  We prove that, in a large class of Banach lattices, the fixed space of each
commuting family of positive linear contractions is a lattice subspace. As
consequences, new cyclicity results for the peripheral point spectra of
positive operators and semigroups are derived; we also pose an open problem
that naturally occurs in this context. Finally, a variety of counterexamples is
presented to point out some limits of our results.
