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.
@article{bwmeta1.element.bwnjournal-article-bcpv43i1p135bwm,
author = {Cipriani, Fabio},
title = {The variational approach to the Dirichlet problem in C*-algebras},
journal = {Banach Center Publications},
volume = {43},
year = {1998},
pages = {135-146},
zbl = {0953.46036},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv43i1p135bwm}
}
Cipriani, Fabio. The variational approach to the Dirichlet problem in C*-algebras. Banach Center Publications, Tome 43 (1998) pp. 135-146. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv43i1p135bwm/
[000] [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. | Zbl 0287.46074
[001] [Bre] H. Brezis, Analyse Fonctionnelle, Masson S.A., Paris, Milan, Barcelone 1993.
[002] [Cip] F. Cipriani, Dirichlet forms and Markovian semigroups on standard forms of von Neumann algebras, J. Funct. Anal. 147 (1997), 259-300. | Zbl 0883.47031
[003] [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. | Zbl 0287.46078
[004] [Con2] A. Connes, Noncommutative Geometry, Academic Press, London New York San Francisco, 1994.
[005] [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. | Zbl 0795.46047
[006] [Haa] U. Haagerup, The standard form of a von Neumann algebra, Math. Scand. 37 (1975), 271-283.
[007] [Ioc] B. Iochum, Cones autopolaires et algèbres de Jordan, Lecture Notes in Mathematics 1049, Springer-Verlag, Berlin-Heidelberg-New York, 1984. | Zbl 0556.46040
[008] [Ped] G. K. Pedersen, C*-Algebras and their Automorphism Groups, Academic Press, London New York San Francisco, 1979.
[009] [Ric] C. E. Rickart, General Theory of Banach Algebras, D. Van Nostrand Company, Inc., Princeton, New Jersey, Toronto New York London, 1960.
[010] [Sau1] J.-L. Sauvageot, Le problème de Dirichlet dans les C*-algèbres, J. Funct. Anal. 101 (1991), 50-73.
[011] [Sau2] J.-L. Sauvageot, Markov quantum semigroups admits covariant Markov C*-dilations, Comm. Math. Phys. 106 (1986), 91-103. | Zbl 0606.60079
[012] [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.
[013] [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.