# Journal of Operator Theory

Volume 85, Issue 1, Winter 2021 pp. 101-133.

Free Stein irregularity and dimension**Authors**: Ian Charlesworth (1), Brent Nelson (2)

**Author institution:**(1) Department of Mathematics, University of California, Berkeley, Berkeley, CA, 94720, U.S.A.

(2) Department of Mathematics, Michigan State University, East Lansing, MI, 48824, U.S.A.

**Summary:**We introduce a free probabilistic quantity called free Stein irregularity, which is defined in terms of free Stein discrepancies. It turns out that this quantity is related via a simple formula to the Murray-von Neumann dimension of the closure of the domain of the adjoint of the non-commutative Jacobian associated to Voiculescu's free difference quotients. We call this dimension the free Stein dimension, and show that it is a $*$-algebra invariant. We relate these quantities to the free Fisher information, the non-microstates free entropy, and the non-microstates free entropy dimension. In the one-variable case, we show that the free Stein dimension agrees with the free entropy dimension, and in the multivariable case compute it in a number of examples.

**DOI:**http://dx.doi.org/10.7900/jot.2019aug29.2271

**Keywords:**free probability, free entropy, von Neumann algebras, operator algebras

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