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A Microstates Approach to Relative Free Entropy

Dimitri Shlyakhtenko
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Abstract

We define and study a relative free entropy quantity, analogous in its properties to Voiculescu's relative free entropy Chi^*(...:B). Our definition uses matricial microstates, unlike his definition, which involves non-commutative Hilbert transform. We prove a change of variable formula and certain maximization results for our quantity. We also exhibit a connection between the free entropy of a matrix with operator entries, relative to the algebra of scalar matrices, with the free entropy of the entries of the matrix.

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Dimitri Shlyakhtenko. 1998-12-14. A Microstates Approach to Relative Free Entropy. https://arxiv.org/abs/math/9812079

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