Subquadratic growth and uniform property \(Γ\)

Abstract

We prove that every unital separable ASH algebra with subquadratic growth has uniform property $Γ$ whenever it has no nonzero finite-dimensional representations. When simple and non-elementary, these algebras therefore satisfy the Toms--Winter regularity conjecture despite the fact that they generally fail its three conjecturally equivalent properties. In light of the second author's recent construction of a unital simple separable AH algebra of quadratic growth which fails uniform property $Γ$, we conclude that the quadratic dimension growth scale (equivalently, the 2-norm slow dimension growth scale) is the precise geometric threshold governing the potential failure of uniform property \(Γ\). We also extend recent work of Elliott--Niu and Vaccaro to the optimal subquadratic scale by proving that separable unital \(C^*\)-algebras with locally tracially subquadratic RSH approximation have uniform property \(Γ\), provided that they have no nonzero finite-dimensional representations.

Disclosure

“lative rank avoidance for arbitrary RSH decompositions and deduces the main RSH-inductive-limit theorem. Acknowledgements. The second author gratefully acknowledges the support of the Simons Foundation (SFI-MPS-TSM-00025606). AI Statement. ChatGPT was used for language proofreading, notational consis- tency, and reference checking. The mathematical content of the paper is due solely to the authors. 2. Uniform tracial ultrapowers and the property (F G) criterion 2.1. Uniform tra”

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2026-08-23T19:09:02.332087Z

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