AI 中文总结
该研究针对\(\vert q \vert < \sqrt{2}-1\)的有限\(q -\)高斯族，先通过图积半圆归一化和超积方法实现非交换多项式强收敛，再用定量张量 - GUE转化为有限维随机矩阵，得出相关\(C^*\) - 代数及扩张半群的性质结论。
AI 中文摘要
我们为\(\vert q \vert < \sqrt{2}-1\)范围内的有限\(q -\)高斯族构造强收敛的有限维随机矩阵模型。构造分两步：首先表明在厄尔多斯 - 雷尼图上的图积半圆的归一化和满足\(q -\)托普利兹关系且算子范数误差依概率收敛到零，利用超积方法使有界次数的非交换多项式完全强收敛；其次用定量张量 - GUE将算子模型转化为有限维随机矩阵，固定多项式次数时，收敛在大于随机矩阵维数的系数维上一致。作为应用，在此\(q\)范围内，有限\(q -\)高斯族生成的\(C^*\) - 代数是MF，且由有限\(q -\)高斯族生成的非平凡\(C^*\) - 代数的布朗 - 道格拉斯 - 菲尔莫尔扩张半群不是群。
英文摘要
We construct strongly convergent finite-dimensional random matrix models for finite $q$-Gaussian family in the range $\vert q \vert < \sqrt{2}-1$. The construction has two stages. First, we show that normalized sums of graph-product semicirculars over an Erdos-Renyi graph satisfy the $q$-Toeplitz relations up to an operator norm error converging to zero in probability. Using ultraproduct methods, this yields complete strong convergence, uniformly over all matrix coefficient dimensions, for noncommutative polynomials with bounded degree. Second, we use a quantitative tensor-GUE for graph-product semicirculars to convert these operator models into finite-dimensional random matrices. For every fixed polynomial degree, the convergence is uniform over coefficient dimensions that may be larger than the dimension of the random matrices. As applications, in the above range of $q$, the C$^\ast$-algebra generated by a finite $q$-Gaussian family is MF, and the Brown-Douglas-Fillmore extension semigroup of the nontrivial C$^\ast$-algebra generated by a finite $q$-Gaussian family is not a group.
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