Let $A$ be the sequence space $(\ell_2, \|\cdot\|)$ with coordinate-wise product
Prove that $(\ell_2, \|\cdot\|)$ with coordinate-wise product is semisimple
My attempt: I define $\delta_k:\ell_2 \to \mathbb {C}$ by $\delta_k(a_n)=a_k$. Therefore $\delta_k$ is a character (non-zero homomorphism) and $\ker\delta_k=\{0\}$ but we cannot say $\operatorname{Rad}A=\{0\}$ since $A$ does not have an identity.
Here is where I am stuck. Could you please tell me how to solve this?