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Orthogonally additive polynomials on non-commutative $L^p$-spaces (1903.10192v1)

Published 25 Mar 2019 in math.OA and math.FA

Abstract: 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 $Lp(\mathcal{M},\tau)$, 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 $Lp(\mathcal{M},\tau)$ can be represented in the form $P(x)=\Phi(xm)$ $(x\in Lp(\mathcal{M},\tau))$ for some continuous linear map $\Phi\colon L{p/m}(\mathcal{M},\tau)\to X$.

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