Advertisement

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

  • Jerónimo Alaminos
  • María L. C. Godoy
  • Armando R. VillenaEmail author
Article
  • 11 Downloads

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

Keywords

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

Mathematics Subject Classification

46L10 46L52 47H60 

Notes

References

  1. 1.
    Alaminos, J., Extremera, J., Villena, A.R.: Orthogonally additive polynomials on Fourier algebras. J. Math. Anal. Appl. 422, 72–83 (2015)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle 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 MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefGoogle 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)CrossRefGoogle Scholar
  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)MathSciNetCrossRefGoogle 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)CrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Nelson, E.: Notes on non-commutative integration. J. Funct. Anal. 15, 103–116 (1974)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Palazuelos, C., Peralta, A.M., Villanueva, I.: Orthogonally additive polynomials on \(C^*\)-algebras. Q. J. Math. 59, 363–374 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Pérez-García, D., Villanueva, I.: Orthogonally additive polynomials on spaces of continuous functions. J. Math. Anal. Appl. 306, 97–105 (2005)MathSciNetCrossRefGoogle 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)zbMATHGoogle Scholar
  14. 14.
    Raynaud, Y., Xu, Q.: On subspaces of non-commutative \(L_p\)-spaces. J. Funct. Anal. 203, 149–196 (2003)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Saito, K.S.: Noncommutative \(L^p\)-spaces with \(0<p<1\). Math. Proc. Camb. Philos. Soc. 89, 405–411 (1981)CrossRefGoogle 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)CrossRefGoogle Scholar
  18. 18.
    Terp, M.: \(L^p\) spaces associated with von Neumann algebras. Notes, Mathematical Institute, Copenhagen University (1981)Google Scholar
  19. 19.
    Villena, A.R.: Orthogonally additive polynomials on Banach function algebras. J. Math. Anal. Appl. 448, 447–472 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Universidad Complutense de Madrid 2019

Authors and Affiliations

  • Jerónimo Alaminos
    • 1
  • María L. C. Godoy
    • 1
  • Armando R. Villena
    • 1
    Email author
  1. 1.Departamento de Análisis Matemático, Facultad de CienciasUniversidad de GranadaGranadaSpain

Personalised recommendations