The crossed product encodes the action of \(G\) on \(A\). In group theory this is called the semidirect-product. We adapt this to create an algebra with the property that there is a copy of \(G\) inside the unitary group of \(A \rtimes_{\alpha} G\) (at least when \(A\) is unital). Further, there is a natural inclusion \(A \subset A \rtimes_{\alpha}G\) such that

1. \(A \rtimes_{\alpha}G\) is generated by \(A\) and \(G\) and
2. \(\alpha_{g}(a) = gag^*\).

For a \(G\)-\(C^*\)-algebra \(A\), we denote by \(C_{c}(G,A)\) the linear space of finitely supported functions on \(G\) with values in \(A\). A typical element \(S \in C_{C}(G,A)\) is written as a finite sum \(S = \sum_{g \in G}a_{g}g\).

We equip \(C_{C}(G,A)\) with an \(α\)-twisted convolution product and \(*\)-operation as follows:

Let \(S = \sum_{g \in G}a_{g}g, T = \sum_{g \in G}b_{g}g\), then \[ S \astr_{\alpha}T = \sum_{s,t \in G}a_{s}\alpha_{s}(b_{t})st \text{ and } S^* = \sum_{s \in G}\alpha_{s^{-1}}(a_{s}^*)s^{-1}. \]