The unitary group of a $\mathrm{II}_1$ factor is SOT-contractible

TL;DR

This paper proves that the unitary group of any SOT-separable $ ext{II}_1$ factor is contractible, extending to certain von Neumann algebras, using free convolution regularization and Popa's theorem.

Contribution

It establishes the SOT-contractibility of the unitary group for SOT-separable $ ext{II}_1$ factors and related von Neumann algebras, a novel topological property.

Findings

Unitary group of SOT-separable $ ext{II}_1$ factors is contractible.

Extension of contractibility to von Neumann algebras without finite type I summands.

Application of free convolution and Popa's theorem in the proof.

Abstract

We show that the unitary group of any SOT-separable $\mathrm{II}_1$ factor $M$, with the strong operator topology, is contractible. Combined with several old results, this implies that the same is true for any SOT-separable von Neumann algebra with no type $\mathrm{I}_n$ direct summands ($n < \infty$). The proof for the $\mathrm{II}_1$-factor case uses regularization via free convolution and Popa's theorem on the existence of approximately free Haar unitaries in $\mathrm{II}_1$ factors.