Download PDF - The variational approach to the Dirichlet problem in C*-algebrasAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

The variational approach to the Dirichlet problem in C*-algebras

Authors 1

Affiliations

  1. Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy

Abstract

The aim of this work is to develop the variational approach to the Dirichlet problem for generators of sub-Markovian semigroups on C*-algebras. KMS symmetry and the KMS condition allow the introduction of the notion of weak solution of the Dirichlet problem. We will then show that a unique weak solution always exists and that a generalized maximum principle holds true.

Bibliography

  1. [Ara] H. Araki, Some properties of the modular conjugation operator of von Neumann algebras and noncommutative Radon-Nikodym theorem with chain rule, Pacific J. Math. 50 (1974), 309-354.
  2. [Bre] H. Brezis, Analyse Fonctionnelle, Masson S.A., Paris, Milan, Barcelone 1993.
  3. [Cip] F. Cipriani, Dirichlet forms and Markovian semigroups on standard forms of von Neumann algebras, J. Funct. Anal. 147 (1997), 259-300.
  4. [Con1] A. Connes, Caractérisation des espaces vectoriels ordonnés sous jacents aux algèbres de von Neumann, Ann. Inst. Fourier (Grenoble) 24 No. 4 (1974), 121-155.
  5. [Con2] A. Connes, Noncommutative Geometry, Academic Press, London New York San Francisco, 1994.
  6. [GL] S. Goldstein and J. M. Lindsay, Beurling-Deny conditions for KMS-symmetric dynamical semigroups, C. R. Acad. Sci. Paris, Ser. I 317 (1993), 1053-1057.
  7. [Haa] U. Haagerup, The standard form of a von Neumann algebra, Math. Scand. 37 (1975), 271-283.
  8. [Ioc] B. Iochum, Cones autopolaires et algèbres de Jordan, Lecture Notes in Mathematics 1049, Springer-Verlag, Berlin-Heidelberg-New York, 1984.
  9. [Ped] G. K. Pedersen, C*-Algebras and their Automorphism Groups, Academic Press, London New York San Francisco, 1979.
  10. [Ric] C. E. Rickart, General Theory of Banach Algebras, D. Van Nostrand Company, Inc., Princeton, New Jersey, Toronto New York London, 1960.
  11. [Sau1] J.-L. Sauvageot, Le problème de Dirichlet dans les C*-algèbres, J. Funct. Anal. 101 (1991), 50-73.
  12. [Sau2] J.-L. Sauvageot, Markov quantum semigroups admits covariant Markov C*-dilations, Comm. Math. Phys. 106 (1986), 91-103.
  13. [Sau3] J.-L. Sauvageot, First exit time: A theory of stopping times in quantum processes, in: Quantum Probability and Applications III, Lect. Notes in Math. 1303, Springer-Verlag, New York, 1988.
  14. [Sau4] J.-L. Sauvageot, Semi-groupe de la chaleur transverse sur la C*-algèbre d'un feuilletage riemannien, J. Funct. Anal. 142 (1996), 511-538.
Banach Center Publications
1998/43/1
Export to PBN, Opens in new tab
Pages:
135-146
Main language of publication
English
Published
1998
Exact and natural sciences