  All physical observations are made relative to a reference frame, which is a
system in its own right. If the system of interest admits a group symmetry, the
reference frame observing it must transform commensurately under the group to
ensure the covariance of the combined system. We point out that the crossed
product is a way to realize quantum reference frames from the bottom-up;
adjoining a quantum reference frame and imposing constraints generates a
crossed product algebra. We provide a top-down specification of crossed product
algebras and show that one cannot obtain inequivalent quantum reference frames
using this approach. As a remedy, we define an abstract algebra associated to
the system and symmetry group built out of relational crossed product algebras
associated with different choices of quantum reference frames. We term this
object the G-framed algebra, and show how potentially inequivalent frames are
realized within this object. We comment on this algebra's analog of the
classical Gribov problem in gauge theory, its importance in gravity where we
show that it is relevant for semiclassical de Sitter and potentially beyond the
semiclassical limit, and its utility for understanding the frame-dependence of
physical notions like observables, density states, and entropies.
