  We show how geometric phases may be used to fully describe quantum systems,
with or without gravity, by providing knowledge about the geometry and topology
of its Hilbert space. We find a direct relation between geometric phases and
von Neumann algebras. In particular, we show that a vanishing geometric phase
implies the existence of a well-defined trace functional on the algebra. We
discuss how this is realised within the AdS/CFT correspondence for the eternal
black hole. On the other hand, a non-vanishing geometric phase indicates
missing information for a local observer, associated to reference frames
covering only parts of the quantum system considered. We illustrate this with
several examples, ranging from a single spin in a magnetic field to Virasoro
Berry phases and the geometric phase associated to the eternal black hole in
AdS spacetime. For the latter, a non-vanishing geometric phase is tied to the
presence of a centre in the associated von Neumann algebra.
