Question: Let be a -factor. Is the unitary group contractible when equipped with the strong operator topology?

**Background/Motivation:** By Kuiper’s theorem, if is an infinite-dimensional Hilbert space, then the unitary group of the -factor is contractible in the norm topology. This was generalized by Breuer [1] to (certain) and 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 is not contractible. Similarly, the unitary group of a factor is not contractible in the norm topology since its fundamental group does not vanish – in fact, it was shown in [3] that for every factor .

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 is contractible in the strong operator topology if is a separable factor such that the associated factor admits a trace scaling one-parameter group of automorphisms. This includes all McDuff factors ( is McDuff if for the hyperfinite factor ) and all factors that satisfy , where 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.