Let denote the cone of positive semidefinite complex matrices. The set of all linear transformations that map into is a convex cone in the real vector space of all linear transformations from into which preserve Hermitian matrices. If is an complex matrix, then mapping of types: (i) and (ii) lie in .
It has been shown by H. Schneider [Numer. Math. 7, 11–17 (1965; Zbl 0158.28003)] that if then all invertible linear transformations in are of one of these forms (with invertible). But it is known that, in general, there are elements of 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 . Moreover, if or , then these extremals are exposed.