A Cuntz-Krieger E-family \(\{ S, P \}\) on \(\mathcal{H}\) consists of a set \(\{ P_{v}: v \in E^0 \}\) of mutually orthogonal projections on \(\mathcal{H}\) and a set \(\{ S_{e}\varepsilon \in E^1 \}\) of partial isometries on \(\mathcal{H}\), such that

\begin{align} S_{e}^* S_{e} &= P_{s(e)} & \text{ for all $e \in E^1$; and }\\ P_{v} &= \sum_{\{ e \in E^1 : r(e) = v \}}S_{e}S^*_{e} &\text{ whenever $v$ is not a source.} \end{align}

The first relation ssays that \(S_{e}\) is a partial isometry with initial space \(P_{s(e)}\mathcal{H}\); the second relation says that the range projection \(S_{e}S_{e}^*\) of \(S_{e}\) is dominated by \(P_{r(e)}\), and hence that \(S_{e}\mathcal{H} \subset P_{r(e)}\mathcal{H}\). Thus \(S_{e}\) is an isometry of \(P_{s(e)}\mathcal{H}\) onto a closed subspace of \(P_{r(e)}\mathcal{H}\); expressing this algebraically gives the relation \[ S_{e } = P_{r(e)}S_{e} = S_{e}P_{s(e)}. \]