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


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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  1. 1.

    Alaminos, J., Extremera, J., Villena, A.R.: Orthogonally additive polynomials on Fourier algebras. J. Math. Anal. Appl. 422, 72–83 (2015)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Alaminos, J., Extremera, J., Godoy, M.L.C., Villena, A.R.: Orthogonally additive polynomials on convolution algebras associated with a compact group. J. Math. Anal. Appl. 472, 285–302 (2019)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Alaminos, J., Godoy, M.L.C., Villena, A.R.: Orthogonally additive polynomials on the algebras of approximable operators. Linear Multilinear Algebra 67, 1922–1936 (2019). https://doi.org/10.1080/03081087.2018.1476445

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Benyamini, Y., Lassalle, S., Llavona, J.G.: Homogeneous orthogonally additive polynomials on Banach lattices. Bull. Lond. Math. Soc. 38, 459–469 (2006)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Blackadar, B.: Operator algebras. Theory of \(C^*\)-algebras and von Neumann algebras. In: Cuntz, J., Jones, V.F.R. (eds.) Encyclopaedia of Mathematical Sciences. Operator Algebras and Non-commutative Geometry. III, vol. 122. Springer, Berlin (2006)

  6. 6.

    Carando, D., Lassalle, S., Zalduendo, I.: Orthogonally additive polynomials over \(C(K)\) are measures: a short proof. Integral Equ. Oper. Theory 56, 597–602 (2006)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Davidson, K.R.: \(C^*\)-Algebras by Example. Fields Institute Monographs, vol. 6. American Mathematical Society, Providence (1996)

    Google Scholar 

  8. 8.

    Dineen, S.: Complex Analysis on Infinite-Dimensional Spaces. Springer Monographs in Mathematics. Springer, London (1999)

    Book  Google Scholar 

  9. 9.

    Ibort, A., Linares, P., Llavona, J.G.: A representation theorem for orthogonally additive polynomials on Riesz spaces. Rev. Mat. Complut. 25, 21–30 (2012)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Nelson, E.: Notes on non-commutative integration. J. Funct. Anal. 15, 103–116 (1974)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Palazuelos, C., Peralta, A.M., Villanueva, I.: Orthogonally additive polynomials on \(C^*\)-algebras. Q. J. Math. 59, 363–374 (2008)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Pérez-García, D., Villanueva, I.: Orthogonally additive polynomials on spaces of continuous functions. J. Math. Anal. Appl. 306, 97–105 (2005)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Pisier, G., Xu, Q.: Non-commutative \(L^p\)-Spaces. Handbook of the Geometry of Banach Spaces, vol. 2, pp. 1459–1517. North-Holland, Amsterdam (2003)

    MATH  Google Scholar 

  14. 14.

    Raynaud, Y., Xu, Q.: On subspaces of non-commutative \(L_p\)-spaces. J. Funct. Anal. 203, 149–196 (2003)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Saito, K.S.: Noncommutative \(L^p\)-spaces with \(0<p<1\). Math. Proc. Camb. Philos. Soc. 89, 405–411 (1981)

    Article  Google Scholar 

  16. 16.

    Sundaresan, K.: Geometry of spaces of homogeneous polynomials on Banach lattices. In: Gritzmann, P., Sturmfels, B. (eds.) Applied Geometry and Discrete Mathematics, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 4, pp. 571–586. American Mathematical Society, Providence (1991)

    Google Scholar 

  17. 17.

    Takesaki, M.: Theory of Operator Algebras. I. Springer, New York (1979)

    Book  Google Scholar 

  18. 18.

    Terp, M.: \(L^p\) spaces associated with von Neumann algebras. Notes, Mathematical Institute, Copenhagen University (1981)

  19. 19.

    Villena, A.R.: Orthogonally additive polynomials on Banach function algebras. J. Math. Anal. Appl. 448, 447–472 (2017)

    MathSciNet  Article  Google Scholar 

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

Mathematics Subject Classification

  • 46L10
  • 46L52
  • 47H60