Glöckner–Neeb multiplication-growth question

The question asks whether the Glöckner--Neeb multiplication-growth condition is automatic for every Mackey-complete continuous inverse algebra. Glöckner and Neeb formulated the relevant regularity framework in 2012; their results gave sufficient conditions, not automaticity.

Known results

• Glöckner--Neeb, 2012: $A^{\times}$ is regular when $A$ is Mackey-complete and locally $m$-convex.
• The 2012 work does not establish that the multiplication-growth condition follows from Mackey completeness.

2026 counterexample

A 2026 arXiv preprint claims to construct a complete complex continuous inverse algebra whose unit group is not even $C^{0}$-semiregular and whose multiplication-growth condition fails. Since completeness is stronger than Mackey completeness, this would give a negative answer; the same work claims infinite-dimensional positively graded examples with failure of $C^{k}$-semiregularity for all $k\in\mathbb{N}_{0}\cup\{\infty\}$.

Current status (as of July 2026): A complete counterexample is claimed in an arXiv preprint, so automaticity has a claimed negative answer; independent verification remains open.