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Polish groups of unitaries

Volume 257 / 2021

Hiroshi Ando, Yasumichi Matsuzawa Studia Mathematica 257 (2021), 25-70 MSC: Primary 22A05; Secondary 22E65. DOI: 10.4064/sm190718-5-6 Published online: 7 August 2020


We study the question of which Polish groups can be realized as subgroups of the unitary group of a separable infinite-dimensional Hilbert space. We also show that for a separable unital C$^*$-algebra $A$, the identity component $\mathcal {U}_0(A)$ of its unitary group has property (OB) of Rosendal (hence it also has property (FH)) if and only if the algebra has finite exponential length (e.g. if it has real rank zero), while in many cases the unitary group $\mathcal {U}(A)$ does not have property (T). On the other hand, the $p$-unitary group $\mathcal {U}_p(M,\tau )$ where $M$ is a properly infinite semifinite von Neumann algebra with separable predual, does not have property (FH) for any $1\le p \lt \infty $. This in particular solves a problem left unanswered in the work of Pestov [Trans. Amer. Math. Soc. 370 (2018)].


  • Hiroshi AndoDepartment of Mathematics and Informatics
    Chiba University
    1-33 Yayoi-cho, Inage, Chiba
    263-8522 Japan
  • Yasumichi MatsuzawaDepartment of Mathematics
    Faculty of Education
    Shinshu University
    6-Ro, Nishi-nagano, Nagano
    380-8544 Japan

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