AI 中文总结
该研究解决了Neeb2006年提出的连续逆代数完备化问题，构造了一个非交换反例，证明其Hausdorff完备化虽保持逆运算连续性但可逆元集不开，破坏了单位元处的局部谱稳定性，明确了交换完备化定理的边界。
AI 中文摘要
2006年，Neeb提出问题：连续逆代数的Hausdorff完备化是否一定仍是连续逆代数？当时交换情形的答案已知为是，非交换情形仍悬而未决，二十年后我们给出否定答案。我们构造了一个Hausdorff可度量化局部m-凸复连续逆代数，其完备化是Fréchet局部m-凸代数，其中逆运算保持连续，但可逆元集合不开放。该构造利用了Jacobson半单Banach代数的稠密幂零子代数中的有限支撑序列，有限支撑使每个元素幂零，从而在幂零指数上无局部一致界的情况下得到连续逆运算；在完备化中，将固定非可逆元移至越来越靠后的坐标，会产生收敛到单位元的非可逆元，因此完备化恰好破坏了单位元处的局部谱稳定性。该反例必为非交换的，明确了交换完备化定理的精确边界。
英文摘要
In 2006 Neeb asked whether the Hausdorff completion of a continuous inverse algebra must again be a continuous inverse algebra. The noncommutative case remained open, while the commutative case was known to be true. We give a negative answer after twenty years. We construct a Hausdorff metrizable locally m-convex complex continuous inverse algebra whose completion is a Fréchet locally m-convex algebra, where inversion stays continuous but the set of invertible elements is not open. The construction uses finite-support sequences in a dense nil subalgebra of a Jacobson-semisimple Banach algebra. Finite support makes every element nilpotent, giving continuous inversion without a locally uniform bound on nilpotence indices. In the completion, shifting a fixed noninvertible element to later and later coordinates produces noninvertible elements converging to the identity. Thus completion destroys exactly the local spectral stability at the identity. The counterexample is necessarily noncommutative and marks the precise boundary of the commutative completion theorem.
