Free Fisher information with respect to a completely positive map and cost of equivalence relations

Dimitri Shlyakhtenko

Given a family of isometries $v_1,\dots,v_n$ in a tracial von Neumann algebra $M$, a unital subalgebra $B\subset M$ and a completely-positive map $\eta:B\to B$ we define the free Fisher information $F^*(v_1,\dots,v_n:B,\eta)$ of $v_1,\dots,v_n$ relative to $B$ and $\eta$. Using this notion, we define the free dimension $\delta^*(v_1,\dots,v_n\shortvdots B)$ of $v_1,\dots,v_n$ 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 $B\cong L^\infty (X)$ be the canonical diffuse subalgebra. If $v_1,\dots,v_n,\dots\in M$ are partial isometries arising from a treeing of this equivalence relation, then $\lim_{n}\delta^*(v_1,\dots,v_n,\dots\shortvdots B)$ is equal to the cost of the equivalence relation in the sense of Gaboriau.