arXiv:1106.1247

Exposed faces for decomposable positive linear maps arising from completely positive maps

Hyun-Suk Choi, Seung-Hyeok Kye

Published Jun 7, 2011• 0 citations

Abstract

Let $D$ be a space of $2\times n$ matrices. Then the face of the cone of all completely positive maps from $M_2$ into $M_n$ given by $D$ is an exposed face of the bigger cone of all decomposable positive linear maps if and only if the set of all rank one matrices in $D$ forms a subspace of $D$ together with zero and $D^\perp$ is spanned by rank one matrices.

DOI: 10.48550/arxiv.1106.1247

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arXiv:1106.1247

Exposed faces for decomposable positive linear maps arising from completely positive maps

Hyun-Suk Choi, Seung-Hyeok Kye

Published Jun 7, 2011• 0 citations

arXiv HTML PDF

Abstract

Let

D

be a space of

2\times n

matrices. Then the face of the cone of all completely positive maps from

M_2

into

M_n

given by

D

is an exposed face of the bigger cone of all decomposable positive linear maps if and only if the set of all rank one matrices in

D

forms a subspace of

D

together with zero and

D^\perp

is spanned by rank one matrices.

DOI: 10.48550/arxiv.1106.1247

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arXiv:2301.05788Linear and Multilinear AlgebraJan 18, 20242 citations

Construction of exposed indecomposable positive linear maps between matrix algebras

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Let

D

be a space of

2\times n

matrices. Then the face of the cone of all completely positive maps from

M_2

into

M_n

given by

D

is an exposed face of the bigger cone of all decomposable positive linear maps if and only if the set of all rank one matrices in

D

forms a subspace of

D

together with zero and

D^\perp

is spanned by rank one matrices.

A factorization property of positive maps on

C^*

-algebras

A_i

C_i

be unital C*-algebras and let

α_i

be positive linear maps from

A_i

C_i,

i=1,2

. We obtain conditions under which any positive map

β

from the minimal C*-tensor product

A_1 \otimes_{min} A_2

C_1 \otimes_{min} C_2

, such that $ α_1 \otimes α_2 \geq β$, factorizes as

β= γ\otimes α_2

for some positive map

γ

. In particular we show that when

α_i \colon A_i \rightarrow B(\mathcal H_i)

are completely positive (CP) maps for some Hilbert spaces

\mathcal H_i

(i=1,2)

, and

α_2

is a pure CP map and

β

is a CP map so that

α_1 \otimes α_2 - β

is also CP, then

β= γ\otimes α_2

for some CP map

γ

. We show that a similar result holds in the context of positive linear maps when

A_2 = C_2 = B(\mathcal H)

and

α_2 = id

. As an application we extend \cite[IX Theorem]{PM}( revisited recently by Huber et al in \cite{HLLM}) to show that for any linear map

τ

from a unital C*-algebra

A

to a C*-algebra

C

, if

τ\otimes id_k

is decomposable for some

k \geq 2

, where

id_k

is the identity map on the algebra

M_k(\mathbb {C} )

k\times k

matrices, then

τ

is completely positive.