FREE FISHER INFORMATION WITH RESPECT TO A COMPLETELY POSITIVE MAP AND COST OF EQUIVALENCE

RELATIONS.

DIMITRI SHLYAKHTENKO

Abstract. Given a family of isometries v_{1}, …, v_{n} in a tracial von Neumann algebra M, a unital subalgebra B ⊂ M and a completely-positive map η : B → B we define the free Fisher information F^{∗}(v_{1}, …, v_{n} : B, η) of v_{1}, …, v_{n} relative to B and η. Using this notion, we define the free dimension δ^{∗}(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 ≅ L^{∞}(X) be the canonical diffuse subalgebra. If v_{1}, …, v_{n}, ⋯ ∈ M are partial isometries arising from a treeing of this equivalence relation, then lim_{n} δ^{∗}(v_{1}, …, v_{n}, … B) is equal to the cost of the equivalence relation in the sense of Gaboriau.