scholarly journals On the product system of a completely positive semigroup

2003 ◽  
Vol 200 (1) ◽  
pp. 237-280 ◽  
Author(s):  
Daniel Markiewicz
Keyword(s):  
Positive Semigroup ◽  
Product System ◽  
Completely Positive ◽  
Completely Positive Semigroup

scholarly journals ON THE INDEX AND DILATIONS OF COMPLETELY POSITIVE SEMIGROUPS

1999 ◽  
Vol 10 (07) ◽  
pp. 791-823 ◽  
Author(s):  
WILLIAM ARVESON
Keyword(s):  
Bounded Operator ◽  
Positive Semigroup ◽  
Completely Positive Maps ◽  
Completely Positive ◽  
Positive Semigroups ◽  
Matrix Algebras ◽  
Positive Maps ◽  
Completely Positive Semigroups ◽  
Numerical Index ◽  
Completely Positive Semigroup

It is known that every semigroup of normal completely positive maps P = {Pt:t≥ 0} of ℬ(H), satisfying Pt(1) = 1 for every t ≥ 0, has a minimal dilation to an E0 acting on ℬ(K) for some Hilbert space K⊇H. The minimal dilation of P is unique up to conjugacy. In a previous paper a numerical index was introduced for semigroups of completely positive maps and it was shown that the index of P agrees with the index of its minimal dilation to an E0-semigroup. However, no examples were discussed, and no computations were made. In this paper we calculate the index of a unital completely positive semigroup whose generator is a bounded operator [Formula: see text] in terms of natural structures associated with the generator. This includes all unital CP semigroups acting on matrix algebras. We also show that the minimal dilation of the semigroup P={ exp tL:t≥ 0} to an E0-semigroup is is cocycle conjugate to a CAR/CCR flow.

COCYCLE PERTURBATION OF QUASIFREE ALGEBRAIC K-FLOW LEADS TO REQUIRED ASYMPTOTIC DYNAMICS OF ASSOCIATED COMPLETELY POSITIVE SEMIGROUP

2000 ◽  
Vol 03 (02) ◽  
pp. 237-246 ◽  
Author(s):  
GRIGORI G. AMOSOV
Keyword(s):  
Separable Hilbert Space ◽  
Positive Semigroup ◽  
Completely Positive ◽  
Parameter Group ◽  
Discrete Parameter ◽  
C Algebra ◽  
Asymptotic Dynamics ◽  
Cocycle Perturbation ◽  
Canonical Anticommutation Relations ◽  
Completely Positive Semigroup

We study the quasifree algebraic K-flow τ on the hyperfinite factor ℳ with the expanding subfactor [Formula: see text] generated by representations π of the C*-algebra of the canonical anticommutation relations (CAR) [Formula: see text] over separable Hilbert space ℋ. The type of ℳ and [Formula: see text] can be II1 or IIIλ, 0<λ<1, depending on π. The K-flow τ is obtained by the quasifree lifting of one-parameter group ST consisting of shifts in ℋ with the discrete parameter T=Z or the continuous one T=R. We prove that acting on τ by a quasifree inner Markovian cocycle, one can get the required asymptotic behavior of the perturbed group restriction on [Formula: see text].

Entropy theory without a past

2000 ◽  
Vol 20 (5) ◽  
pp. 1355-1370 ◽  
Author(s):  
E. GLASNER ◽  
J.-P. THOUVENOT ◽  
B. WEISS
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Amenable Group ◽  
Entropy Theory ◽  
Positive Entropy ◽  
Product System ◽  
Completely Positive ◽  
Component Systems ◽  
General Setup

This paper treats the Pinsker algebra of a dynamical system in a way which avoids the use of an ordering on the acting group. This enables us to prove some of the classical results about entropy and the Pinsker algebra in the general setup of measure-preserving dynamical systems, where the acting group is a discrete countable amenable group. We prove a basic disjointness theorem which asserts the relative disjointness in the sense of Furstenberg, of $0$-entropy extensions from completely positive entropy (c.p.e.) extensions. This theorem is used to prove several classical results in the general setup. For example, we show that the Pinsker factor of a product system is equal to the product of the Pinsker factors of the component systems. Another application is to obtain a generalization (as well as a simpler proof) of the quasifactor theorem for $0$-entropy systems of Glasner and Weiss.

scholarly journals The index of product systems of Hilbert modules: Two equivalent definitions

2015 ◽  
Vol 97 (111) ◽  
pp. 49-56
Author(s):  
Biljana Vujosevic
Keyword(s):  
Positive Definite ◽  
Hilbert Modules ◽  
Product System ◽  
Initial Product ◽  
Positive Definite Kernel ◽  
Completely Positive ◽  
C Algebra ◽  
Product Systems ◽  
The One ◽  
Definition Of

We prove that a conditionally completely positive definite kernel, as the generator of completely positive definite (CPD) semigroup associated with a continuous set of units for a product system over a C*-algebra B, allows a construction of a Hilbert B?B module. That construction is used to define the index of the initial product system. It is proved that such definition is equivalent to the one previously given by Keckic and Vujosevic [On the index of product systems of Hilbert modules, Filomat, to appear, ArXiv:1111.1935v1 [math.OA] 8 Nov 2011]. Also, it is pointed out that the new definition of the index corresponds to the one given earlier by Arveson (in the case B = C).

scholarly journals NOAA Satellite Soil Moisture Operational Product System (SMOPS) Version 3.0 Generates Higher Accuracy Blended Satellite Soil Moisture

2020 ◽  
Vol 12 (17) ◽  
pp. 2861
Author(s):  
Jifu Yin ◽  
Xiwu Zhan ◽  
Jicheng Liu
Keyword(s):  
Soil Moisture ◽  
Real Time ◽  
Land Surface ◽  
Vital Role ◽  
Product System ◽  
Global Soil ◽  
One Stop ◽  
Satellite Sensors ◽  
Blended Data

Soil moisture plays a vital role for the understanding of hydrological, meteorological, and climatological land surface processes. To meet the need of real time global soil moisture datasets, a Soil Moisture Operational Product System (SMOPS) has been developed at National Oceanic and Atmospheric Administration to produce a one-stop shop for soil moisture observations from all available satellite sensors. What makes the SMOPS unique is its near real time global blended soil moisture product. Since the first version SMOPS publicly released in 2010, the SMOPS has been updated twice based on the users’ feedbacks through improving retrieval algorithms and including observations from new satellite sensors. The version 3.0 SMOPS has been operationally released since 2017. Significant differences in climatological averages lead to remarkable distinctions in data quality between the newest and the older versions of SMOPS blended soil moisture products. This study reveals that the SMOPS version 3.0 has overwhelming advantages of reduced data uncertainties and increased correlations with respect to the quality controlled in situ measurements. The new version SMOPS also presents more robust agreements with the European Space Agency’s Climate Change Initiative (ESA_CCI) soil moisture datasets. With the higher accuracy, the blended data product from the new version SMOPS is expected to benefit the hydrological, meteorological, and climatological researches, as well as numerical weather, climate, and water prediction operations.

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