Microstates free entropy and cost of equivalence relations. (English) Zbl 1032.37003

Shlyakhtenko, Dimitri

Duke Math. J. 118, No. 3, 375-425 (2003).

Voiculescu’s free entropy dimension is a number \(\delta^{\omega}_{0,k}(u_1 ,...,u_n)\) associated to an \(n\)-tuple of unitaries in a tracial von Neumann algebra (\(\omega\) and \(k\) are certain parameters). The cost \(C(R)\) of an equivalence relation \(R\) is a measure of the least number of generators of the associated groupoid. The cost of a group \(C(G)\) is the infimum of all \(C(R_{\alpha})\) over all free measure preserving actions \(\alpha\) of \(G\) (D. Gaboriau).
Main results of the paper: 1. An invariant of a finite measure preserving \(r\)-discrete measure groupoid \(G\) denoted by \(\delta (G)\) is defined. This invariant is Voiculescu’s free entropy dimension \(\delta\) in the case where \(G\) is a group. In the case where \(G\) is an equivalence relation, \(\delta\) is an invariant for the pair consisting of the von Neumann algebra of \(G\) and the canonical Cartan subalgebra. 2. For a general equivalence relation \(R\) possessing a finite graphing \(\delta (R)\leq C(R)\). 3. For a treeable equivalence relation \(R\) \(\delta (R)=C(R).\) The equality \(\delta (R)=C(R)\) for general \(R\) is unknown. The equality for arbitrary \(R\) would imply that all equivalence relations can be embedded into the ultrapower of the hyperfinite \(II_1\)-factor (problem of A. Connes).

Reviewer: Victor Sharapov (Volgograd)

MSC:

37A20	Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations
46L54	Free probability and free operator algebras

Keywords:

tracial von Neumann algebra; cost of equivalence relation; Voiculescu’s free entropy dimension

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