Unitarity restoration in the presence of closed timelike curves

A proposal is made for a mathematically unambiguous treatment of evolution in the presence of closed timelike curves. In contrast to other proposals for handling the naively nonunitary evolution that is often present in such situations, this proposal is causal, linear in the initial density matrix, and preserves probability. It provides a physically reasonable interpretation of invertible nonunitary evolution by redefining the final Hilbert space so that the evolution is unitary or equivalently by removing the nonunitary part of the evolution operator using a polar decomposition.