Free entropy with respect to a completely positive map

Given an element X in a tracial von Neumann algebra ( M , τ), a unital subalgebra B ⊂ M , and a completely positive map η: B → B , we define the free Fisher information Φ*( X : B , η) of X relative to B with respect to η. The definition is a generalization of Voiculescu's definition of Φ*( X : B ), which corresponds to η = τ. We show that many facts about Φ*( X : B ) generalize to Φ*( X : B , η). As an application, we show that if B is commutative and X is singular with respect to B , then Φ*( X : B ) is infinite. Φ*( X : B , η) is minimized when X is a B -valued semicircular variable with covariance η. Therefore, Φ*(· : B , η) is an appropriate quantity to study B -valued semicircular systems with arbitrary covariance. Associating to a pair of completely positive maps μ, η: B → B the quantity Φ*( X μ : B , η), where X μ is a B -valued semicircular variable with covariance μ, allows us to measure "absolute continuity" of μ and η. To a completely positive map μ we can naturally associate the number Φ*( X μ : B , id). If μ is the conditional expectation onto a subfactor A ⊂ B of a II 1 factor B , we show that Φ*( X E : B , id) is equal to the Jones' index B ⊂ A .