Analysis Mathematica

, Volume 43, Issue 4, pp 603–627 | Cite as

Noncommutative potential theory

  • L. LempertEmail author


We propose to view hermitian metrics on trivial holomorphic vector bundles E → Ω as noncommutative analogs of functions defined on the base Ω, and curvature as the notion corresponding to the Laplace operator or ∂∂̅. We discuss noncommutative generalizations of basic results of ordinary potential theory, mean value properties, maximum principle, Harnack inequality, and the solvability of Dirichlet problems.


curvature of hermitian metrics maximum principle mean value property Dirichlet problem 

Mathematics Subject Classification

31C05 31C10 47A56 47A68 


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  1. [1]
    S. Axelrod, S. Della Pietra and E. Witten, Geometric quantization of Chern–Simons gauge theory, J. Diff. Geom., 33 (1991), 787–902.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    R. Berman and J. Keller, Bergman Geodesics, in: Complex Monge–Ampère Equations and Geodesics in the Space of Kähler Metrics, Lecture Notes in Math., Vol. 2038, Springer (Heidelberg, 2012), pp. 283–302.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    B. Berndtsson, Curvature of vector bundles associated to holomorphic fibrations. Ann. of Math. (2), 169 (2009), 531–160.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    R. R. Coifman and S. Semmes, Interpolation of Banach spaces, Perron processes, and Yang–Mills, Amer. Math. J., 115 (1993), 243–278.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    J. Conway, A Course in Functional Analysis, 2nd ed., Springer (New York, 1990).zbMATHGoogle Scholar
  6. [6]
    A. Devinatz, The factorization of operator valued functions, Ann. of Math., 73 (1961), 458–495.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    S. Donaldson, Boundary value problems for Yang–Mills fields, J. Geom. Phys., 8 (1992), 89–122.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    R. G. Douglas, On factoring positive operator functions, J. Math. Mech., 16 (1966), 119–126.MathSciNetzbMATHGoogle Scholar
  9. [9]
    H. Helson, Lectures on Invariant Subspaces, Academic Press (New York, 1964).zbMATHGoogle Scholar
  10. [10]
    R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd ed., Cambridge University Press (New York, N.Y., 2013).Google Scholar
  11. [11]
    L. Lempert, La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. France, 109 (1981), 427–474.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    L. Lempert, A maximum principle for Hermitian (and other) metrics, Proc. Amer. Math. Soc., 143 (2015), 2193–2200.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    L. Lempert, Extrapolation, a technique to estimate, in: Functional Analysis, Harmonic Analysis, and Image Processing: a Collection of Papers in Honor of Björn Jawerth, Contemp. Math., 693, Amer. Math. Soc. (Providence, RI, 2017), pp. 271–281.MathSciNetGoogle Scholar
  14. [14]
    L. Lempert, Analytic cohomology groups of infinite dimensional complex manifolds, J. Math. Anal. Appl., 445 (2017), 1428–1446.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    L. Lempert and R. Szőke, Direct images, fields of Hilbert spaces, and geometric quantization, Comm. Math. Phys., 327 (2014), 49–99.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    E. H. Lieb and M. B. Ruskai, Some operator inequalities of the Schwarz type, Adv. Math., 12 (1974), 269–273.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    I. Privalov, Sur les fonctions conjugées, Bull. Soc. Math. France, 44 (1916), 100–103.MathSciNetGoogle Scholar
  18. [18]
    C. E. Rickart, General Theory of Banach Algebras, The University Series in Higher Mathematics, D. van Nostrand (Princeton, N.J.–Toronto–London–New York, 1960).Google Scholar
  19. [19]
    F. Riesz and B. Sz. -Nagy, Le¸cons d’analyse fonctionelle, 4th ed., Gauthiers–Villars (Paris, 1965).Google Scholar
  20. [20]
    R. Rochberg, Interpolation of Banach spaces and negatively curved vector bundles, Pacific J. Math., 110 (1984), 355–376.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    M. Rosenblum, Vectorial Toeplitz operators and the Fejér–Riesz theorem, J. Math. Anal. Appl., 23 (1968), 139–147.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    B. Sz. -Nagy and C. Foiaş, Sur les contractions de l’espace de Hilbert IX. Factorisation de la fonction caractéristique. Sous–espaces invariants, Acta Sci. Math. (Szeged), 25 (1964), 283–316.MathSciNetzbMATHGoogle Scholar
  23. [23]
    N. Wiener and P. Masani, The prediction theory of multivariate stochastic processes, Acta Math., 98 (1957), 111–150.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2017

Authors and Affiliations

  1. 1.Department of MathematicsPurdue UniversityWest LafayetteUSA

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