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A Beurling-Blecher-Labuschagne Theorem for Noncommutative Hardy Spaces Associated with Semifinite von Neumann Algebras

Abstract

We prove a Beurling-Blecher-Labuschagne theorem for \({H^\infty}\)-invariant spaces of \({L^p(\mathcal{M},\tau)}\) when \({0 < p \leq\infty}\), using Arveson’s non-commutative Hardy space \({H^\infty}\) in relation to a von Neumann algebra \({\mathcal{M}}\) with a semifinite, faithful, normal tracial weight \({\tau}\). Using the main result, we are able to completely characterize all \({H^\infty}\)-invariant subspaces of \({L^p(\mathcal{M} \rtimes_\alpha \mathbb{Z},\tau)}\), where \({\mathcal{M} \rtimes_\alpha \mathbb{Z} }\) is a crossed product of a semifinite von Neumann algebra \({\mathcal{M}}\) by the integer group \({\mathbb{Z}}\), and \({H^\infty}\) is a non-selfadjoint crossed product of \({\mathcal{M}}\) by \({\mathbb{Z}^+}\). As an example, we characterize all \({H^\infty}\)-invariant subspaces of the Schatten p-class \({S^p(\mathcal{H})}\), where \({H^\infty}\) is the lower triangular subalgebra of \({B(\mathcal{H})}\), for each \({0 < p \leq\infty}\).

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References

  1. 1

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

    MathSciNet  Article  MATH  Google Scholar 

  2. 2

    Bekjan T.N.: Noncommutative Hardy space associated with semi-finite subdiagonal algebras. J. Math. Anal. Appl. 429(2), 1347–1369 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3

    Bekjan T.N.: Noncommutative symmetric Hardy spaces. Integral Equ. Oper. Theory 81, 191–212 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4

    Bekjan T.N., Xu Q.: Riesz and Szegö type factorizations for noncommutative Hardy spaces. J. Oper. Theory 62, 215–231 (2009)

    MathSciNet  MATH  Google Scholar 

  5. 5

    Beurling A.: On two problems concerning linear transformations in Hilbert space. Acta Math. 81, 239–255 (1949)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6

    Blecher D., Labuschagne L.E.: A Beurling theorem for noncommutative L p. J. Operator Theory 59, 29–51 (2008)

    MathSciNet  MATH  Google Scholar 

  7. 7

    Bochner S.: Generalized conjugate and analytic functions without expansions. Proc. Nat. Acad. Sci. USA 45, 855–857 (1959)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8

    Davidson, K.: Nest Algebras: Triangular Forms for Operator Algebras on Hilbert Space. Longman Scientific & Technical (1988)

  9. 9

    Dodds, P., Dodds, T.: Some properties of symmetric operator spaces, Proc. Centre Math. Appl. Austral. Nat. Univ., 29, Austral. Nat. Univ., Canberra (1992)

  10. 10

    Dodds P., Dodds T., Pagter B.: Noncommutative Banach function spaces. Mathematische Zeitschrift 201, 583–597 (1989)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11

    Dodds P., Dodds T., Pagter B.: Noncommutative Köthe duality. Trans. Am. Math. Soc. 339, 717–750 (1993)

    MathSciNet  MATH  Google Scholar 

  12. 12

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

    MathSciNet  Article  MATH  Google Scholar 

  13. 13

    Fack T., Kosaki H.: Generalized s-numbers of \({\tau}\)-measurable operators. Pac. J. Math. 2, 269–300 (1986)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14

    Fang J., Hadwin D., Nordgren E., Shen J.: Tracial gauge norms on finite von Neumann algebras satisfying the weak Dixmier property. J. Funct. Anal. 255, 142–183 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15

    Halmos P.: Shifts on Hilbert spaces. J. Reine Angew. Math. 208, 102–112 (1961)

    MathSciNet  MATH  Google Scholar 

  16. 16

    Helson H.: Lectures on Invariant Subspaces. Academic Press, New York (1964)

    MATH  Google Scholar 

  17. 17

    Helson H., Lowdenslager D.: Prediction theory and Fourier series in several variables. Acta Math. 99, 165–202 (1958)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18

    Hoffman K.: Analytic functions and logmodular Banach algebras. Acta Math. 108, 271–317 (1962)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19

    Kadison, R., Ringrose J.: Fundamentals of the Theory of Operator Algebras. Advanced Theory, vol. 2. Academic Press, Inc. (1986)

  20. 20

    Kunze R.A.: L p-Fourier transforms on locally compact unimodular groups. Trans. Am. Math. Soc. 89, 519–540 (1958)

    MathSciNet  MATH  Google Scholar 

  21. 21

    Ji G.: Maximality of semi-finite subdiagonal algebras. J. Shaanxi Normal Univ. Sci. Ed. 28, 15–17 (2000)

    MathSciNet  MATH  Google Scholar 

  22. 22

    Junge M., Sherman D.: Noncommutative L p-modules. J. Oper. Theory 53, 3–34 (2005)

    MathSciNet  MATH  Google Scholar 

  23. 23

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

    MathSciNet  MATH  Google Scholar 

  24. 24

    McCarthy C.A.: C p . Israel J. Math. 5, 249–271 (1967)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25

    McAsey, M., Muhly, P., Saito, K.: Nonselfadjoint crossed products (invariant subspaces and maximality). Trans. Am. Math. Soc. 248(2), 381–409 (1979)

  26. 26

    Nakazi T., Watatani Y.: Invariant subspace theorems for subdiagonal algebras. J. Oper. Theory 37, 379–395 (1997)

    MathSciNet  MATH  Google Scholar 

  27. 27

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

    Article  MATH  Google Scholar 

  28. 28

    Neumann J.: Some matrix-inequalities and metrization of matric-space. Tomsk Univ. Rev. 1, 286–300 (1937)

    MATH  Google Scholar 

  29. 29

    Pisier, G., Xu, Q. Noncommutative Lp-spaces. Handbook of the geometry of Banach spaces, pp. 1459–1517. Amsterdam, 2 (2003)

  30. 30

    Sakai, S.: C*-algebras and W*-algebras. Springer, New York (1971)

  31. 31

    Saito K.S.: A note on invariant subspaces for finite maximal subdiagonal algebras. Proc. Am. Math. Soc. 77, 348–352 (1979)

    MathSciNet  Article  MATH  Google Scholar 

  32. 32

    Saito K.S.: A simple approach to the invariant subspace structure of analytic crossed products. J. Oper. Theory 27(1), 169–177 (1992)

    MathSciNet  MATH  Google Scholar 

  33. 33

    Segal I.: A noncommutative extension of abstract integration. Ann. Math. 57, 401–457 (1952)

    Article  MATH  Google Scholar 

  34. 34

    Simon, B.: Trace ideals and their applications, London Mathematical Society Lecture Note Series, vol. 35, Cambridge University Press, Cambridge (1979)

  35. 35

    Srinivasan T.P.: Simply invariant subspaces. Bull. Am. Math. Soc. 69, 706–709 (1963)

    MathSciNet  Article  MATH  Google Scholar 

  36. 36

    Srinivasan, T., Wang, J.K.: Weak*-Dirichlet algebras. Proceedings of the International Symposium on Function Algebras, Tulane University, 1965 (Chicago), Scott-Foresman, 216–249 (1966)

  37. 37

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

  38. 38

    Xu Q.: On the maximality of subdiagonal algebras. J. Oper. Theory 54(1), 137–146 (2005)

    MathSciNet  MATH  Google Scholar 

  39. 39

    Xu, Q.: Operator spaces and noncommutative Lp, The part on noncommutative Lp-spaces. Lectures in the Summer School on Banach spaces and Operator spaces, Nankai University China, July 16–20 (2007)

  40. 40

    Yeadon F.: Noncommutative L p-spaces. Math. Proc. Cambridge Philos. Soc. 77, 91–102 (1975)

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Lauren B. M. Sager.

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Sager, L.B.M. A Beurling-Blecher-Labuschagne Theorem for Noncommutative Hardy Spaces Associated with Semifinite von Neumann Algebras. Integr. Equ. Oper. Theory 86, 377–407 (2016). https://doi.org/10.1007/s00020-016-2308-z

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Mathematics Subject Classification

  • Primary 32A35
  • 46L52
  • Secondary 47A15
  • 47L65

Keywords

  • Beurling theorem
  • Schatten p-classes
  • Semifinite von Neuman algebra
  • Non-commutative Hardy space
  • Crossed products of von Neumann algebras
  • \({L^p}\) spaces
  • Invariant subspaces