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.