Contractibility of unitary groups of II-1 factors

Question: Let $M$ be a $\mathrm{II}_1$ -factor. Is the unitary group $\mathcal{U}(M)$ contractible when equipped with the strong operator topology?

Background/Motivation: By Kuiper’s theorem, if $H$ is an infinite-dimensional Hilbert space, then the unitary group of the $\mathrm{I}_\infty$ -factor $\mathcal{B}(H)$ is contractible in the norm topology. This was generalized by Breuer [1] to (certain) $\mathrm{I}_\infty$ and $\mathrm{II}_\infty$ von Neumann algebras, and eventually Brüning-Willgerodt [2] showed that the unitary group of every properly infinite von Neumann algebra is contractible in the norm topology. This is no longer true for finite von Neumann algebras: The unitary group of a finite matrix algebra $M_n(\mathbb{C})$ is not contractible. Similarly, the unitary group of a $\mathrm{II}_1$ factor is not contractible in the norm topology since its fundamental group does not vanish – in fact, it was shown in [3] that $\pi_1(\mathcal{U}(M))\cong\mathbb{R}$ for every $\mathrm{II}_1$ factor $M$ .

As noted in the introduction of [4], the unitary group of every properly infinite von Neumann algebra is also contractible in the strong operator topology. This naturally leads to the above question, which was considered by Popa-Takesaki in [4]. They showed that $\mathcal{U}(M)$ is contractible in the strong operator topology if $M$ is a separable $\mathrm{II}_1$ factor such that the associated $\mathrm{II}_\infty$ factor $M\bar{\otimes} \mathcal{B}(H)$ admits a trace scaling one-parameter group of automorphisms. This includes all McDuff factors ( $M$ is McDuff if $M\cong M\bar{\otimes}\mathcal{R}$ for the hyperfinite $\mathrm{II}_\infty$ factor $\mathcal{R}$ ) and all factors that satisfy $M\cong M\bar{\otimes} L(\mathbb{F}_\infty)$ , where $L(\mathbb{F}_\infty)$ is the group von Neumann algebra of the free group on infinitely many generators.

[1]
M. Breuer, On the homotopy type of the group of regular elements of semifinite von Neumann algebras, Math. Ann. (1970) 61–74. https://doi.org/10.1007/bf01350761.
[2]
J. Brüning, W. Willgerodt, Eine Verallgemeinerung eines Satzes von N. Kuiper, Math. Ann. (1976) 47–58. https://doi.org/10.1007/bf01354528.
[3]
H. Araki, M.-S.B. Smith, L. Smith, On the homotopical significance of the type of von Neumann algebra factors, Commun.Math. Phys. (1971) 71–88. https://doi.org/10.1007/bf01651585.
[4]
S. Popa, M. Takesaki, The topological structure of the unitary and automorphism groups of a factor, Commun.Math. Phys. (1993) 93–101. https://doi.org/10.1007/bf02100051.

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