  We prove that the split property is a stable feature for spin chain states
which are related by composition with *-automorphisms generated by power-law
decaying interactions. We apply this to the theory of the $\mathbb{Z}_2$-index
for gapped ground states of symmetry protected topological phases to show that
the $\mathbb{Z}_2$-index is an invariant of gapped classification of phases
containing fast-decaying interactions.
