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 $C^*(\mathcal{G})$ stands for the C*-algebra generated by the subset $\mathcal{G}$. Thank you in advance for your help.

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.

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 $\displaystyle\oplus_{k\in \mathbb{N}}A_{k}$. 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 $C^*(\mathcal{G})$ stands for the $C^*$-algebra generated by the subset $\mathcal{G}$. Thank you in advance for your help.

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 $C^*(\mathcal{G})$ stands for the C*-algebra generated by the subset $\mathcal{G}$. Thank you in advance for your help.

I don't know how to begin to address the following problem: Let (A_k)_{k\in \mathbb{N}} a family of C-algebras and let \mathcal{G} a subset of \oplus_{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}) stands for the C-algebra generated by the subset \mathcal{G}. Thank you in advance for any ideas.

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 $\displaystyle\oplus_{k\in \mathbb{N}}A_{k}$. 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 $C^*(\mathcal{G})$ stands for the $C^*$-algebra generated by the subset $\mathcal{G}$. Thank you in advance for your help.