In this article, "Spectral measures in C∗-algebras of singular integral operators with shifts", in chapter 3.1.
They have a $C^*$ algebra $U$, and an unitary representation $\pi$ of a discrete group $G$ of unitary elements.
I'm having difficulties understanding what they mean by the minimal $C^∗$ algebra containing the $C^∗$ algebra $A$ and the group $\pi(G)$ (which is the $C^*$ algebra generated by them).
If someone could clarify it to me I would be very grateful.
There is some context missing, but I think the situation is the following: $A$ is a $C^\ast$-algebra acting on $H$ and $\pi$ is a unitary representation of $G$ on $H$. Then both $A$ and $\pi(G)$ consist of bounded linear operators on $H$ and one can form the smallest $C^\ast$-algebra containing $A\cup \pi(G)$, which is the intersection of all $C^\ast$-subalgebras of $B(H)$ that contain both $A$ and $\pi(G)$.