C*-algebra generated by the symmetric on 3 elements

Question

I want compute $C^*(S_3)$ where $S_3$ is the symmetric group on $\{1,2,3\}$ and $C^*(S_3)$ is the (full) C*-algebra generated by $S_3$.

My attempt: Since $S_3$ is a finite group, $C^*(S_3)=C_c(S_3)$ and its dimension is equal to the group order. So, I was trying to representate this group C*-algebra in a algebra of the form $\oplus_{i=1}^nM_{d_i}$. Since $S_3$ has order $6$, there are only two possibilities:

• $C^*(S_3)=\oplus_{i=1}^6\mathbb{C}$
• $C^*(S_3)=M_2(\mathbb{C})\oplus\mathbb{C}\oplus\mathbb{C}$

Can anyone help me identify which one is it?

Answer 1

The first one is commutative, so...