K-Groups of a C^{*}-Algebra Generated by a Single Operator

In this paper, we compute \(K\) -groups \(\{K_{n}(C^{*}(x))\}_{n=0}^{\infty }\) of the \(C^{*}\) -subalgebra \(C^{*}(x)\) of \(B(H),\) generated by a single operator \(x,\) where \(H\) is a separable infinite dimensional Hilbert space, and \(B(H)\) is the operator algebra consisting of all (bounded linear) operators on \(H.\) These computations not only provide nice examples in \(K\) -theory, but also characterize-and-classify projections in a \(C^{*}\) -algebra generated by a single operator. The main result of this paper shows that: the \(K\) -groups of \(C^{*}(x)\) are completely characterized by those of \(C^{*}(q),\) where \(q\) is the positive-operator part of \(x\) in the polar decomposition of \(x.\)