What is a C*-algebra generated by a subset of a direct sum of C*-algebras equal to?

Question

I'm studying C-algebras and I don't know how to address the following question: let $(A_k)_{k\in \mathbb{N}}$ a family of C-algebras and let $\mathcal{G}$ a subset of $\displaystyle\bigoplus_{k\in \mathbb{N}}A_{k}$. Consider $m\in \mathbb{N}$ and define $B_{m}(\mathcal{G}):=\{a_{m}:a \in \mathcal{G}\}$. Under what conditions does the following equality hold?: $$B_{m}(C^*(\mathcal{G}))= C^*(B_{m}(\mathcal{G})).$$ Where $C^*(\mathcal{G})$ and $C^*(B_{m}(\mathcal{G}))$ stands for the C*-algebra generated by the subsets $\mathcal{G}$ and $B_{m}(\mathcal{G})$ respectively. Thank you in advance for your help.

Answer 1

This is always true. The key point is that the projection map $a \mapsto a_m$ from the direct sum onto $A_m$ is a $*$-homomorphism, so its restriction to $C^*(G)$ is a $*$-homomorphism, and the image of any C${}^*$-algebra under a $*$-homomorphism is a C*-algebra.