Orthogonally additive polynomials on non-commutative $L^p$-spaces

Let $\mathcal{M}$ be a von Neumann algebra with a normal semifinite faithful trace $\tau$. We prove that every continuous $m$-homogeneous polynomial $P$ from $L^p(\mathcal{M},\tau)$, with $0<p<\infty$, into each topological linear space $X$ with the property that $P(x+y)=P(x)+P(y)$ whenever $x$ and $y$ are mutually orthogonal positive elements of $L^p(\mathcal{M},\tau)$ can be represented in the form $P(x)=\Phi(x^m)$ $(x\in L^p(\mathcal{M},\tau))$ for some continuous linear map $\Phi\colon L^{p/m}(\mathcal{M},\tau)\to X$.