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Contractivity of Projections Commuting with Inner Derivations on JBW*-triples

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Abstract

It is shown that if P is a weak*-continuous projection on a JBW*-triple A with predual A *, such that the range PA of P is an atomic subtriple with finite-dimensional Cartan-factors, and P is the sum of coordinate projections with respect to a standard grid of PA, then P is contractive if and only if it commutes with all inner derivations of PA. This provides characterizations of 1-complemented elements in a large class of subspaces of A * in terms of commutation relations.

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References

  1. Braun H., Koecher M.: Jordan-Algebren. Springer, Berlin (1966)

    MATH  Google Scholar 

  2. Bunce L.J.: Norm preserving extensions in JBW*-triple preduals. Quart. J. Math. 52, 133–136 (2000)

    Article  MathSciNet  Google Scholar 

  3. Dang T., Friedman Y.: Classifiaction of JBW*-triple factors and applications. Math. Scand. 61, 292–330 (1987)

    MATH  MathSciNet  Google Scholar 

  4. Edwards C.M., Hügli R.V.: Order structure of GL-projections on the predual of a JBW*-triple. Atti. Sem. Math. Fis. Univ. Modena LIII, Fasc 2, 271–287 (2005)

    Google Scholar 

  5. Edwards C.M., Hügli R.V.: M-orthogonality and holomorphic rigidity in complex Banach spaces. Acta Sci. Math. (Szeged) 70, 237–264 (2004)

    MATH  MathSciNet  Google Scholar 

  6. Edwards C.M., Hügli R.V., Rüttimann G.T.: A geometric characterization of structural projections on a JBW*-triple. J. Funct. Anal. 202, 174–194 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Edwards C.M., Rüttimann G.T.: Orthogonal faces of the unit ball in a Banach space. Atti Sem. Math. Fis. Univ. Modena 49, 473–493 (2001)

    MATH  Google Scholar 

  8. Edwards C.M., Rüttimann G.T.: Structural projections on JBW*-triples. J. Lond. Math. Soc. 53, 354–368 (1996)

    MATH  Google Scholar 

  9. Effros E.G., Størmer E.: Positive projections and Jordan structure in operator algebras. Math. Scand. 45, 127–138 (1979)

    MATH  MathSciNet  Google Scholar 

  10. Faraut, J., Kaneyuki, S., Korányi, A., Lu, Q.-K., Roos, G.: Analysis and Geometry on Complex Homogeneous Domains, Progress in Mathematics. Birkhäuser, Boston, Basel, Berlin (2000)

  11. Friedman Y., Russo B.: Conditional expectation and bicontractive projections on Jordan C*-algebras and their generalisations. Math. Z. 194, 227–236 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  12. Friedman Y., Russo B.: Structure of the predual of a JBW*-triple. J. Reine Angew. Math. 356, 67–89 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  13. Friedman , Russo B.: Conditional expectation without order. Pacif. J. Math. 115, 351–360 (1984)

    MATH  MathSciNet  Google Scholar 

  14. Friedman, Y., Scarr, T.: Physical Applications of Homogeneous Balls. Progress in Mathematical Physics, vol. 40, Birkhäuser, Boston (2005)

  15. Hanche-Olsen H., Størmer E.: Jordan Operator Algebras. Pitman Advanced, Boston/London/Melbourne (1984)

    MATH  Google Scholar 

  16. Harris, L.A.: Bounded symmetric domains in infinite dimensional spaces. In: Hayden.Suffridge.inf.hol Hayden T.L., Suffridge T.J. (eds.) Proceedings on Infinite Dimensional Holomorphy, Lecture Notes in Mathematics, vol. 364(13–40), Springer Berlin/Heidelberg/New York (1974)

  17. Harris L.A.: A generalisation of C*-algebras. Proc. Lond. Math. Soc. 42, 331–361 (1981)

    Article  MATH  Google Scholar 

  18. Horn G.: Characterisation of the predual and ideal structure of a JBW*-triple . Math. Scand. 61, 117–133 (1983)

    MathSciNet  Google Scholar 

  19. Hügli R.V.: Structural projections on a JBW*-triple and GL-projections on its predual. J. Korean Math. Soc. 41(1), 107–130 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  20. Hügli R.V.: Characterizations of tripotents in JB*-triples. Math. Scand. 99, 147–160 (2006)

    MATH  MathSciNet  Google Scholar 

  21. Hügli R.V.: Collinear systems and normal contractive projection in JBW*-triples. Int. Equ. Oper. Theory 58, 315–339 (2007)

    Article  MATH  Google Scholar 

  22. Hügli R.V., Mackey M.: Transitivity of inner automorphisms on manifolds of tripotents. Math. Z 262, 125–141 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  23. Kaup W.: Contractive projections on Jordan C*-algebras and generalisations. Math. Scand. 54, 95–100 (1984)

    MATH  MathSciNet  Google Scholar 

  24. Kaup W.: A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces. Math. Z. 183, 503–529 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  25. Lewicki, G., Odyniec, W.: Minimal projections in Banach spaces. In: Springer Lecture Notes in Mathematics, vol. 1449. Springer, Berlin/Heidelberg (1987)

  26. Loos O.: Bounded Symmetric Domains and Jordan Pairs, Mathematical Lectures. University of California, Irvine (1977)

    Google Scholar 

  27. Neal M., Ricard É, Russo B.: Classification of contractively complemented Hilbertian operator spaces. J. Funct. Anal. 237, 589–616 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  28. Neal M., Russo B.: Contractive projections and operator spaces. Trans. Am. Math. Soc. 355(6), 2223–2262 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  29. Neher, E.: Jordan triple systems by the grid approach. In: Lecture Notes in Mathematics, vol. 1280. Springer, Berlin/Heidelberg/New York (1987)

  30. Satake, I.: Algebraic Structures of Symmetric Domains. Publ. Math. Soc. Japan, Kanô Memorial Lectures. Iwanami Shoten and Princeton Univ. Press (1980)

  31. Stachò L.L.: A projection principle concerning biholomorphic automorphisms. Acta Sci. Math. 44, 99–124 (1982)

    MATH  Google Scholar 

  32. Upmeier, H.: Symmetric Banach manifolds and Jordan*-algebras. In: North-Holland Math. Studies, vol. 104, p. 444 (1985)

  33. Upmeier, H.: Jordan algebras in analysis, operator theory and quantum mechanics. Providence Am. Math. Soc. 81–85 (1986)

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  1. UCD School of Physics, Condensed Matter Theory group, University College Dublin, Belfield D4, Dublin, Ireland

    Remo V. Hügli

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  1. Remo V. Hügli
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Correspondence to Remo V. Hügli.

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Partly supported by the Science Foundation of Ireland (grant no. 05/IN1/I853).

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Hügli, R.V. Contractivity of Projections Commuting with Inner Derivations on JBW*-triples. Integr. Equ. Oper. Theory 70, 101–123 (2011). https://doi.org/10.1007/s00020-010-1860-1

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