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 $Pcolon L^1(G)to X$ be an orthogonally additive, continuous $n$-homogeneous polynomial. Then we show that there exists a unique continuous linear map $Phicolon L^1(G)to X$ such that $P(f)=Phi bigl(faststackrel{n}{cdots}ast f bigr)$ for each $fin L^1(G)$. We also seek analogues of this result about $L^1(G)$ for various other convolution algebras, including $L^p(G)$, for $1< pleinfty$, and $C(G)$.