Sofic Approximations and the Classification of Amenable Actions on II1 Factors via Non-Commutative Entropy

This monograph establishes a rigorous framework for the classification of amenable actions of countable sofic groups on type II1 von Neumann factors, utilizing a non-commutative extension of sofic entropy. We construct a microstates-based entropy invariant for automorphisms of II1 factors associated with sofic approximations, generalizing the classical Connes-Størmer entropy and the topological sofic entropy of Kerr and Li. Our main result demonstrates that for amenable groups, this non-commutative sofic entropy coincides with the classical dynamical entropy, thereby providing a new operator-algebraic proof of the uniqueness of the hyperfinite II1 factor's amenable actions up to cocycle conjugacy. Furthermore, we investigate the rigidity of Bernoulli actions of non-amenable sofic groups, proving that the sofic entropy is an invariant of cocycle conjugacy for a broad class of deformation-rigid factors. These results bridge the gap between Popa's deformation/rigidity theory and the measurable entropy theory of Bowen, offering a unified perspective on the isomorphism problem for group measure space factors.