A Radon-Nikodym theorem for von Neumann algebras

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…

Maharam traces on von Neumann algebras

Traces $Φ$ on von Neumann algebras with values in complex order complete vector lattices are considered. The full description of these traces is given for the case when $Φ$ is the Maharam trace. The version of Radon-Nikodym-type theorem for Maharam traces is established.

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。

Exponential arcs in the manifold of vector states on a sigma-finite von Neumann algebra

This paper introduces the notion of exponential arcs in Hilbert space and of exponential arcs connecting vector states on a sigma-finite von Neumann algebra in its standard representation. Results from Tomita-Takesaki theory form an essential ingredient. Starting point is a non-commutative Radon-Nikodym theorem that involves positive operators affiliated with the commutant algebra. It is shown that exponential arcs are differentiable and that parts of an exponential arc are again exponential arcs. Special cases of probability theory and of quantum probability are used to illustrate the approac…

Von Neumann algebra conditional expectations with applications to generalized representing measures for noncommutative function algebras

We establish several deep existence criteria for conditional expectations on von Neumann algebras, and then apply this theory to develop a noncommutative theory of representing measures of characters of a function algebra. Our main cycle of results describes what may be understood as a `noncommutative Hoffman-Rossi theorem' giving the existence of weak* continuous `noncommutative representing measures' for so-called D-characters. These results may also be viewed as `module' Hahn-Banach extension theorems for weak* continuous `characters' into possibly noninjective von Neumann algebras. In clos…

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.