Duke Mathematical Journal

Microstates free entropy and cost of equivalence relations

Dimitri Shlyakhtenko

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Abstract

We define an analog of Voiculescu's free entropy for $n$-tuples of unitaries $u\sb 1,\ldots u\sb n$ in a tracial von Neumann algebra $M$ which normalize a unital subalgebra $L\sp \infty[0,1]=B\subset M$. Using this quantity, we define the free dimension $\delta\sb 0(u\sb 1,\ldots,u\sb n\between B)$. This number depends on $u\sb 1,\ldots u\sb n$ only up to orbit equivalence over $B$. In particular, if $R$ is a measurable equivalence relation on $[0,1]$ generated by $n$ automorphisms $\alpha\sb 1,\ldots \alpha\sb n$, let $u\sb 1,\ldots u\sb n$ be the unitaries implementing $\alpha\sb 1,\ldots \alpha\sb n$ in the Feldman-Moore crossed product algebra $M=W\sp \ast([0,1],R)\supset B=L\sp \infty[0,1]$. Then the number $\delta(R)=\delta\sb 0(u\sb 1,\ldots u\sb n\between B)$ is an invariant of the equivalence relation $R$. If $R$ is treeable, $\delta(R)$ coincides with the cost $C(R)$ of $R$ in the sense of D. Gaboriau. In particular, it is $n$ for an equivalence relation induced by a free action of the free group $\mathbb {F}\sb n$. For a general equivalence relation $R$ possessing a finite graphing of finite cost, $\delta(R)\leq C(R)$. Using the notion of free dimension, we define a dynamical entropy invariant for an automorphism of a measurable equivalence relation (or, more generally, of an $r$-discrete measure groupoid) and give examples.

Article information

Source
Duke Math. J. Volume 118, Number 3 (2003), 375-425.

Dates
First available: 23 April 2004

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1082744674

Mathematical Reviews number (MathSciNet)
MR1983036

Digital Object Identifier
doi:10.1215/S0012-7094-03-11831-1

Zentralblatt MATH identifier
1032.37003

Subjects
Primary: 46Lxx: Selfadjoint operator algebras ($C^*$-algebras, von Neumann ($W^*$-) algebras, etc.) [See also 22D25, 47Lxx]

Citation

Shlyakhtenko, Dimitri. Microstates free entropy and cost of equivalence relations. Duke Mathematical Journal 118 (2003), no. 3, 375--425. doi:10.1215/S0012-7094-03-11831-1. http://projecteuclid.org/euclid.dmj/1082744674.


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