AI 中文总结
本研究针对无限维李理论的基础开放问题，构造反例证明以完备局部凸空间为模型的李群未必正则，否定了相关学者关于乘法增长条件自动成立的猜想。
AI 中文摘要
我们构造了一个反例，解决了无限维李理论中的一个基础开放问题：每个以完备局部凸空间为模型的李群是否都是正则的。该反例是一个可缩复解析BCH-李群H，以完备Silva空间φ_C为模型，其指数映射是同胚，但H甚至不是C^0-半正则的。在一个固定有限维子空间中趋于零的光滑控制不存在H取值的C^1演化，尽管每个在[0,1)上都有唯一的光滑演化。该群是一个完备复连续逆代数的主单位群。该构造还否定了Glöckner-Neeb提出的问题：他们的乘法增长条件对于Mackey-完备连续逆代数是否自动成立，即使在更强的完备性假设下也不成立。
英文摘要
We give negative solutions to the two basic completeness problems for locally convex Lie groups: whether a Lie group modelled on a Mackey-complete locally convex space must be regular, and whether it must at least possess a smooth exponential function. Both counterexamples have complete model spaces. We first construct a contractible complex analytic BCH--Lie group $H$, modelled on the complete Silva space $\mathbb C^{(\mathbb N)}$, for which $\exp_H$ is a global homeomorphism although $H$ is not $C^0$-semiregular. The failure is witnessed by smooth controls converging to zero in a fixed finite-dimensional subspace; their evolutions exist uniquely on $[0,1)$ but have no endpoint at time $1$. The obstruction is a multiplicative graded escape in the principal unit group of a complete continuous inverse algebra. We then suspend one such control: the translation action on $C^\infty(S^1,H)$ converts the time-dependent obstruction into a single element of a semidirect-product Lie algebra. The resulting Lie group is modelled on a complete Hausdorff locally convex space and contains an element which generates no one-parameter subgroup. Hence it admits no exponential function. Thus completeness forces neither non-autonomous evolution nor autonomous exponentiation.
Comments16 pages; comments welcome! v2 adds the construction of a Lie group without exponential modelled on a complete locally convex space. The initial ideas and inspiration for this work came from the authors and were further developped and completed by AI. The authors have manually checked the manuscript and concluded it sound. Please refer to the acknowledgements-
