A note on the groups of finite type and the Hartman-Mycielski construction

Abstract

Ando, Matsuzawa, Thom, and Törnquist have resolved a problem by Sorin Popa by constructing an example of a Polish group of unitary operators with the strong operator topology, whose left and right uniform structures coincide, but which does not embed into the unitary group of a finite von Neumann algebra. The question remained whether such a group can be connected. Here we observe that a connected (in fact, homeomorphic to the Hilbert space) example is obtained from the example of the above authors via the Hartman--Mycielski construction.

arXiv comment5 pp., latex, a version accepted by Colloq. Math. The title has been changed compared to v.1, and some very minor changes made in the text