K-Group Of The Commutants Of One Kind Of Operators

In recent years, K-theory has played an important role in studying the Operator Algebras. C. L. Jiang, J. S. Fang and Y. Cao find the connection between the uniquely strongly irreducible decomposition up to similarity of Type I operators and the K₀-groups of the commutants of the operators in 1990s. So one begins to concern the calculation of K₀-groups.A bounded linear operator K on a Hilbert space H is compact if the image of the unit ball of H under K has compact closure. Let C be the complex plane and I be the identity operator on H. Let A'(S) denote the commutant of a bounded linear operator S on H and R denote the collection of all operators in A'(S) which are compact and upper triangular operators with diagonal sequence being zeros. In this paper, we show that if A'(S) = CI + R, the K₀-group of A' (S) is isomorphic to the integer group.