---
address: 'Department of Mathematics, UCLA, Los Angeles, CA 90095'
author:
- Dimitri Shlyakhtenko
title: A Microstates Approach to Relative Free Entropy
---

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Introduction.
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Let $ {X_{1},\dots ,X_{n}}\in (M,\tau ) $ be a family of non-commutative random variables in a tracial $ W^{*} $-probability space, and let $ B\subset M $ be a unital subalgebra. Voiculescu has introduced in [@dvv:entropy5] a free entropy quantity $$\chi ^{*}({X_{1},\dots ,X_{n}}:B).$$ His approach involved non-commutative Hilbert transform and is algebraic in nature. In the case that $ B=\mathbb C $, this quantity is denoted $ \chi ^{*}({X_{1},\dots ,X_{n}}) $, and its properties are very similar to those of the free entropy $ \chi ({X_{1},\dots ,X_{n}}) $ introduced by Voiculescu in [@dvv:entropy2] using microstates; in fact, it may very well be that the two quantities coinside.

Using the microstates approach to free entropy, we introduce in this paper a quantity $ \chi ({X_{1},\dots ,X_{n}}|B) $ which has several properties in common $ \chi ^{*}({X_{1},\dots ,X_{n}}:B) $.

The infinitesimal change of variables formula for $ \chi  $ involves conjugate variables, introduced by Voiculescu in order to define $ \chi ^{*} $. Both $ \chi  $ and $ \chi ^{*} $ have the same behavior 