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 .

scholarly journals FACIAL STRUCTURES FOR VARIOUS NOTIONS OF POSITIVITY AND APPLICATIONS TO THE THEORY OF ENTANGLEMENT

2013 ◽  
Vol 25 (02) ◽  
pp. 1330002 ◽  
Author(s):  
SEUNG-HYEOK KYE
Keyword(s):  
Convex Cones ◽  
Edge States ◽  
Completely Positive ◽  
Matrix Algebras ◽  
Linear Maps ◽  
Entanglement Witnesses ◽  
Positive Linear Maps

In this expository note, we explain facial structures for the convex cones consisting of positive linear maps, completely positive linear maps, and decomposable positive linear maps between matrix algebras, respectively. These will be applied to study the notions of entangled edge states with positive partial transposes and optimality of entanglement witnesses.

Tracial Positive Linear Maps of C ∗ -Algebras

1983 ◽  
Vol 87 (1) ◽  
pp. 57
Author(s):  
Man-Duen Choi ◽  
Sze-Kai Tsui
Keyword(s):  
C Algebras ◽  
Linear Maps ◽  
Positive Linear Maps

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)).

scholarly journals TRACIAL EQUIVALENCE FOR $C^*$-ALGEBRAS AND ORBIT EQUIVALENCE FOR MINIMAL DYNAMICAL SYSTEMS

2005 ◽  
Vol 48 (3) ◽  
pp. 673-690
Author(s):  
Huaxin Lin
Keyword(s):  
Dynamical Systems ◽  
Crossed Products ◽  
Rank Zero ◽  
C Algebras ◽  
Orbit Equivalent ◽  
Linear Maps ◽  
Positive Linear Maps ◽  
Tracial Topological Rank ◽  
Minimal Dynamical Systems ◽  
Cantor Minimal Systems

AbstractWe introduce the notion of tracial equivalence for $C^*$-algebras. Let $A$ and $B$ be two unital separable $C^*$-algebras. If they are tracially equivalent, then there are two sequences of asymptotically multiplicative contractive completely positive linear maps $\phi_n:A\to B$ and $\psi_n:B\to A$ with a tracial condition such that $\{\phi_n\circ\psi_n\}$ and $\{\psi_n\circ\phi_n\}$ are tracially approximately inner. Let $A$ and $B$ be two unital separable simple $C^*$-algebras with tracial topological rank zero. It is proved that $A$ and $B$ are tracially equivalent if and only if $A$ and $B$ have order isomorphic ranges of tracial states. For the Cantor minimal systems $(X_1,\sigma_1)$ and $(X_2,\sigma_2)$, using a result of Giordano, Putnam and Skau, we show that two such dynamical systems are (topological) orbit equivalent if and only if the associated crossed products $C(X_1)\times_{\sigma_1}\mathbb{Z}$ and $C(X_2)\times_{\sigma_2}\mathbb{Z}$ are tracially equivalent.

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