Let $A$ be a C*-algebra and $A_0\subset A$. Then it is known that the $\mathbb{C}$-algebra generated by $A_0$ (i.e. the intersection of all sub-$\mathbb{C}$-algebras containing it ) is just the vector space of finite, linear elements on the form $T_1\cdots T_n$ where $n\in \mathbb{N}$ and $T_j \in A_0$. And the Banach algebra generated by $A_0$ is the norm closure of the above.
Is there a corresponding neat result for C*-algebras?
I assume that $A$ is a $C^\star$-algebra and that $A_0$ is a subset of $A$? Then the generated $C^\star$-subalgebra is the norm-closure of the linear span of the multplicative closure of the $\star$-closure of $A_0$. That is, it is the norm-closure of the generated $*$-subalgebra $\{\sum_i \lambda_i a^i_1 \dotsc a^i_n : \lambda_i \in \mathbb{C}, a^i_k \in A_0 \cup A_0^\star\}$.