For a compact group $\mathbb{ G}$ acting continuously on a Banach Lie group $\mathbb{ U}$, we prove that maps $\mathbb{ G}\to \mathbb{ U}$ close to being 1-cocycles for the action can be deformed analytically into actual 1-cocycles. This recovers Hyers-Ulam stability results of Grove-Karcher-Ruh (trivial $\mathbb{ G}$-action, compact Lie $\mathbb{ G}$ and $\mathbb{ U}$) and de la Harpe-Karoubi (trivial $\mathbb{ G}$-action, $\mathbb{ U}:=$invertible elements of a Banach algebra). The obvious analogues for higher cocycles also hold for abelian $\mathbb{ U}$.
Categories: math.GR math.FA math.GN