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Extremals and exposed faces of the cone of positive maps. (English) Zbl 1076.15021
Let $\text{PSD}_{n}$ denote the cone of $n\times n$ positive semidefinite complex matrices. The set\break $\pi(\text{PSD}_{m},\text{PSD}_{n})$ of all linear transformations that map $\text{PSD}_{m}$ into $\text{PSD}_{n}$ is a convex cone in the real vector space of all linear transformations from $\Bbb{C}^{m\times m}$ into $\Bbb{C}^{n\times n}$ which preserve Hermitian matrices. If $A$ is an $n\times m$ complex matrix, then mapping of types: (i) $X\mapsto AXA^{\ast}$ and (ii) $X\mapsto AX^{T}A^{\ast}$ lie in $\pi(\text{PSD}_{m},\text{PSD}_{n})$. It has been shown by {\it H. Schneider} [Numer. Math. 7, 11--17 (1965; Zbl 0158.28003)] that if $m=n$ then all invertible linear transformations in $\pi(\text{PSD}_{n},\text{PSD}_{n})$ are of one of these forms (with $A$ invertible). But it is known that, in general, there are elements of $\pi(\text{PSD}_{m},\text{PSD}_{n})$ which are not sums of mappings of the forms (i) and (ii) [see {\it M. D. Choi}, “Positive linear maps”, Proc. Symp. Pure Math. 38, 583--590 (1982; Zbl 0522.46037)]. In the present paper the authors show that all maps of type (i) or (ii) are extremal in the cone $\pi(\text{PSD}_{m},\text{PSD}_{n})$. Moreover, if $\text{rank}(A)=1$ or $m$, then these extremals are exposed.

15B48Positive matrices and their generalizations; cones of matrices
15A04Linear transformations, semilinear transformations (linear algebra)
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