  We introduce positive cones on algebras with involution. These allow us to
prove analogues of Artin's solution to Hilbert's 17th problem, the
Artin-Schreier theorem characterizing formally real fields, and to define
signatures with respect to positive cones. We consider the space of positive
cones of an algebra with involution and investigate its topological properties,
showing in particular that it is a spectral space. As an application we solve
the problem of the existence of positive involutions.
