Microstates Free Entropy and Cost of Equivalence Relations

We define an analog of Voiculescu’s free entropy for n-tuples of unitaries u1, . . . , un in a tracial von Neumann algebra M which normalize a unital subalgebra L[0, 1] = B ⊂ M. Using this quantity, we define the free dimension δ0(u1, . . . , un G B). This number depends on u1, . . . , un only up to orbit equivalence over B. In particular, if R is a measurable equivalence relation on [0, 1] generated by n automorphisms α1, . . . , αn , let u1, . . . , un be the unitaries implementing α1, . . . , αn in the FeldmanMoore crossed product algebra M = W ([0, 1], R) ⊃ B = L[0, 1]. Then the number δ(R) = δ0(u1, . . . , un G B) is an invariant of the equivalence relation R. If R is treeable, δ(R) coincides with the cost C(R) of R in the sense of D. Gaboriau. In particular, it is n for an equivalence relation induced by a free action of the free group Fn . For a general equivalence relation R possessing a finite graphing of finite cost, δ(R) ≤ C(R). Using the notion of free dimension, we define a dynamical entropy invariant for an automorphism of a measurable equivalence relation (or, more generally, of an r-discrete measure groupoid) and give examples.