Free fisher information with respect to a completely positive map and costof equivalence relations

Given a family of isometries v(1),..., v(n) in a tracial von Neumann algebr a M, a unital subalgebra B subset of M and a completely-positive map eta : B --> B we define the free Fisher information F*(v(1),..., v(n) : B, eta) o f v(1),..., v(n) relative to B and eta. Using this notion, we define the fr ee dimension delta*(v(1),..., v(n))( B) of v(1),..., v(n) relative to B. id . Let R be a measurable equivalence relation on a finite measure space X. Let M be the von Neumann algebra associated to R, and let B congruent to L-inf inity(X) be the canonical diffuse subalgebra. If v(1),..., v(n), ... is an element of M are partial isometries arising from a treeing of this equivale nce relation, then lim(n) delta*(v(1),..., v(n),... )( B) is equal to the c ost of the equivalence relation in the sense of Gaboriau and Levitt.