On a tensor-analogue of the Schur product

• Published in: Positivity : an international journal devoted to the theory and applications of positivity in analysis Vol. 20; no. 3; pp. 621 - 624
• Main Authors: Sumesh, K., Sunder, V. S.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-09-2016; Springer Nature B.V
• Subjects: Algebra; Calculus of Variations and Optimal Control; Optimization; Econometrics; Fourier Analysis; Linear algebra; Mathematics; Mathematics and Statistics; Operator Theory; Potential Theory; Studies; algebras; Choi’s theorem; Completely positive maps; 47.030; 46.025
• ISSN: 1385-1292; 1572-9281
• DOI: 10.1007/s11117-015-0377-x

We consider the
tensorial Schur product
R
∘
⊗
S
=
[
r
i
j
⊗
s
i
j
]
for
R
∈
M
n
(
A
)
,
S
∈
M
n
(
B
)
,
with
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
ϕ
:
M
n
→
M
d
is completely positive if and only if
[
ϕ
(
E
i
j
)
]
∈
M
n
(
M
d
)
+
, where of course
{
E
i
j
:
1
≤
i
,
j
≤
n
}
denotes the usual system of matrix units in
M
n
(
:
=
M
n
(
C
)
)
. We also discuss some other corollaries of the main result.