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Dilation Theory Yesterday and Today

  • William Arveson
Part of the Operator Theory Advances and Applications book series (OT, volume 207)

Abstract

Paul Halmos’ work in dilation theory began with a question and its answer: Which operators on a Hilbert space H can be extended to normal operators on a larger Hilbert space KH? The answer is interesting and subtle.

The idea of representing operator-theoretic structures in terms of conceptually simpler structures acting on larger Hilbert spaces has become a central one in the development of operator theory and, more generally, non-commutative analysis. The work continues today. This article summarizes some of these diverse results and their history.

Mathematics Subject Classification (2000)

46L07 

Keywords

Dilation theory completely positive linear maps 

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References

  1. [Arv69]
    W. Arveson. Subalgebras of C*-algebras. Acta Math., 123:141–224, 1969.CrossRefzbMATHMathSciNetGoogle Scholar
  2. [Arv72]
    W. Arveson. Subalgebras of C*-algebras II. Acta Math., 128:271–308, 1972.CrossRefzbMATHMathSciNetGoogle Scholar
  3. [Arv01]
    W. Arveson. A Short Course on Spectral Theory, volume 209 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2001.Google Scholar
  4. [Arv02]
    W. Arveson. Generators of noncommutative dynamics. Erg. Th. Dyn. Syst., 22:1017–1030, 2002. arXiv:math.OA/0201137.CrossRefzbMATHMathSciNetGoogle Scholar
  5. [Arv03]
    W. Arveson. Noncommutative Dynamics and E-semigroups. Monographs in Mathematics. Springer-Verlag, New York, 2003.Google Scholar
  6. [Bha99]
    B.V.R. Bhat. Minimal dilations of quantum dynamical semigroups to semigroups of endomorphisms of C*-algebras. J. Ramanujan Math. Soc., 14(2): 109–124, 1999.zbMATHMathSciNetGoogle Scholar
  7. [BLM04]
    D. Blecher and C. Le Merdy. Operator algebras and their modules, volume 30 of LMS Monographs. Clarendon Press, Oxford, 2004.Google Scholar
  8. [BP94]
    B.V.R. Bhat and K.R. Parthasarathy. Kolmogorov’s existence theorem for Markov processes in C*-algebras. Proc. Indian Acad. Sci. (Math. Sci.), 104:253–262, 1994.CrossRefzbMATHMathSciNetGoogle Scholar
  9. [Bra55]
    J. Bram. Subnormal operators. Duke Math. J., 22(1):75–94, 1955.CrossRefzbMATHMathSciNetGoogle Scholar
  10. [Dix64]
    J. Dixmier. Les C*-algèbres et leurs Représentations. Gauthier—Villars, Paris, 1964.Google Scholar
  11. [ER00]
    E.G. Effros and Z.-J. Ruan. Operator Spaces. Oxford University Press, Oxford, 2000.Google Scholar
  12. [Hal50]
    P.R. Halmos. Normal dilations and extensions of operators. Summa Brasil., 2:125–134, 1950.MathSciNetGoogle Scholar
  13. [Hal67]
    P. Halmos. A Hilbert space problem book. Van Nostrand, Princeton, 1967.Google Scholar
  14. [MS02]
    P. Muhly and B. Solel. Quantum Markov processes (correspondences and dilations). Int. J. Math., 13(8):863–906, 2002.CrossRefzbMATHMathSciNetGoogle Scholar
  15. [Nai43a]
    M.A. Naimark. On the representation of additive operator set functions. C.R. (Dokl.) Acad. Sci. URSS, 41:359–361, 1943.Google Scholar
  16. [Nai43b]
    M.A. Naimark. Positive definite operator functions on a commutative group. Bull. (Izv.) Acad. Sci. URSS (Ser. Math.), 7:237–244, 1943.zbMATHGoogle Scholar
  17. [Par91]
    K.R. Parthasarathy. An introduction to quantum stochastic calculus, volume I. Birkhäuser Verlag, Basel, 1991.Google Scholar
  18. [Pau86]
    V. Paulsen. Completely bounded maps and dilations. Wiley, New York, 1986.Google Scholar
  19. [Pau02]
    V. Paulsen. Completely bounded maps and operator algebras. Cambridge University Press, Cambridge, UK, 2002.Google Scholar
  20. [SN53]
    B. Sz.-Nagy. Sur les contractions de l’espace de Hilbert. Acta Sci. Math. Szeged, 15:87–92, 1953.zbMATHMathSciNetGoogle Scholar
  21. [SNF70]
    B. Sz.-Nagy and C. Foias. Harmonie analysis of operators on Hilbert space. American Elsevier, New York, 1970.Google Scholar
  22. [Sti55]
    W.F. Stinespring. Positive functions on C*-algebras. Proc. Amer. Math. Soc., 6:211–216, 1955.zbMATHMathSciNetGoogle Scholar
  23. [Szf77]
    F.H. Szfraniec. Dilations on involution semigroups. Proc. Amer. Math. Soc., 66(1):30–32, 1977.CrossRefMathSciNetGoogle Scholar
  24. [vN51]
    J. von Neumann. Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes. Math. Nachr., 4:258–281, 1951.zbMATHMathSciNetGoogle Scholar
  25. [Wit81]
    G. Wittstock. Ein operatorwertiger Hahn-Banach Satz. J. Funct. Anal., 40:127–150, 1981.CrossRefzbMATHMathSciNetGoogle Scholar
  26. [Wit84]
    G. Wittstock. On matrix order and convexity. In Functional Analysis: Surveys and recent results, volume 90 of Math. Studies, pages 175–188. North-Holland, 1984.Google Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • William Arveson
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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