Compact group Rohlin actions and $G$-kernels on von Neumann algebras

We provide a new construction of a topological group model for the string group of a compact, simple, and simply-connected Lie group, by solving the obstruction realization problem for compact group $G$-kernels on full factors. Furthermore, we introduce the Rohlin property for actions and cocycle actions of compact groups in order to establish cohomology vanishing theorems.

💡 Research Summary

The paper by Takumi Nishihara tackles two intertwined problems in the theory of operator algebras: (1) the realization of arbitrary third‑cohomology classes as obstruction classes for compact‑group kernels on full factors, and (2) the introduction of a Rohlin property for compact‑group actions (and cocycle actions) on von Neumann algebras, leading to cohomology vanishing results.

Obstruction Realization (Theorem A).
For any compact group (G) and any measurable 3‑cocycle (c\in Z^{3}(G;\mathbb T)), the author constructs a full type II(_1) factor (M) together with a measurable lift (\alpha:G\to\operatorname{Aut}(M)) such that the induced homomorphism (\kappa(g)=\varepsilon_M(\alpha_g)) is a faithful (G)-kernel whose obstruction class (\operatorname{Ob}(\kappa)) equals (