The K-group Of Toeplitz Algebra

In this paper,from the six-term exact sequence,it is proved that the direct sum of Z and K1 -group of a Toeplitz algebra is always isomorphic to the topological K1 -group of the boundary of the relative domain.Further more,we prove that the cohomotopy groups is isomorphic to K1 -group of the continuous algebra on topological boundary of bounded domains in C. And also we discuss the K-group of the Toeplitz algebras on the torus. Finally we discuss the collectively compact Toeplitz operator sequences on Dirichlet space.it extend the single compact operator.