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Abstract for MSRI Preprint 1999-030

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

Dimitri Shlyakhtenko

Given a family of isometries v1,,vn in a tracial von Neumann algebra M, a unital subalgebra BM and a completely-positive map η:BB we define the free Fisher information F(v1,,vn:B,η) of v1,,vn relative to B and η. Using this notion, we define the free dimension δ(v1,,vnB) of v1,,vn 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 BL(X) be the canonical diffuse subalgebra. If v1,,vn,M are partial isometries arising from a treeing of this equivalence relation, then limnδ(v1,,vn,B) is equal to the cost of the equivalence relation in the sense of Gaboriau.