scholarly journals Isometries between matrix algebras

2004 ◽  
Vol 77 (1) ◽  
pp. 1-16 ◽  
Author(s):  
Wai-Shun Cheung ◽  
Chi-Kwong Li ◽  
Yiu-Tung Poon
Keyword(s):  
Numerical Range ◽  
Complex Matrices ◽  
Linear Map ◽  
Matrix Algebras ◽  
Linear Maps ◽  
Positive Linear Maps ◽  
Linear Isometries

AbstractAs an attempt to understand linear isometries between C*-algebras without the surjectivity assumption, we study linear isometries between matrix algebras. Denote by Mm the algebra of m × m complex matrices. If k ≥ n and φ: Mn → Mk has the form X ↦ U[X ⊕ f(X)] V or X ↦ U[X1 ⊕ f(X)]V for some unitary U, V ∈ Mk and contractive linear map f: Mn → Mk, then ║φ(X)║ = ║X║ for all X ∈ Mn. We prove that the converse is true if k ≤ 2n - 1, and the converse may fail if k ≥ 2n. Related results and questions involving positive linear maps and the numerical range are discussed.

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.

scholarly journals A Simple Proof of the Jamiołkowski Criterion for Complete Positivity of Linear Maps

2005 ◽  
Vol 12 (01) ◽  
pp. 55-64 ◽  
Author(s):  
D. Salgado ◽  
J. L. Sánchez-Gómez ◽  
M. Ferrero
Keyword(s):  
Simple Proof ◽  
Direct Proof ◽  
Complete Positivity ◽  
Systematic Method ◽  
Completely Positive ◽  
Linear Map ◽  
Matrix Algebras ◽  
Linear Maps ◽  
Kraus Decomposition

We give a simple direct proof of the Jamiołkowski criterion to check whether a linear map between matrix algebras is completely positive or not. This proof is more accessible for physicists than other ones found in the literature and provides a systematic method to give any set of Kraus matrices of the Kraus decomposition.

scholarly journals Some matrix power and Karcher means inequalities involving positive linear maps

Filomat ◽  
2018 ◽  
Vol 32 (7) ◽  
pp. 2625-2634
Author(s):  
Monire Hajmohamadi ◽  
Rahmatollah Lashkaripour ◽  
Mojtaba Bakherad
Keyword(s):  
Weight Vector ◽  
Positive Definite ◽  
Matrix Inequalities ◽  
Positive Definite Matrices ◽  
Linear Map ◽  
Linear Maps ◽  
Karcher Mean ◽  
The Matrix ◽  
Positive Linear Maps ◽  
Matrix Power

In this paper, we generalize some matrix inequalities involving the matrix power means and Karcher mean of positive definite matrices. Among other inequalities, it is shown that if A = (A1,...,An) is an n-tuple of positive definite matrices such that 0 < m ? Ai ? M (i = 1,...,n) for some scalars m < M and ? = (w1,...,wn) is a weight vector with wi ? 0 and ?n,i=1 wi=1, then ?p (?n,i=1 wiAi)? ?p?p(Pt(?,A)) and ?p (?n,i=1 wiAi) ? ?p?p(?(?,A)), where p > 0,? = max {(M+m)2/4Mm,(M+m)2/42p Mm}, ? is a positive unital linear map and t ? [-1,1]\{0}.

scholarly journals Inequalities for sector matrices and positive linear maps

2019 ◽  
Vol 35 ◽  
pp. 418-423 ◽  
Author(s):  
Fuping Tan ◽  
Huimin Che
Keyword(s):  
Geometric Mean ◽  
Positive Definite ◽  
Norm Inequalities ◽  
Linear Map ◽  
Linear Maps ◽  
Positive Linear Map ◽  
Positive Linear Maps

Ando proved that if A, B are positive definite, then for any positive linear map Φ, it holds Φ(A#λB) ≤ Φ(A)#λΦ(B), where A#λB, 0 ≤ λ ≤ 1, means the weighted geometric mean of A, B. Using the recently defined geometric mean for accretive matrices, Ando’s result is extended to sector matrices. Some norm inequalities are considered as well.    

Positive linear maps of matrix algebras

2012 ◽  
Vol 98 ◽  
pp. 293-302
Author(s):  
W. A. Majewski
Keyword(s):  
Matrix Algebras ◽  
Linear Maps ◽  
Positive Linear Maps

Duality for positive linear maps in matrix algebras

2000 ◽  
Vol 86 (1) ◽  
pp. 130 ◽  
Author(s):  
Myoung-Hoe Eom ◽  
Seung-Hyeok Kye
Keyword(s):  
Matrix Algebras ◽  
Linear Maps ◽  
Positive Linear Maps
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