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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 $n×n$ positive semidefinite complex matrices. The set$\pi \left({\text{PSD}}_{m},{\text{PSD}}_{n}\right)$ 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 ${ℂ}^{m×m}$ into ${ℂ}^{n×n}$ which preserve Hermitian matrices. If $A$ is an $n×m$ complex matrix, then mapping of types: (i) $X↦AX{A}^{*}$ and (ii) $X↦A{X}^{T}{A}^{*}$ lie in $\pi \left({\text{PSD}}_{m},{\text{PSD}}_{n}\right)$.

It has been shown by H. Schneider [Numer. Math. 7, 11–17 (1965; Zbl 0158.28003)] that if $m=n$ then all invertible linear transformations in $\pi \left({\text{PSD}}_{n},{\text{PSD}}_{n}\right)$ are of one of these forms (with $A$ invertible). But it is known that, in general, there are elements of $\pi \left({\text{PSD}}_{m},{\text{PSD}}_{n}\right)$ which are not sums of mappings of the forms (i) and (ii) [see 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 $\pi \left({\text{PSD}}_{m},{\text{PSD}}_{n}\right)$. Moreover, if $\text{rank}\left(A\right)=1$ or $m$, then these extremals are exposed.

MSC:
 15A48 Positive matrices and their generalizations (MSC2000) 15A04 Linear transformations, semilinear transformations (linear algebra)