Boundedness exponents of unitary function groups and measured full groups

Let $X$ be an infinite compact metrizable space and let $M\ne0$ be a von Neumann algebra with separable predual. For the strong topology on $\mathcal{U}(M)$, we classify boundedness of $C(X,\mathcal{U}(M))$: finite exponents occur exactly when $M$ is properly infinite, with least open-exhaustion exponent five and least finite-neighbourhood exponent three, and ordinary Property $(\mathrm{OB})$ holds exactly when $\dim X=0$ or $M$ has no matrix direct summand. Continuous fixed-involution factors give the finite-exponent result; compact-uniform adaptations of spectral regularization and central splitting give ordinary boundedness. The classification concerns function-group boundedness, while the corresponding target contractibility criterion is due to Jekel. Related four-factor target-unitary and type III full-group bounds are upper bounds.

This paper develops material from the author's monograph Property (OB) and the Coarse Geometry of Polish Groups. It is one of eight separate papers intended to replace the monograph.

GitHub repository: https://github.com/eulogica/property-ob-independent-papers