  Maximum principles and uniform anti-maximum principles are a ubiquitous topic
in PDE theory that is closely tied to the Krein--Rutman theorem and kernel
estimates for resolvents.
  We take up a classical idea of Tak\'a\v{c} - to prove (anti-)maximum
principles in an abstract operator theoretic framework - and combine it with
recent ideas from the theory of eventually positive operator semigroups. This
enables us to derive necessary and sufficient conditions for (anti-)maximum
principles in a very general setting. Consequently, we are able to either prove
or disprove (anti-)maximum principles for a large variety of concrete
differential operators. As a bonus, for several operators that are already
known to satisfy or to not satisfy anti-maximum principles, our theory gives a
very clear and concise explanation of this behaviour.
