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On peak phenomena for non-commutative H ∞

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Abstract

A non-commutative extension of Amar and Lederer’s peak set result is given. As its simple applications it is shown that any non-commutative H ∞-algebra H ∞(M, τ) has unique predual, and moreover some restriction in some of the results of Blecher and Labuschagne are removed, making them hold in full generality.

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References

  1. Akemann C.A.: The general Stone-Weierstrass problem. J. Funct. Anal. 4, 277–294 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  2. Akemann C.A.: Left ideal structure of C *-algebras. J. Funct. Anal. 6, 305–317 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  3. Amar E., Lederer A.: Points exposés de la boúle unitié de H ∞(D). C. R. Acad. Sci. Paris Ser A 272, 1449–1452 (1971)

    MATH  MathSciNet  Google Scholar 

  4. Ando T.: On the predual of H ∞. Comment. Math. Spec. issue 1, 33–40 (1978)

    MathSciNet  Google Scholar 

  5. Arveson W.B.: Analyticity in operator algebras. Amer. J. Math. 89, 578–642 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  6. Barbey K.: Ein Satz über abstrakte analytische Funktionen. Arch. Math. (Basel) 26, 521–527 (1975)

    MATH  MathSciNet  Google Scholar 

  7. Blecher D.P., Labuschagne L.E.: Noncommutative function theory and unique extensions. Stud. Math. 178, 177–195 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  8. Blecher, D.P., Labuschagne, L.E.: von Neumann algebraic H p theory in Function spaces, Contemp. Math., pp. 89–114, vol. 435, American Mathematical Society, Providence (2007)

  9. Chaumat J.: Unicité du prédual. C. R. Acad. Sci. Paris 288, 411–414 (1979)

    MATH  MathSciNet  Google Scholar 

  10. Dixmier J.: Formes linéaires sur un anneau d’opérateurs. Bull. Soc. Math. France 81, 9–39 (1953)

    MATH  MathSciNet  Google Scholar 

  11. Exel R.: Maximal subdiagonal algebras. Am. J. Math. 110, 775–782 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  12. Foguel S.R.: Power of contraction in Hilbert space. Pacific J. Math. 13, 551–562 (1963)

    MATH  MathSciNet  Google Scholar 

  13. Godefroy G.: Sous-espaces bien disposes de L 1-applications. Trans. Am. Math. Soc. 286, 227–249 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  14. Godefroy G., Talagrand M.: Classes d’espaces de Banach à prédual unique. C. R. Acad. Sci. Paris Ser. I Math. 292, 323–325 (1981)

    MATH  MathSciNet  Google Scholar 

  15. Grothendieck A.: Une caractérisation vectorielle-mérique des espaces L1. Canad. J. Math. 7, 552–561 (1955)

    MATH  MathSciNet  Google Scholar 

  16. Hay D.M.: Closed projections and peak interpolation for operator algebras. Int. Equ. Oper. Theory 57, 491–512 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  17. Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces. II. Function spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 97, pp. 1979 x+243. Springer, Berlin

  18. Marsalli M., West G.: Noncommutative H p spaces. J. Oper. Theory 40, 339–355 (1998)

    MATH  MathSciNet  Google Scholar 

  19. Pełczyński A.: Banach spaces on which every unconditionally converging operator is weakly compact. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. Phys. 10, 641–648 (1962)

    MATH  Google Scholar 

  20. Pełczyński, A.: Banach spaces of analytic functions and absolutely summing operators, CBMS Regional Conference Series in Mathematics, vol. 30. American Mathematical Society, Providence (1977)

  21. Pfitzner H.: L-summands in their biduals have Pełczyński’s property (V*). Stud. Math. 104, 91–98 (1963)

    MathSciNet  Google Scholar 

  22. Pfitzner H.: Separable L-embedded Banach spaces are preduals. Bull. Lond. Math. Soc. 39, 1039–1044 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  23. Pisier, G., Xu, Q.: Non-commutative L p-spaces. In: Johnson, W.B., Lindenstrauss, J. (eds.) Handbook on Banach spaces, vol. 2. North-Holland, Amsterdam (2003)

  24. Randrianantoanina N.: Hilbert transform associated with finite maximal subdiagonal algebras. J. Austral. Math. Soc. Ser. A 65(3), 388–404 (1998)

    MATH  MathSciNet  Google Scholar 

  25. Sakai S.: A characterization of W *-algebras. Pacific J. Math. 6, 763–773 (1956)

    MATH  MathSciNet  Google Scholar 

  26. Sakai S.: On linear functionals of W *-algebras. Proc. Jpn. Acad. 34, 571–574 (1958)

    MATH  Google Scholar 

  27. Sewatari S.: Master thesis, Kyushu University (2007)

  28. Strătilă Ş., Zsidó L.: Lectures on von Neumann Algebras. Abacus Press, Tunbridge Wells (1979)

    MATH  Google Scholar 

  29. Takesaki M.: On the conjugate space of operator algebras. Tohôku Math. Jour. 10, 194–203 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  30. Takesaki M.: On the singularity of a positive linear functional on operator algebra. Proc. Jpn. Acad. 30, 365–366 (1959)

    MathSciNet  Google Scholar 

  31. Tomita M.: Spectral theory of operator algebras, I. Math. J. Okayama Univ. 9, 63–98 (1959)

    MathSciNet  Google Scholar 

  32. Wojtaszczyk P.: On projections in spaces of bounded analytic functions with applications. Stud. Math. 65, 147–173 (1979)

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Graduate School of Mathematics, Kyushu University, Fukuoka, 810-8560, Japan

    Yoshimichi Ueda

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  1. Yoshimichi Ueda
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Correspondence to Yoshimichi Ueda.

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Dedicated to Professor Fumio Hiai on the occasion of his 60th birthday.

Yoshimichi Ueda was supported in part by Grant-in-Aid for Young Scientists (B)17740096.

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Ueda, Y. On peak phenomena for non-commutative H ∞ . Math. Ann. 343, 421–429 (2009). https://doi.org/10.1007/s00208-008-0277-5

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  • DOI: https://doi.org/10.1007/s00208-008-0277-5

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