Toeplitz algebras on strongly pseudoconvex domains

Abstract

In the present paper, it is proved that the $K_{0}$-group of a Toeplitz algebra on any strongly pseudoconvex domain is always isomorphic to the $K_{0}$-group of the relative continuous function algebra, and is thus isomorphic to the topological $K^{0}$-group of the boundary of the relative domain. Further there exists a ring isomorphism between the $K_{0}$-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.