On Free Stein Dimension

TL;DR

This paper explores properties of the free Stein dimension, an invariant in free probability, including its behavior under algebraic operations, its invariance in certain algebras, and its extension to von Neumann algebras.

Contribution

It provides formulas for free Stein dimension under algebraic operations, establishes its invariance in specific cases, and extends its definition to von Neumann algebras.

Findings

Free Stein dimension behaves predictably under direct sums and tensor products.

In separable abelian von Neumann algebras, free Stein dimension is an invariant.

Under mild conditions, L^2-rigidity implies free Stein dimension equals one.

Abstract

We establish several properties of the free Stein dimension, an invariant for finitely generated unital tracial $*$-algebras. We give formulas for its behaviour under direct sums and tensor products with finite dimensional algebras. Among a given set of generators, we show that (approximate) algebraic relations produce (non-approximate) bounds on the free Stein dimension. Particular treatment is given to the case of separable abelian von Neumann algebras, where we show that free Stein dimension is a von Neumann algebra invariant. In addition, we show that under mild assumptions $L^2$-rigidity implies free Stein dimension one. Finally, we use limits superior/inferior to extend the free Stein dimension to a von Neumann algebra invariant -- which is substantially more difficult to compute in general -- and compute it in several cases of interest.