Completeness of the Model Space Does Not Force Regularity for Infinite-Dimensional Lie Groups

Abstract

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.

arXiv comment16 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-