Proposition 34.2.2.label Let $A$ be a unital $C^{*}$-algebra, $S \subset A$, and $\mathcal{S}= S \cup \bracs{x^*|x \in S}$, then

1. (1)

   The linear span

   \[\text{span}\bracs{\prod_{j = 1}^n x_j \bigg | \seqf{x_j} \subset \mathcal{S}}\]

   
   

   is dense in $A[S]$.

2. (2)

   If for any $x, y \in \mathcal{S}$, $xy = yx$, then $A[S]$ is commutative.

3. (3)

   For any normal element $x \in A$, $A[x]$ is commutative.