# Free Stein Irregularity and Dimension

## TL;DR

This paper introduces the free Stein irregularity and free Stein dimension, connecting them to key free probability concepts and invariants, and computes these dimensions in various examples.

## Contribution

It defines the free Stein irregularity and free Stein dimension, establishing their relationships with existing free probability invariants and demonstrating their properties and computations.

## Key findings

- Free Stein dimension equals free entropy dimension in one-variable case.
- The free Stein dimension is a $*$-algebra invariant.
- Computed free Stein dimension in multiple multivariable examples.

## Abstract

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.