Orthogonally additive polynomials on convolution algebras associated with a compact group

Abstract

Let $G$ be a compact group, let $X$ be a Banach space, and let $P:L^1(G)\to X$ be an orthogonally additive, continuous $n$-homogeneous polynomial. Then we show that there exists a unique continuous linear map $\Phi :L^1(G)\to X$ such that $P(f)=\Phi (f\ast \stackrel{n}{\cdots }\ast f)$ 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 $C(G)$.