Explore Ogata's construction of invariants for Symmetry Protected Topological (SPT) phases in quantum spin chains through this comprehensive lecture. Delve into the simplest situation of spin chains and discover the connection with systems featuring Matrix Product State (MPS) ground states. Learn about the index previously established by other authors for these special cases. Gain insights from Bruno Nachtergaele's explanation of this complex topic, presented as part of the Thematic Programme on "Tensor Networks: Mathematical Structures and Novel Algorithms" at the Erwin Schrödinger International Institute for Mathematics and Physics (ESI).