Skip to main content
SpringerLink
Log in

Hardy algebras, W*-correspondences and interpolation theory

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract.

Given a von Neumann algebra M and a W*-correspondence E over M, we construct an algebra H∞(E) that we call the Hardy algebra of E. When M= =E, H∞(E) is the classical Hardy space H∞ of bounded analytic functions on the unit disc. When M= and E= H∞(E) is the free semigroup algebra studied by Popescu, Davidson and Pitts and many others. We show that given any faithful normal representation σ of M on a Hilbert space H there is a natural correspondence Eσ over the commutant σ(M)′, called the σ-dual of E, and that H∞(E) can be realized in terms of (B(H)-valued) functions on the open unit ball ((Eσ)*) in the space of adjoints of elements in Eσ. We prove analogues of the Nevanlinna-Pick theorem in this setting and discover other aspects of the value ‘‘distribution theory’’ for elements in H∞(E). We also analyze the ‘‘boundary behavior’’ of elements in H∞(E) and obtain generalizations of the Sz.-Nagy–Foiaş functional calculus and the functional calculus of Popescu for c.n.c. row contractions. The correspondence Eσ has a dual that is naturally isomorphic to E and the commutants of certain, so-called induced representations of H∞(E) can be viewed as induced representations of H∞(Eσ). For these induced representations a double commutant theorem is proved.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anoussis, M.: Interpolating operators in nest algebras. Proc. Amer. Math. Soc. 114, 707–710 (1992)

    MathSciNet  MATH  Google Scholar 

  2. Anoussis, M., Katsoulis, E.G., Moore, R.L., Trent, T.T.: Interpolation problems for ideals in nest algebras. Math. Proc. Cambridge Philos. Soc. 111, 151–160 (1992)

    MathSciNet  MATH  Google Scholar 

  3. Arias, A., Popescu, G.: Factorization and reflexivity on Fock spaces. Integral Equations Operator Theory 23, 268–286 (1995)

    MathSciNet  MATH  Google Scholar 

  4. Arias, A., Popescu, G.: Noncommutative interpolation and Poisson transforms. Israel J. Math. 115, 205–234 (2000)

    MathSciNet  MATH  Google Scholar 

  5. Arveson, W.: Subalgebras of C*-algebras. Acta Math. 123, 141–224 (1969)

    MATH  Google Scholar 

  6. Arveson, W.: Interpolation problems in nest algebras. J. Funct. Anal. 20, 208–233 (1975)

    MATH  Google Scholar 

  7. Baillet, M., Denizeau, Y., Havet, J.-F.: Indice d’une espérance conditionnelle. Compositio Math. 66, 199–236 (1988)

    MathSciNet  MATH  Google Scholar 

  8. Ball, J., Gohberg, I.: A commutant lifting theorem for triangular matrices with diverse applications. Integral Equations Operator Theory 8, 205–267 (1985)

    MathSciNet  MATH  Google Scholar 

  9. Ball, J., Vinnikov, V.: Lax-Phillips scattering and conservative linear systems: A Cuntz-algebra multidimensional setting. Preprint

  10. Bratteli, O., Jorgensen, P.: Wavelets through a Looking Glass. Applied and Numerical Harmonic Analysis, Birkhäuser, Boston, 2002.

  11. Choi, M.-D.: Completely positive linear maps on complex matrices. Linear Algebra Appl. 10, 285–290 (1975)

    Article  Google Scholar 

  12. Davidson, K.: Nest Algebras. Vol. 191 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, 1988

  13. Davidson, K.: Free semigroup algebras: A survey. In: Systems, Approximation, Singular Integral Operators, and Related Topics (Bordeaux, 2000) (Borichev, A.A., Nikolski, N.K., Eds.), pp. 209–240. Vol. 129 of Operator Theory: Advances and Applications, Birkhäuser, Basel, 2001

  14. Davidson, K., Pitts, D.: The algebraic structure of non-commutative analytic Toeplitz algebras. Math. Ann. 311, 275–303 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. Davidson, K., Pitts, D.: Nevanlinna-Pick interpolation for non-commutative analytic Toeplitz algebras. Integral Equations Operator Theory 31, 321–337 (1998)

    MathSciNet  MATH  Google Scholar 

  16. Davidson, K., Pitts, D.: Invariant subspaces and hyper-reflexivity for free semigroup algebras. Proc. London Math. Soc. (3) 78, 401–430 (1999)

    Google Scholar 

  17. Davidson, K., Katsoulis, E., Pitts, D.: The structure of free semigroup algebras. J. Reine Angew. Math. 533, 99–125 (2001)

    MathSciNet  MATH  Google Scholar 

  18. Exel, R.: A new look at the crossed-product of a C*-algebra by an endomorphism. Preprint (arXiv:math.OA/0012084)

  19. Jo, Y.S., Kang, J.H., Moore, R.L., Trent, T.T.: Interpolation in self-adjoint settings. Proc. Amer. Math. Soc. 130, 3269–3281 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jones, V., Sunder, V.S.: Introduction to Subfactors. Vol. 234 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1997

  21. Katsoulis, E.G.: Remarks on the interpolation and the similarity problem for nest subalgebras of von Neumann algebras. J. Math. Anal. Appl. 190, 755–762 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  22. Katsoulis, E.G., Moore, R.L., Trent, T.T.: Interpolation in nest algebras and applications to operator corona theorems. J. Operator Theory 29, 115–123 (1993)

    MathSciNet  MATH  Google Scholar 

  23. Kribs, D., Power, S.: Free semigroupoid algebras. Preprint (arXiv:math.OA/0309394)

  24. Lance, E.C.: Some properties of nest algebras. Proc. London Math. Soc. (3) 19, 45–68 (1969)

    Google Scholar 

  25. Lance, E.C.: Hilbert C*-Modules: A Toolkit for Operator Algebraists. Vol. 210 in London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1994

  26. Muhly, P., Solel, B.: Tensor algebras over C*-correspondences: representations, dilations, and C*-envelopes. J. Funct. Anal. 158, 389–457 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  27. Muhly, P., Solel, B.: Tensor algebras, induced representations, and the Wold decomposition. Canad. J. Math. 51, 850–880 (1999)

    MathSciNet  MATH  Google Scholar 

  28. Muhly, P., Solel, B.: On the Morita equivalence of tensor algebras. Proc. London Math. Soc. (3) 81, 113–168 (2000)

    Google Scholar 

  29. Muhly, P., Solel, B.: Quantum Markov processes (correspondences and dilations). Internat. J. Math. 13, 863–906 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  30. Paschke, W.: Inner product modules over B*-algebras. Trans. Amer. Math. Soc. 182, 443–468 (1973)

    MATH  Google Scholar 

  31. Paulsen, V.: Completely Bounded Maps and Dilations. Vol. 146 in Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, 1986

  32. Popescu, G.: Isometric dilations for infinite sequences of noncommuting operators. Trans. Amer. Math. Soc. 316, 523–536 (1989)

    MathSciNet  MATH  Google Scholar 

  33. Popescu, G.: von Neumann inequality for Math. Scand. 68, 292–304 (1991)

    Google Scholar 

  34. Popescu, G.: Multi-analytic operators on Fock spaces. Math. Ann. 303, 31–46 (1995)

    MathSciNet  MATH  Google Scholar 

  35. Popescu, G.: Functional calculus for noncommuting operators. Michigan Math. J. 42, 345–356 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  36. Popescu, G.: Non-commutative disc algebras and their representations. Proc. Amer. Math. Soc. 124, 2137–2148 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  37. Popescu, G.: Interpolation problems in several variables. J. Math. Anal. Appl. 227, 227–250 (1998)

    Article  MathSciNet  Google Scholar 

  38. Popescu, G.: Poisson transforms on some C*-algebras generated by isometries. J. Funct. Anal. 161, 27–61 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  39. Popescu, G.: Commutant lifting, tensor algebras, and functional calculus. Proc. Edinburgh Math. Soc. (2) 44, 389–406 (2001)

    Google Scholar 

  40. Popescu, G.: Multivariable Nehari problem and interpolation. J. Funct. Anal. 200, 536–581 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  41. Power, S.C.: An elementary approach to the matricial Nevanlinna-Pick interpolation criterion. Proc. Edinburgh Math. Soc. (2) 32, 121–126 (1989)

    Google Scholar 

  42. Rieffel, M.: Induced representations of C*-algebras. Adv. Math. 13, 176–257 (1974)

    MATH  Google Scholar 

  43. Rieffel, M.: Morita equivalence for C*-algebras and W*-algebras. J. Pure Appl. Algebra 5, 51–96 (1974)

    Article  MATH  Google Scholar 

  44. Rosenblum, M., Rovnyak, J.: Hardy Classes of Operator Theory. Dover Publications, Mineola, NY, 1997

  45. Sarason, D.: Generalized interpolation in H∞. Trans. Amer. Math. Soc. 127 179–203 (1967)

    Google Scholar 

  46. Skeide, M.: Commutants of von Neumann modules, representations of (E) and other topics related to product systems of Hilbert modules. In: Advances in Quantum Dynamics (Price, G., et al., Eds.), pp. 253–262. Vol. 335 of Contemporary Mathematics, American Mathematical Society, Providence, 2003

  47. Skeide, M.: von Neumann modules, intertwiners and self-duality. Preprint, Bangalore, Jul 2003 (arXiv:math.OA/0308230), to appear in J. Operator Theory

  48. Sz.-Nagy, B., Foiaş, C.: Harmonic Analysis of Operators on Hilbert Space. North-Holland, Amsterdam, 1970

  49. Sz.-Nagy, B., Korányi, A.: Relations d’un problème de Nevanlinna et Pick avec la théorie des opérateurs de l’espace hilbertien. Acta Math. Acad. Sci. Hungar. 7, 295–303 (1956)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Iowa, Iowa City, IA, 52242, USA

    Paul S. Muhly

  2. Department of Mathematics, Technion, 32000, Haifa, Israel

    Baruch Solel

Authors
  1. Paul S. Muhly
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Baruch Solel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paul S. Muhly.

Additional information

Supported in part by grants from the National Science Foundation and from the U.S.-Israel Binational Science Foundation.

Supported in part by the U.S.-Israel Binational Science Foundation and by the Fund for the Promotion of Research at the Technion.

Revised version: 11 March 2004

Rights and permissions

Reprints and permissions

About this article

Cite this article

Muhly, P., Solel, B. Hardy algebras, W*-correspondences and interpolation theory. Math. Ann. 330, 353–415 (2004). https://doi.org/10.1007/s00208-004-0554-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-004-0554-x

Keywords

Search

Navigation