  We investigate the star-free closure, which associates to a class of
languages its closure under Boolean operations and marked concatenation. We
prove that the star-free closure of any finite class and of any class of groups
languages with decidable separation (plus mild additional properties) has
decidable separation. We actually show decidability of a stronger property,
called covering. This generalizes many results on the subject in a unified
framework. A key ingredient is that star-free closure coincides with another
closure operator where Kleene stars are also allowed in restricted contexts.
