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

TL;DR

This work solves the obstruction realization problem for $G$-kernels on full von Neumann factors by constructing a full II$_1$ factor $M$ that realizes any given class in $\mathrm{H}^3(G;\mathbb{T})$ as the lifting obstruction of a $G$-kernel, extending Wassermann's results to non-inner-amenable cases. It then builds a Polish group model for the string group $\mathrm{String}(G)$ as a central extension $P\mathcal{U}(M)\hookrightarrow \widetilde{G} \twoheadrightarrow G$, with $\widetilde{G}$ encoding the generator of $\mathrm{H}^3(G;\mathbb{Z})$. The paper introduces the Rohlin property for compact-group actions on von Neumann algebras, classifies Rohlin actions, and proves strong cohomology vanishing results: 1- and 2-cohomology vanish for Rohlin/cocycle actions, yielding robust conjugacy and perturbation results. Together, these results give a new geometric model for $\mathrm{String}(G)$ and extend Rohlin-type classification and cohomology vanishing to non-abelian compact group actions on von Neumann algebras, with applications to crossed products, amplifications, and modular settings.

Abstract

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.

Theorems & Definitions (73)

• Theorem A
• Corollary B
• Theorem C
• Theorem D
• Definition 1.1
• Lemma 1.2: Shapiro lemma
• proof
• Definition 1.3: cocycle action
• Definition 1.4: $G$-kernel
• Example 1.5: Wassermann