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arXiv · 2201.00062

On Free Stein Dimension

Ian Charlesworth·Brent Nelson
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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.

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Ian Charlesworth, Brent Nelson. 2022-01-04. On Free Stein Dimension. https://arxiv.org/abs/2201.00062

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