  Building on a recent proposal for a quantum reduction to spherical symmetry
from full loop quantum gravity, we investigate the relation between a
quantisation of spherically symmetric general relativity and a reduction at the
quantum level. To this end, we generalise the previously proposed quantum
reduction by dropping the gauge fixing condition on the radial diffeomorphisms,
thus allowing to make direct contact between previous work on reduced
quantisation. A dictionary between spherically symmetric variables and
observables with respect to the reduction constraints in the full theory is
discussed, as well as an embedding of reduced quantum states to a sub sector of
the quantum symmetry reduced full theory states. On this full theory sub
sector, the quantum algebra of the mentioned observables is computed and shown
to qualitatively reproduce the quantum algebra of the reduced variables in the
large quantum number limit for a specific choice of regularisation.
Insufficiencies in recovering the reduced algebra quantitatively from the full
theory are attributed to the oversimplified full theory quantum states we use.

General Relativity and Quantum Cosmology