  Based on a recent purely geometric construction of observables for the
spatial diffeomorphism constraint, we propose two distinct quantum reductions
to spherical symmetry within full 3+1-dimensional loop quantum gravity. The
construction of observables corresponds to using the radial gauge for the
spatial metric and allows to identify rotations around a central observer as
unitary transformations in the quantum theory. Group averaging over these
rotations yields our first proposal for spherical symmetry. Hamiltonians of the
full theory with angle-independent lapse preserve this spherically symmetric
subsector of the full Hilbert space. A second proposal consists in implementing
the vanishing of a certain vector field in spherical symmetry as a constraint
on the full Hilbert space, leading to a close analogue of diffeomorphisms
invariant states. While this second set of spherically symmetric states does
not allow for using the full Hamiltonian, it is naturally suited to implement
the spherically symmetric midisuperspace Hamiltonian, as an operator in the
full theory, on it. Due to the canonical structure of the reduced variables,
the holonomy-flux algebra behaves effectively as a one parameter family of
2+1-dimensional algebras along the radial coordinate, leading to a diagonal
non-vanishing volume operator on 3-valent vertices. The quantum dynamics thus
becomes tractable, including scenarios like spherically symmetric dust
collapse.
