Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-13T06:45:12.838Z Has data issue: false hasContentIssue false
  • English
  • Français

Toeplitz Algebras on Strongly Pseudoconvex Domains

Published online by Cambridge University Press:  11 January 2016

Guangfu Cao
Guangfu Cao*
Affiliation:
Department of Mathematics, Guangzhou University, Guangzhou 510006, China, guangfucao@163.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In the present paper, it is proved that the K0-group of a Toeplitz algebra on any strongly pseudoconvex domain is always isomorphic to the K0-group of the relative continuous function algebra, and is thus isomorphic to the topological K0-group of the boundary of the relative domain. Further there exists a ring isomorphism between the K0-groups of Toeplitz algebras and the Chern classes of the relative boundaries of strongly pseudoconvex domains. As applications of our main result, K-groups of Toeplitz algebras on some special strongly pseudoconvex domains are computed. Our results show that the Toeplitz algebras on strongly pseudoconvex domains have rich structures, which deeply depend on the topological structures of relative domains. In addition, the first cohomology groups of Toeplitz algebras are also computed.

Keywords

47B35
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 185 , 2007 , pp. 171 - 186
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2007

References

[1] Douglas, R. G., Banach algebra techniques in operator theory, Academic Press, 1972.Google Scholar
[2] Axler, S., Conway, J. B. and McDonald, G., Toeplitz operators on Bergman spaces, Can. J. Math., XXXIV (1982), no. 2, 466–483.Google Scholar
[3] Cao, G. F., Composition and Toeplitz operators on general domains, Tohoku Math. J., 57 (2005), 11–22.CrossRefGoogle Scholar
[4] Cao, G. F., Cohomotopy groups and K0-groups on connected domains, preprint.Google Scholar
[5] Coburn, L. A., Singular integral operators and Toeplitz operators on odd spheres, Indiana Univ. Math. J., 23 (1973), 433–439.Google Scholar
[6] Venugopalkrishna, U., Fredholm operators associated with strongly pseudoconvex domains, J. Funct. Anal., 9 (1972), 349–373.Google Scholar
[7] Upmeier, H., Toeplitz operators and index theory in several complex variables, Proceedings of Symposia in pure mathematics (Part I), 51 (1990), 585–598.Google Scholar
[8] Salinas, N. A. Sheu and Upmeier, H., Toeplitz operators on pseudoconvex domains and foliation C*-algebras, Ann. of Math., 120 (1989), 531–565.Google Scholar
[9] Jewell, N. P. and Krantz, S. G., Toeplitz operators and related function algebras on certain pseudoconvex domains, Trans. Amer. Math. Soc., 252 (1979), 297–312.Google Scholar
[10] Raeburn, I., On Toeplitz operators associated with strongly pseudoconvex domains, Studia Math., 63 (1979), 253–258.Google Scholar
[11] Jewell, N. P., Fredholm Toeplitz operators on strongly pseudoconvex domains, Studia Math., T.LXVIII (1980), 25–34.Google Scholar
[12] Janas, J., Toeplitz operators related to certain domains in ℂn , Studia Math., T.LIV (1975), 73–79.Google Scholar
[13] Sato, H. and Yabuta, K., Toeplitz operators on strongly pseudoconvex domains in Stein spaces, Tohoku Math. J., 30 (1978), 153–162.Google Scholar
[14] Yabuta, K., A remark to a paper of Janas “Toeplitz operators related to certain domains in ℂn”, Studia Math., T.LXII (1978), 73–74.Google Scholar
[15] Boutet, L. Monvel, de and Guillemin, V., The spectral theory of Toeplitz operators, Princeton U. Press, Annals of Math. series no. 99, 1981.Google Scholar
[16] Guillemin, V., Toeplitz operators in n-dimensions, Integral Equations and Operator Theory, 7 (1984), 145–205.Google Scholar
[17] Boutet, L. Monvel, de, On the index of Toeplitz operators of several complex variables, Inventiones Math., 50 (1979), 249–272.Google Scholar
[18] Krantz, S. G., Function theory of several complex variables, Tohn Wiley & Sons, 1982.Google Scholar
[19] Curto, R. E., Fredholm and invertible tuples of bounded linear operators, Trans. Amer. Math. Soc., 266 (1981), 129–159.Google Scholar
[20] Cao, G. F. and Zhong, Ch. Y., Automorphism groups of Toeplitz algebras on certain pseudoconvex domains, Northeast. Math. J., 14 (1998), no. 1, 112–118.Google Scholar
[21] Dugundji, J., Topology, Allyn and Bacon, Inc., Boston, 1966.Google Scholar
[22] Rørdam, M., Larsen, F. and Laustsen, N., An introduction to K-theory for C*-algebras, Cambridge Univ. Press, 2000.CrossRefGoogle Scholar
[23] Douglas, R. G., C∗-algebra extensions and K-homology, Princeton University Press and University of Tokyo Press, 1980.CrossRefGoogle Scholar
[24] Sakai, S., On a problem of Kaplansky, Tohoku Math. J., 12 (1960), 31–33.Google Scholar
[25] Kadison, R. V., Notes on the Gelfand-Neumark theorem, Contemporary Math., 167 (1994), 21–53.Google Scholar
[26] Helemskii, A. Ya., The homology of Banach and topological algebras, Kluwer Academic Publishers, 1989.Google Scholar
[27] Kadison, R. V., Derivations of operator algebras, Ann. of Math., 83 (1966), 280–293.Google Scholar