Some 40 years ago, Vaughan Jones showed the amazing fact that symmetries
of II_1 factors are quantized, and that under fusion-product they generate quantized groups.
This revolutionary discovery complemented the theory of von Neumann algebras with a striking
set of new mathematical objects and new phenomena, that nobody expected. This
led over the years to a series of deep results and a huge diversity of beautiful problems.
I will make some reflections on all this, especially on results and problems pertaining to analysis aspects.
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