We prove that if a unital Banach lattice algebra has sufficiently many one-dimensional elements and if its unit element has sufficiently many components then its positive elements have spectral properties analogous to those of positive operators on Banach lattices. In particular, if a positive element is irreducible (in the sense that (1—e)xe> 0 for all componentseof 1 satisfying 0 ≠e≠ 1) and compact, its spectral radius is positive and its spectrum shows cyclic behaviour.