MF actions and K-theoretic dynamics

We study the interplay of C⁎C⁎-dynamics and K  -theory. Notions of chain recurrence for transformations groups (X,Γ)(X,Γ) and MF actions for non-commutative C⁎C⁎-dynamical systems (A,Γ,α)(A,Γ,α) are translated into K-theoretical language, where purely algebraic conditions are shown to be necessary and sufficient for a reduced crossed product to admit norm microstates. We are particularly interested in actions of free groups on AF algebras, in which case we prove that a K-theoretic coboundary condition determines whether or not the reduced crossed product is a matricial field (MF) algebra. One upshot is the equivalence of stable finiteness and being MF for these reduced crossed product algebras.