The authors show that, if \(\mathcal{A}\) is a \(C^*\)-algebra (resp., a \(C^*\)-algebra of real rank zero with an increasing approximate unit of projections), \(X\) is a locally convex space (resp., topological vector space), and \(P:\mathcal{A}\to X\) is a continuous \(m\)-homogeneous polynomial, then the following are equivalent: (i) there is a continuous linear mapping \(\Phi:\mathcal{A}\to X\) such that \(P(x)=\Phi(x^m)\) for all \(x\) in \(\mathcal{A}\); (ii) \(P\) is orthogonally additive on \(\mathcal{A}_{sa}\), the self-adjoint part of \(\mathcal{A}\); (iii) \(P\) is orthogonally additive on \(\mathcal{A}_+\), the positive part of \(\mathcal A\).
Using these results, they show that, if \(H\) is a Hilbert space of dimension greater than or equal to \(2\), then there are no non-zero orthogonally additive \(m\)-homogeneous polynomials from \(\mathcal{B}(H)\) into any topological vector space \(X\).
When \(\mathcal{M}\) is a von Neumann algebra with a normal semifinite faithful trace \(\tau\) and \(P\) is a continuous \(m\)-homogeneous polynomial from \(L^p(\mathcal{M},\tau)\) into a topological vector space \(X\), \(0< p<\infty\), the authors show that the following are equivalent: (i) there is a continuous linear mapping \(\Phi: L^{p/m}(\mathcal{M},\tau)\to X\) such that \(P(x)=\Phi(x^m)\) for all \(x\) in \(L^p(\mathcal{M},\tau)\); (ii) \(P\) is orthogonally additive on \(L^p(\mathcal{M},\tau)_{sa}\), the self-adjoint part of \(L^p(\mathcal{M},\tau)\); (iii) \(P\) is orthogonally additive on \(S(\mathcal{M},\tau)_+\).
This allows the authors to present a number of results concerning orthogonally additive polynomials on the Schatten \(p\)-class, \(S^p(H)\), and the space of compact operators on a Hilbert space \(H\), \(\mathcal{K}(H)\). From this it follows that, if \(H\) is a Hilbert space of dimension at least \(2\), then for \(0< p<\infty\), there are no non-zero orthogonally additive polynomials on \(S^p(H)\).