Orthogonally additive polynomials on non-commutative \(L^p\)-spaces

Abstract

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 \(\varPhi :L^{p/m}({{\mathscr {M}}},\tau )\rightarrow X\).

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Correspondence to Armando R. Villena.

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The authors were supported by MINECO Grant PGC2018–093794–B–I00. and Junta de Andalucía Grant FQM–185. The second named author was supported by Contrato Predoctoral FPU, Plan propio de Investigación y Transferencia 2018, University of Granada.

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Alaminos, J., Godoy, M.L.C. & Villena, A.R. Orthogonally additive polynomials on non-commutative \(L^p\)-spaces. Rev Mat Complut 33, 835–858 (2020). https://doi.org/10.1007/s13163-019-00330-1

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Keywords

  • Non-commutative \(L^p\)-space
  • Schatten classes
  • Orthogonally additive polynomial

Mathematics Subject Classification

  • 46L10
  • 46L52
  • 47H60