Contractibility of the automorphism group of a von Neumann algebra

TL;DR

The article proves that the approximately inner automorphism group $\overline{\mathrm{Int}}(M)$ of a separable strongly stable von Neumann algebra $M$ is contractible in the $u$-topology, with a key corollary that $\mathrm{Aut}(\mathcal{R}_{III_1})$ is contractible. The authors employ a convexity-based deformation (à la Dadarlat–Pennig) to establish Theorem A, deforming automorphisms toward $\mathrm{Aut}(M_0)\otimes\mathrm{id}_{\mathcal{R}}$ while preserving approximate innerness. For Theorem B, they develop a cross-section/selection approach using Michael's theorem to produce a continuous $\mathcal{U}(N)$-equivariant retraction from $\mathcal{U}(M)$ onto $\mathcal{U}(N)$ in the strongly stable setting, and extend this to the general case. These results advance Popa–Takesaki's program on contractibility of automorphism groups and have potential implications for geometric topology and mathematical physics. $\overline{\mathrm{Int}}(M)$ and $\mathrm{Aut}(\mathcal{R}_{III_1})$ are central objects, linked through the $u$-topology and strong self-absorption properties of hyperfinite factors.

Abstract

We prove that the approximately inner automorphism group of a separable strongly stable von Neumann algebra is contractible in the u-topology. Thus the automorphism group of the hyperfinite type III_1 factor is contractible.

Theorems & Definitions (27)

• Theorem A
• Corollary
• Theorem B
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4