Linear maps between C*-algebras whose adjoints preserve extreme points of the dual ball

We give a structural characterisation of linear operators from one $C^\ast$% -algebra into another whose adjoints map extreme points of the dual ball onto extreme points. We show that up to a $\ast$-isomorphism, such a map admits of a decomposition into a degenerate and a non-degenerate part, the non-degenerate part of which appears as a Jordan $\ast$-morphism followed by a ``rotation'' and then a reduction. In the case of maps whose adjoints preserve pure states, the degenerate part does not appear, and the ``rotation'' is but the identity. In this context the results concerning such pure state preserving maps depend on and complof St\o rmer [St\o 2; 5.6 \& 5.7]. In conclusion we consider the action of maps with ``extreme point preserving'' adjoints on some specific $C^\ast$-algebras.