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

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.

arXiv comment6 pages; v2, v3 minor corrections