Schwarz inequalities and the decomposition of positive maps on C*-algebras

1983 ◽  
Vol 94 (2) ◽  
pp. 291-296 ◽  
Author(s):  
A. Guyan Robertson
Keyword(s):  
Partial Ordering ◽  
Considerable Progress ◽  
Structure Theory ◽  
Complete Positivity ◽  
Completely Positive ◽  
Positive Order ◽  
C Algebras ◽  
Positive Maps ◽  
Linear Maps ◽  
Algebraic Formula

In recent years there has been considerable progress in the study of certain linear maps of C*-algebras which preserve the natural partial ordering. The most tractable such maps, the completely positive ones, have proved to be of great importance in the structure theory of C*-algebras(4). However general positive (order-preserving) linear maps are (at present) very intractable. For example, there is no algebraic formula which enables one to construct a general positive map, even on the algebra of 3 3 complex matrices. It is therefore of interest to study conditions stronger than positivity, but weaker than complete positivity.

Positive Linear Maps on C*-Algebras

1972 ◽  
Vol 24 (3) ◽  
pp. 520-529 ◽  
Author(s):  
Man-Duen Choi
Keyword(s):  
Vector Spaces ◽  
Complex Numbers ◽  
Complex Matrices ◽  
Completely Positive ◽  
C Algebras ◽  
Linear Maps ◽  
Positive Linear Maps

The objective of this paper is to give some concrete distinctions between positive linear maps and completely positive linear maps on C*-algebras of operators.Herein, C*-algebras possess an identity and are written in German type . Capital letters A, B, C stand for operators, script letters for vector spaces, small letters x, y, z for vectors. Capital Greek letters Φ, Ψ stand for linear maps on C*-algebras, small Greek letters α, β, γ for complex numbers.We denote by the collection of all n × n complex matrices. () = ⊗ is the C*-algebra of n × n matrices over .

MODULES OVER MONOTONE COMPLETE C*-ALGEBRAS

1992 ◽  
Vol 03 (02) ◽  
pp. 185-204 ◽  
Author(s):  
MASAMICHI HAMANA
Keyword(s):  
Completely Positive Maps ◽  
Completely Positive ◽  
C Algebra ◽  
C Algebras ◽  
Positive Maps

The main result asserts that given two monotone complete C*-algebras A and B, B is faithfully represented as a monotone closed C*-subalgebra of the monotone complete C*-algebra End A(X) consisting of all bounded module endomorphisms of some self-dual Hilbert A-module X if and only if there are sufficiently many normal completely positive maps of B into A. The key to the proof is the fact that each pre-Hilbert A-module can be completed uniquely to a self-dual Hilbert A-module.

scholarly journals BURES DISTANCE FOR COMPLETELY POSITIVE MAPS

2013 ◽  
Vol 16 (04) ◽  
pp. 1350031 ◽  
Author(s):  
B. V. RAJARAMA BHAT ◽  
K. SUMESH
Keyword(s):  
Von Neumann Algebra ◽  
Rigidity Theorem ◽  
Completely Positive Maps ◽  
Completely Positive ◽  
Von Neumann ◽  
C Algebras ◽  
Positive Maps ◽  
Normal States ◽  
Bures Distance

Bures had defined a metric on the set of normal states on a von Neumann algebra using GNS representations of states. This notion has been extended to completely positive maps between C*-algebras by Kretschmann, Schlingemann and Werner. We present a Hilbert C*-module version of this theory. We show that we do get a metric when the completely positive maps under consideration map to a von Neumann algebra. Further, we include several examples and counter examples. We also prove a rigidity theorem, showing that representation modules of completely positive maps which are close to the identity map contain a copy of the original algebra.

Universal $C^∗$-algebras with the local lifting property

2021 ◽  
Vol 127 (2) ◽  
pp. 361-381
Author(s):  
Kristin E. Courtney
Keyword(s):  
Completely Positive Maps ◽  
Completely Positive ◽  
C Algebras ◽  
Positive Maps

The Local Lifting Property (LLP) is a localized version of projectivity for completely positive maps between $\mathrm{C}^*$-algebras. Outside of the nuclear case, very few $\mathrm{C}^*$-algebras are known to have the LLP\@. In this article, we show that the LLP holds for the algebraic contraction $\mathrm{C}^*$-algebras introduced by Hadwin and further studied by Loring and Shulman. We also show that the universal Pythagorean $\mathrm{C}^*$-algebras introduced by Brothier and Jones have the Lifting Property.

scholarly journals Extremal problems for completely positive maps

1994 ◽  
Vol 17 (3) ◽  
pp. 607-608
Author(s):  
Mingze Yang
Keyword(s):  
Extremal Problems ◽  
Completely Positive Maps ◽  
Completely Positive ◽  
Positive Maps ◽  
Linear Maps ◽  
Positive Linear Maps ◽  
Convex Subsets

In this note, we study the faces of some convex subsets ofCPc(A,B(ℋ))(the continuous completely positive linear maps from pro-C*-algebraAtoB(ℋ)).

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