The stability of the quadratic functional equation on amenable groups (Q1433369)

Let \(G\) be a group, and let \(E\) be a Banach space. The author defines \((G,E)\) to be QS (short for \((G,E)\) has the property of the stability of the quadratic functional equation) if, for every function \(f : G \to E\) such that \[ \sup_{x,y \in G} \| f(xy) + f(xy^{-1}) - 2f(x) -2f(y) \| := \delta < \infty, \] there are a function \(q : G \to E\) satisfying \[ q(xy) + q(xy^{-1}) - 2q(x)-2q(y) = 0 \qquad (x,y \in G) \] and \(\delta' > 0\) (depending only on \(\delta\)) such that \(\| f(x) - q(x) \| < \delta'\) for all \(x \in G\). There are two theorems in the paper. The first one asserts that if \((G,\mathbb{C})\) is QS, then \((G,E)\) is QS for every complex Banach space \(E\) (and analogously with \(\mathbb{C}\) replaced by \(\mathbb{R}\)). Secondly, the author proves that \((G,\mathbb{C})\) is QS for every amenable (discrete) group \(G\), so that \((G,E)\) is QS for every Banach space in this particular situation.