整数集Z²在算子范数下是灵活稳定的
$\mathbb{Z}^2$ is flexibly stable in the operator norm
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中文总结 AI 辅助
研究整数集Z²在算子范数下的稳定性，通过渐近可忽略地扩大维度证明其灵活稳定，这是首个非稳定的灵活稳定群的例子，还基于埃克哈特构造说明某些群在算子范数下有类似区分。
中文摘要 AI 辅助
稳定性理论的一个基石是沃伊库列斯库1983年的反例：他构造了一系列酉矩阵对，其换位子在算子范数下收敛到零，但它们与可交换酉对集合的距离仍有界远离零。即整数集Z²在算子范数下不稳定。我们证明，在维度渐近可忽略的扩大后稳定性得以恢复。即整数集Z²在算子范数下是灵活稳定的。这在任何背景下提供了第一个非稳定的灵活稳定群的例子。基于埃克哈特的构造，我们表明相同的群在算子范数下也有类似的区分：它们是非常灵活稳定但不是灵活稳定的。
英文摘要
A cornerstone of stability theory is Voiculescu's 1983 counterexample: he constructed a sequence of pairs of unitary matrices whose commutators converge to zero in the operator norm, but whose distances from the set of commuting unitary pairs remain bounded away from zero. Namely, the group $\mathbb{Z}^2$ is not stable in the operator norm. We prove, somewhat surprisingly, that stability is restored after an asymptotically negligible enlargement of the dimension. That is, the group $\mathbb{Z}^2$ is flexibly stable in the operator norm. This provides the first example, in any context, of a flexibly stable group that is not stable. Building on a construction of Eckhardt, who produced finitely generated amenable groups that are very-flexibly stable but not flexibly stable in the normalized Hilbert-Schmidt norm, we show that the same groups exhibit the analogous separation in the operator norm: they are very-flexibly stable but not flexibly stable.
