Question

Find an example of a real commutative banach algebra with identity which does not admit a nonzero real multiplicative linear func tional.

Answer 1

Let $\mathbb{C}$ with it's usual structure of real commutative unital Banach algebra. If $\varphi:\mathbb{C}\rightarrow\mathbb{R}$ is linear and multiplicative, then $\varphi(1)=\varphi(1)^2$, hence $\varphi(1)\in\left\{0,1\right\}$. Since $-\varphi(1)=\varphi(-1)=\varphi(i^2)=\varphi(i)^2$, then $\varphi(1)$ cannot be equal to $1$. Therefore, $\varphi(1)=0$ and $\varphi=0$.

by_Gopal siba paul.

please follow me.✌✌✌✌✌✌✌✌