Microstates free entropy and cost of equivalence relations

We define an analog of Voiculescuʹs free entropy for n-tuples of unitaries u1, ... , un in a tracial von Neumann algebra M which normalize a unital subalgebra L^(infinity)[0,1]= B(subset)M. Using this quantity, we define the free dimension (delta)(u1, ... , un)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 (alpha)1, ... , (alpha)n, let u1, ... , un be the unitaries implementing alpha)1, ... , (alpha)n in the Feldman-Moore crossed product algebra M= W^* ([0,1], R) (subset) B= L^(infinity) [0,1]. Then the number (delta)(R)=(delta)0(u1, ... , un) B is an invariant of the equivalence relation R. If R is treeable, (delta)(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, (delta)(R) (less than) 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.