  We study the limits of inductive sequences (A_i,\phi_i) where each A_i is a
direct sum of full matrix algebras over compact metric spaces and each partial
map of \phi_i is diagonal. We give a new characterisation of simplicity for
such algebras, and apply it to prove that the said algebras have stable rank
one whenever they are simple and unital. Significantly, our results do not
require any dimension growth assumption.
