Hölder continuous retractions and amenable semigroups of \n uniformly Lipschitzian mappings in Hilbert spaces

Suppose that $S$ is a left amenable semitopological semigroup.\nWe prove that\nif $\\mathcal{S}=\\{ T_{t}:t\\in S\\} $ is a uniformly $k$-Lipschitzian semigroup on a bounded\nclosed and convex subset $C$ of a\nHilbert space and $k< \\sqrt{2}$, then the set of fixed points of $\\mathcal{S}$\nis a Hölder continuous retract of $C$. This gives a qualitative\ncomplement to the Ishihara-Takahashi fixed point existence theorem.