# Mathematics Colloquia and Seminars

The “split property” is an interpretation of statistical independence of states with respect to the left and right infinite subsystems of the spin chain quasi-local algebra. States with the split property are important in the study of symmetry protected topological (SPT) phases, cf. recent work by Y. Ogata, and by F. Pollmann, E. Berg, A.M. Turner and M. Oshikawa. In this talk, 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 Z2-index for gapped ground states of SPT phases to show that the Z2-index is an invariant of gapped classification of phases containing fast-decaying interactions.