  We develop a general operator algebraic method which focuses on projective
representations of symmetry group for proving Lieb-Schultz-Mattis type
theorems, i.e., no-go theorems that rule out the existence of a unique gapped
ground state (or, more generally, a pure split state), for quantum spin chains
with on-site symmetry. We first prove a theorem for translation invariant spin
chains that unifies and extends two theorems proved by two of the authors in
[OT1]. We then prove a Lieb-Schultz-Mattis type theorem for spin chains that
are invariant under the reflection about the origin and not necessarily
translation invariant.
