  We show that the standard representation of homogeneous isotropic loop
quantum cosmology (LQC) is the GNS-representation that corresponds to the
unique state on the reduced quantum holonomy-flux $^*$-algebra that is
invariant under residual diffeomorphisms $-$ both when the standard algebra is
used as well as when one uses the extended algebra proposed by Fleischhack.
More precisely, we find that in both situations the GNS-Hilbert spaces
coincide, and that in the Fleischhack case the additional algebra elements are
just mapped to zero operators. In order for the residual diffeomorphisms to
have a well-defined action on the quantum algebra, we have let them act on the
fiducial cell as well as on the dynamical variables, thereby recovering
covariance. Consistency with Ashtekar and Campiglia in the Bianchi I case is
also shown.
