On a tensor-analogue of the Schur product
Tóm tắt
We consider the tensorial Schur product
$$R \circ ^\otimes S = [r_{ij} \otimes s_{ij}]$$
for
$$R \in M_n(\mathcal {A}), S\in M_n(\mathcal {B}),$$
with
$$\mathcal {A}, \mathcal {B}~\text{ unital }~ C^*$$
-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
$$\phi :M_n \rightarrow M_d$$
is completely positive if and only if
$$[\phi (E_{ij})] \in M_n(M_d)^+$$
, where of course
$$\{E_{ij}:1 \le i,j \le n\}$$
denotes the usual system of matrix units in
$$M_n (:= M_n(\mathbb C))$$
. We also discuss some other corollaries of the main result.
Tài liệu tham khảo
Choi, M.D.: Completely positive linear maps on complex matrices. Linear Algebra Appl. 10, 285–290 (1975)
Choi, M.D., Effros, E.G.: Injectivity and operator spaces. J. Funct. Anal. 24(2), 156–209 (1977)
Pisier, G.: Introduction to Operator Space Theory. LMS Lecture Note Series 294. Cambridge University Press, Cambridge (2003)