2026/07/30 by Zongjian Han, Fungo Liu
Mathematics · #math.DG #msc:16N40 #msc:22E65 #msc:46A13 #msc:46H05
14 pages; comments welcome! 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 will have it rewritten by hand for publication. Please refer to the acknowledgements-
We construct a counterexample resolving a fundamental open problem in infinite-dimensional Lie theory: whether every Lie group modelled on a complete locally convex space is regular. The example is a contractible complex analytic BCH--Lie group H, modelled on the complete Silva space φ\mathbb C, whose exponential map is a homeomorphism, yet H is not even C0-semiregular. Smooth controls tending to zero in one fixed finite-dimensional subspace have no H-valued C1 evolution, although each has a unique smooth evolution on [0,1). The group is the principal unit group of a complete complex continuous inverse algebra. The construction also answers negatively the Glöckner--Neeb question whether their multiplication-growth condition is automatic for Mackey-complete continuous inverse algebras, even under the stronger assumption of completeness.