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

Loading next page...
Springer International Publishing
Copyright © 2015 by Springer Basel
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
Publisher site
See Article on Publisher Site


You’re reading a free preview. Subscribe to read the entire article.

DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Unlimited reading

Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere.

Stay up to date

Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates.

Organize your research

It’s easy to organize your research with our built-in tools.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

Monthly Plan

  • Read unlimited articles
  • Personalized recommendations
  • No expiration
  • Print 20 pages per month
  • 20% off on PDF purchases
  • Organize your research
  • Get updates on your journals and topic searches


Start Free Trial

14-day Free Trial

Best Deal — 39% off

Annual Plan

  • All the features of the Professional Plan, but for 39% off!
  • Billed annually
  • No expiration
  • For the normal price of 10 articles elsewhere, you get one full year of unlimited access to articles.



billed annually
Start Free Trial

14-day Free Trial