C *-Subalgebras Generated by a Single Operator in B(H)

Abstract

In this paper, we characterize a C*-subalgebra C*(x) of B(H), generated by a single operator x. We show that if x is polar-decomposed by aq, where a is the partial isometry part and q is the positive operator part of x, then C*(x) is *-isomorphic to the groupoid crossed product algebra \(\mathcal{A}_{q}\times_{\alpha }\mathbb{G}_{a}\) , where \(\mathcal{A}_{q}=C^{*}(q)\) and \(\mathbb{G}_{a}\) is the graph groupoid induced by a partial isometry part a of x.