Microstates free entropy and cost of equivalence relations

Abstract

We define an analog of Voiculescu's free entropy for n-tuples of unitaries (u_{1},...,u_{n}) in a tracial von Neumann algebra M, normalizing a unital diffuse abelian subalgebra B in M. Using this quantity, we define the free dimension \delta_{0}(u_{1},..,u_{n}\btw B). This number depends on (u_{1},... ,u_{n}) only up ``orbit equivalence'' over B. In particular, if R is an measirable equivalence relation on [0,1] generated by n automorphisms \alpha_{1},...,\alpha_{n}, let u_{1},..., u_{n} be the unitaries implementing \alpha_{1},..,\alpha_{n} in the Feldman-Moore crossed product algebra M=W^{*}([0,1],R), and let B be the canonical copy of L^\infty functions on [0,1] inside M. In this way, we obtain an invariant \delta (R)=\delta _{0}(u_{1},...,u_{n},\btw B) of the equivalence relation (R). If R is treeable, \delta (R) coincides with the cost C(R) of R in the sense of Gaboriau. For a general equivalence relation R posessing a finite graphing, \delta(R)\leq C(R). Using the notion of free dimension, we define an dynamical entropy invariant for an automorphism of a measurable equivalence relation (or more generally of an r-discrete measure groupoid), and give examples.