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May 13, 2013 at 10:23 comment added Nicolas Boerger I am Thinking mainly about Isometric actions. Thanks for the comments.
May 13, 2013 at 9:22 comment added Daniele Zuddas If you mean only topological actions, then this is the same of actions on Hilbert spaces, because the infinite-dimensional unitary sphere is diffeomorphic with the Hilbert space, see C. Bessaga, "Every infinite-dimensional Hilbert space is diffeomorphic with its unit sphere", Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 14 1966 27-31.
May 13, 2013 at 8:55 comment added Mariano Suárez-Álvarez Every torsion free discrete group $G$ acts properly and discontinuousy on the unit sphere of $L^2(G)$, no?
May 13, 2013 at 8:38 history asked Nicolas Boerger CC BY-SA 3.0