Free Fisher Information¶with Respect to a Completely Positive Map¶and Cost of Equivalence Relations

Abstract:

Given a family of isometries in a tracial von Neumann algebra M, a unital subalgebra B⊂M and a completely-positive map we define the free Fisher information of relative to B and η. Using this notion, we define the free dimension of 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 be the canonical diffuse subalgebra. If are partial isometries arising from a treeing of this equivalence relation, then is equal to the cost of the equivalence relation in the sense of Gaboriau and Levitt.