operator Models for Hilbert Locally $C^*$-Modules

本文通过局部有界算子刻画了局部 $C^*$-代数上的具体 Hilbert 模概念，为所有 Hilbert 局部 $C^*$-模构造了算子模型，并基于一般膨胀定理导出了外张量积的直接构造及两个 Stinespring 型定理。

On discrete twisted C*-dynamical systems, Hilbert C*-modules and regularity

We first give an overview of the basic theory for discrete unital twisted C*-dynamical systems and their covariant representations on Hilbert C*-modules. After introducing the notion of equivariant representations of such systems and their product with covariant representations, we prove a kind of Fell absorption principle saying that the product of an induced regular equivariant representation with a covariant faithful representation is weakly equivalent to an induced regular covariant representation. This principle is the key to our main result, namely that a certain property, formally weake…

Covariant representations of subproduct systems: Invariant subspaces and curvature

Let $X=(X(n))_{n \in \mathbb{Z_+}}$ be a standard subproduct system of $C^*$-correspondences over a $C^*$-algebra $\mathcal M.$ Assume $T=(T_n)_{n \in \mathbb{Z_+}}$ to be a pure completely contractive, covariant representation of $X$ on a Hilbert space $\mathcal H,$ and $\mathcal S$ to be a non-trivial closed subspace of $\mathcal H.$ Then $\mathcal{S}$ is invariant for $T$ if and only if there exist a Hilbert space $\mathcal{D},$ a representation $π$ of $\mathcal M$ on $\mathcal D,$ and a partial isometry $Π: \mathcal{F}_X\bigotimes_π\mathcal{D}\to \mathcal{H} $ such that $$Π(S_n(ζ)\otimes I…

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…

Martingales, endomorphisms, and covariant systems of operators in Hilbert space

We show that a class of dynamical systems induces an associated operator system in Hilbert space. The dynamical systems are defined from a fixed finite-to-one mapping in a compact metric space, and the induced operators form a covariant system in a Hilbert space of L^2-martingales. Our martingale construction depends on a prescribed set of transition probabilities, given by a non-negative function. Our main theorem describes the induced martingale systems completely. The applications of our theorem include wavelets, the dynamics defined by iterations of rational functions, and sub-shifts in sy…

On the index of product systems of Hilbert modules

In this note we prove that the set of all uniformly continuous units on a product system over a C* algebra B can be endowed with the structure of left right B - B Hilbert module after identifying similar units by the suitable equivalence relation. We use this construction to define the index of the initial product system, and prove that it is the generalization of earlier defined indices by Arveson (in the case B=C) and Skeide (in the case of spatial product system). We prove that such defined index is a covariant functor from the category od continuous product systems to the category of B bim…

Kaplansky-Hilbert 模上酉群表示的分解定理与 Furstenberg-Zimmer 结构定理

A Decomposition Theorem for Unitary Group Representations on Kaplansky-Hilbert Modules and the Furstenberg-Zimmer Structure Theorem

本文建立了 Stone 代数上 Kaplansky-Hilbert 模的酉群表示分解定理，并基于算子理论将其应用于推广著名的 Furstenberg-Zimmer 结构定理至任意群在任意概率空间上的保测作用情形。

局部紧量子群的协变 Stone-von Neumann 定理

The Covariant Stone-von Neumann Theorem for Locally Compact Quantum Groups

本文将 Stone-von Neumann 定理推广至 Hilbert 模块上的局部紧量子动力学系统，引入一对模表示以涵盖多种模型，并提出了识别强正则局部紧量子群的新准则。

Cuntz-Krieger uniqueness theorem for crossed products by Hilbert bimodules

It is shown that a C*-algebra generated by any faithful covariant representation of a Hilbert bimodule X is canonically isomorphic to the crossed product associated to X provided that Rieffel's induced representation functor X-ind is topologically free. It is discussed how this result could be applied to universal C*-algebras generated by relations with a circle gauge action. In particular, it leads to generalizations of isomorphism theorems for various crossed products, and is shown to be equivalent to Cuntz-Krieger uniqueness theorem for finite graph C*-algebras (on that occasion an intrigui…

对称性导致的熵界

An entropy bound due to symmetries

本文在满足Bisognano-Wichmann性质的时空局部网中，推导出仅依赖Poincaré群表示和PCT反幺正算子的局部熵上界，并在Möbius协变网及$U(1)$-流导数网中验证了该界与熵增量。