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

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Let $${{\mathscr {M}}}$$ be a von Neumann algebra with a normal semifinite faithful trace $$\tau $$ . We prove that every continuous m-homogeneous polynomial P from $$L^p({{\mathscr {M}}},\tau )$$ , 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 $$L^p({{\mathscr {M}}},\tau )$$ can be represented in the form $$P(x)=\varPhi (x^m)$$ $$(x\in L^p({{\mathscr {M}}},\tau ))$$ for some continuous linear map $$\varPhi :L^{p/m}({{\mathscr {M}}},\tau )\rightarrow X$$ .