Operator-valued distributions. I. Characterizations of freeness

By Alexandru Nica, D. Shlyakhtenko, Roland Speicher. 2001

77 citations recorded by the source corpus.

Let $M$ be a $B$-probability space. Assume that $B$ itself is a $D$-probability space; then $M$ can be viewed as $D$-probability space as well. Let $X$ be in $M$. We look at the question of relating the properties of $X$ as $B$-valued random variable to its properties as $D$-valued random variable. We characterize freeness of $X$ from $B$ with amalgamation over $D$: (a) in terms of a certain factorization condition linking the $B$-valued and $D$-valued cumulants of $X$, and (b) for $D$ finite-dimensional, in terms of linking the $B$-valued and the $D$-valued Fisher information of $X$. We give an application to random matrices. For the second characterization we derive a new operator-valued description of the conjugate variable and introduce an operator-valued version of the liberation gradient.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/ebd4e2ddc7ab4ad3c0936b49786944ca42a116b1

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In Discrete Mathematics and Combinatorics

In Statistics and Probability

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Impact of antenna correlation on the capacity of multiantenna channels (2005, background): “Roland Speicher (Queens University, Canada) for providing reference [61], which led to the idea of mean regularity.”