Orthogonally additive polynomials on convolution algebras associated with a compact group

Let \(G\) be a compact group, let \(X\) be a Banach space, and let \(P\colon L^1(G)\to X\) be an orthogonally additive, continuous \(n\)-homogeneous polynomial. Then we show that there exists a unique continuous linear map \(\Phi\colon L^1(G)\to X\) such that \(P(f)=\Phi \bigl(f\ast\stackrel{n}{\cdots}\ast f \bigr)\) for each \(f\in L^1(G)\). We also seek analogues of this result about \(L^1(G)\) for various other convolution algebras, including \(L^p(G)\), for \(1< p\le\infty\), and \(C(G)\).