  M. Krein proved in 1948 that if T is a continuous operator on a normed space
leaving invariant an open cone, then its adjoint T* has an eigenvector. We
present generalizations of this result as well as some applications to
C*-algebras, operators on l_1, operators with invariant sets, contractions on
Banach lattices, the Invariant Subspace Problem, and von Neumann algebras.
