# On a tensor-analogue of the Schur product

On a tensor-analogue of the Schur product We consider the tensorial Schur product $$R \circ ^\otimes S = [r_{ij} \otimes s_{ij}]$$ R ∘ ⊗ S = [ r i j ⊗ s i j ] for $$R \in M_n(\mathcal {A}), S\in M_n(\mathcal {B}),$$ R ∈ M n ( A ) , S ∈ M n ( B ) , with $$\mathcal {A}, \mathcal {B}~\text{ unital }~ C^*$$ A , B unital C ∗ -algebras, verify that such a ‘tensorial Schur product’ of positive operators is again positive, and then use this fact to prove (an apparently marginally more general version of) the classical result of Choi that a linear map $$\phi :M_n \rightarrow M_d$$ ϕ : M n → M d is completely positive if and only if $$[\phi (E_{ij})] \in M_n(M_d)^+$$ [ ϕ ( E i j ) ] ∈ M n ( M d ) + , where of course $$\{E_{ij}:1 \le i,j \le n\}$$ { E i j : 1 ≤ i , j ≤ n } denotes the usual system of matrix units in $$M_n (:= M_n(\mathbb C))$$ M n ( : = M n ( C ) ) . We also discuss some other corollaries of the main result. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# On a tensor-analogue of the Schur product

, Volume 20 (3) – Oct 14, 2015
4 pages
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Publisher
Springer International Publishing
Copyright
Copyright © 2015 by Springer Basel
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-015-0377-x
Publisher site
See Article on Publisher Site

### Abstract

We consider the tensorial Schur product $$R \circ ^\otimes S = [r_{ij} \otimes s_{ij}]$$ R ∘ ⊗ S = [ r i j ⊗ s i j ] for $$R \in M_n(\mathcal {A}), S\in M_n(\mathcal {B}),$$ R ∈ M n ( A ) , S ∈ M n ( B ) , with $$\mathcal {A}, \mathcal {B}~\text{ unital }~ C^*$$ A , B unital C ∗ -algebras, verify that such a ‘tensorial Schur product’ of positive operators is again positive, and then use this fact to prove (an apparently marginally more general version of) the classical result of Choi that a linear map $$\phi :M_n \rightarrow M_d$$ ϕ : M n → M d is completely positive if and only if $$[\phi (E_{ij})] \in M_n(M_d)^+$$ [ ϕ ( E i j ) ] ∈ M n ( M d ) + , where of course $$\{E_{ij}:1 \le i,j \le n\}$$ { E i j : 1 ≤ i , j ≤ n } denotes the usual system of matrix units in $$M_n (:= M_n(\mathbb C))$$ M n ( : = M n ( C ) ) . We also discuss some other corollaries of the main result.

### Journal

PositivitySpringer Journals

Published: Oct 14, 2015

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