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- W2326155077 abstract "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*(Xμ : B, η), where Xμ 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*(Xμ : B, id). If μ is the conditional expectation onto a subfactor A ⊂ B of a II1 factor B, we show that Φ*(XE : B, id) is equal to the Jones' index [B ⊂ A]." @default.
- W2326155077 created "2016-06-24" @default.
- W2326155077 creator A5085505149 @default.
- W2326155077 date "2000-01-01" @default.
- W2326155077 modified "2023-10-17" @default.
- W2326155077 title "Free entropy with respect to a completely positive map" @default.
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- W2326155077 doi "https://doi.org/10.1353/ajm.2000.0005" @default.
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