  Motivated by our previous results, we investigate structural properties of
probability measure-preserving actions of sofic groups relative to their
Pinsker factor. We also consider the same properties relative to the Outer
Pinsker factor, which is another generalization of the Pinsker factor in the
nonamenable case. The Outer Pinsker factor is motivated by extension entropy
for extensions, which fixes some of the "pathological" behavior of sofic
entropy: namely increase of entropy under factor maps. We show that an
arbitrary probability measure-preserving action of a sofic group is mixing
relative to its Pinsker and Outer Pinsker factors and, if the group is
nonamenable, it has spectral gap relative to its Pinsker and Outer Pinsker
factors. Our methods are similar to those we developed in "Polish models and
sofic entropy" and based on representation-theoretic techniques. One crucial
difference is that instead of considering unitary representations of a group we
must consider *-representations of algebraic crossed products.
