AI 中文总结
研究具有次二次增长的单位可分ASH代数，证明其无非零有限维表示时具一致性质\(Γ\)，得出二次维数增长尺度是\(Γ\)潜在失效阈值，还将相关工作扩展到最优次二次尺度。
AI 中文摘要
我们证明，每个具有次二次增长且无非零有限维表示的单位可分ASH代数都具有一致性质\(Γ\)。当这些代数是简单且非初等时，尽管它们通常不满足三个推测等价的性质，但仍满足Toms-Winter正则性猜想。鉴于第二作者最近构造了一个具有二次增长且不具有一致性质\(Γ\)的单位简单可分AH代数，我们得出二次维数增长尺度（等价地，2-范数慢维数增长尺度）是控制一致性质\(Γ\)潜在失效的精确几何阈值。我们还将Elliott-Niu和Vaccaro的近期工作扩展到最优次二次尺度，证明了具有局部迹次二次RSH逼近的可分单位\(C^*\) - 代数具有一致性质\(Γ\)，前提是它们没有非零有限维表示。
英文摘要
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.
Comments28 pages, 1 figure
