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 $P\colon L^1(G)\to X$beanorthogonallyadditive,continuous$n$-homogeneous polynomial. Then we show that there exists a unique continuous linear map $\mathrm{\Phi }\text{\hspace{0.17em}⁣}:L^1(G)\to X$ such that $P(f)=\Phi \bigl(f\ast\stackrel{n}{\cdots}\ast f \bigr)$foreach$f\in L^1(G)$. We also seek analogues of this result about $L^1(G)$ for various other convolution algebras, including $L^p(G)$, for $1<p\le \mathrm{\infty }$, and $C(G)$.