In the 1920s, von Neumann introduced the notion of amenability for groups, and he showed that a group is nonamenable whenever it contains the free group . The question of whether this characterizes (non)amenability became known as von Neumann’s problem, and it was finally answered negatively by Olshanskii in 1980: There exist nonamenable groups that…
Question: Is every element of trace zero in a factor a commutator? Update (August 2024): A positive answer to the questions was recently given In the following article: S. Wen, J. Fang, Z. Yao. A stronger version of Dixmier’s averaging theorem and some applications. J. Funct. Anal. 287 (2024). Background on commutators: It is a…
Surprisingly, the following question is open: Question: Are maximal ideals in C*-algebras closed? Let be a C*-algebra. By an ideal in we mean a two-sided ideal that is not necessarily closed. We say that is a maximal ideal if the only ideals satisfying are and . An ideal is proper if . If is unital,…
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