A Radon-Nikodym theorem for completely n-positive linear maps on pro-C*-algebras and its applications

A Radon-Nikodym theorem for completely multi-positive linear maps and applications

052<p type="texpara" tag="Body Text" et="abstract" >A completely $n$ -positive linear map from a locally $C^{\ast}$-algebra $A$ to another locally $C^{\ast}$-algebra $B $is an $n\times n$ matrix whose elements are continuous linear maps from $A$ to $B$ and which verifies the condition of completely positivity. In this paper we prove a Radon-Nikodym type theorem for strict completely $n$-positive linear maps which describes the order relation on the set of all strict completely $n$ -positive linear maps from a locally $C^{\ast }$-algebra $A$ to a $C^{\ast}$-algebra $B$, in terms of a self-dual …

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。

Radon-Nikodym derivatives of quantum operations

Given a completely positive (CP) map $T$, there is a theorem of the Radon-Nikodym type [W.B. Arveson, Acta Math. {\bf 123}, 141 (1969); V.P. Belavkin and P. Staszewski, Rep. Math. Phys. {\bf 24}, 49 (1986)] that completely characterizes all CP maps $S$ such that $T-S$ is also a CP map. This theorem is reviewed, and several alternative formulations are given along the way. We then use the Radon-Nikodym formalism to study the structure of order intervals of quantum operations, as well as a certain one-to-one correspondence between CP maps and positive operators, already fruitfully exploited in m…

不变块多重线性完全正映射的 Russo-Dye 型定理、Stinespring 表示与 Radon-Nikodym 导数

Russo-Dye type Theorem, Stinespring representation,and Radon Nikodym dervative for invariant block multilinear completely positive maps

本文研究了不变多重线性完全正映射的性质，证明了其 Russo-Dye 型定理、自动对称性与完全有界性，引入了不变块 CP 映射的多重线性版本，推导出极小 Stinespring 膨胀并获得了 Radon Nikodym 定理。

量子期望的弱*分解与 Radon-Nikodym 定理

Weak* decomposition and Radon-Nikodym theorem for quantum expectations

本文定义了非交换概率空间上量子期望的弱*连续性与奇异性，利用 von Neumann 代数理论提出弱*分解，证明其在 KMS 条件下与 AGKL 分解等价，从而计算 Radon-Nikodym 导数，并探讨了向 Cuntz-Toeplitz 算子系统的扩展。

Radon--Nikodym representations of Cuntz--Krieger algebras and Lyapunov spectra for KMS states

We study relations between $(H,β)$--KMS states on Cuntz--Krieger algebras and the dual of the Perron--Frobenius operator $\mathcal{L}_{-βH}^{*}$. Generalising the well--studied purely hyperbolic situation, we obtain under mild conditions that for an expansive dynamical system there is a one--one correspondence between $(H,β)$--KMS states and eigenmeasures of $\mathcal{L}_{-βH}^{*}$ for the eigenvalue 1. We then consider representations of Cuntz--Krieger algebras which are induced by Markov fibred systems, and show that if the associated incidence matrix is irreducible then these are $\ast$--is…

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…

C*-代数上态的支配性与等价性：拟不变态

Dominance and equivalence for states on $C^*$-algebras: Quasi-Invariant states

本文研究C*-代数上态的支配性与等价性的非交换推广，推导相关Radon-Nikodym导数，探讨群作用下拟不变态的性质，给出群作用的酉实现并与Pedersen-Takesaki构造对比。

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.

On the origin of non-decomposable maps

The Radon-Nikodym formalism is used to study the structure of the set of positive maps from $\mathcal{B}(\mathcal{H})$ into itself, where $\mathcal{H}$ is a finite dimensional Hilbert space. In particular, this formalism was employed to formulate simple criteria which ensure that certain maps are non decomposable. In that way, a recipe for construction of non decomposable maps was obtained.