Positive semigroups are infinite dimensional generalizations of matrix exponents exp(tA) with positive entries. They describe solutions to evolution equations that preserve positivity of initial conditions, like vector flows and the heat equation. Since Perron and Frobenius it is known that matrix exponents are positive if and only if their generators A have non-negative entries off of the main diagonal. In 1980's Arendt, Chernoff and Kato combined convex analysis and spectral theory to show that this criterion generalizes to continuous positive semigroups on some Banach spaces, so called order unit spaces. Self-adjoint parts of C* and von Neumann algebras with unity are examples of such spaces. We will present an alternative approach to their criterion based on positivity and order arguments, particularly on a non-spectral condition for the generator to have a bounded negative inverse.