  There are two parts to this work:
  First, we study the error correction properties of the real-space
renormalization group (RG). The long-distance operators are the (approximately)
correctable operators encoded in the physical algebra of short-distance
operators. This is closely related to modeling the holographic map as a quantum
error correction code. As opposed to holography, the real-space RG of a
many-body quantum system does not have the complementary recovery property. We
discuss the role of large $N$ and a large gap in the spectrum of operators in
the emergence of complementary recovery.
  Second, we study the operator algebra exact quantum error correction for any
von Neumann algebra. We show that similar to the finite dimensional case, for
any error map in between von Neumann algebras the Petz dual of the error map is
a recovery map if the inclusion of the correctable subalgebra of operators has
finite index.
