  We present a generalization of quantum error correction to
infinite-dimensional Hilbert spaces. The generalization yields new classes of
quantum error correcting codes that have no finite-dimensional counterparts.
The error correction theory we develop begins with a shift of focus from states
to algebras of observables. Standard subspace codes and subsystem codes are
seen as the special case of algebras of observables given by finite-dimensional
von Neumann factors of type I. Our generalization allows for the correction of
codes characterized by any von Neumann algebra and we give examples, in
particular, of codes defined by infinite-dimensional algebras.
