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

Fabio Cipriani

Banach Center Publications (1998)

  • Volume: 43, Issue: 1, page 135-146
  • ISSN: 0137-6934

Abstract

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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.

How to cite

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Cipriani, Fabio. "The variational approach to the Dirichlet problem in C*-algebras." Banach Center Publications 43.1 (1998): 135-146. <http://eudml.org/doc/208832>.

@article{Cipriani1998,
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.},
author = {Cipriani, Fabio},
journal = {Banach Center Publications},
keywords = {maximum principle; weak solution; noncommutative Dirichlet problem; sub-Markovian semigroup; KMS condition; Dirichlet forms on standard von Neumann algebras},
language = {eng},
number = {1},
pages = {135-146},
title = {The variational approach to the Dirichlet problem in C*-algebras},
url = {http://eudml.org/doc/208832},
volume = {43},
year = {1998},
}

TY - JOUR
AU - Cipriani, Fabio
TI - The variational approach to the Dirichlet problem in C*-algebras
JO - Banach Center Publications
PY - 1998
VL - 43
IS - 1
SP - 135
EP - 146
AB - 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.
LA - eng
KW - maximum principle; weak solution; noncommutative Dirichlet problem; sub-Markovian semigroup; KMS condition; Dirichlet forms on standard von Neumann algebras
UR - http://eudml.org/doc/208832
ER -

References

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  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. Zbl0287.46078
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  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. Zbl0606.60079
  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. 

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