The K-Groups Of Algebras

K theory has a profound influence on the study of operator algebra. K theory contains a lot of information about a single C* algebra.We can understand the structure of the operator by operating its K group. Let L(H) be a separable complex infinite dimensional Hilbert space and let L(H) be the collection of bounded linear operators on H. In this paper, we calculate the K group of several types of algebra.This article consists of four chapters.The first chapter introduce the back-ground of the paper topic.The second chapter introduce some preliminary knowledge needed in this paper.In the third chapter,we compute the K0 group of the algebra A(D)={f:f is anaiytic in the open disk D and continuous in the D}.we discuss the properties of the adjoint operator T of unilateral weighted shift,proved that T is strongly irreducible Cowen-Douglas operator, and compute the K0 group of the commutant algebra of T.