Sum of Self-adjoint elements is self-adjoint

Question

Let $A$ be a commutative $C^*$- algebra. Let $a_1,a_2\in A$, where $a_1=a_1^*$ and $a_2=a_2^*$.

Let $a = a_1+a_2$. Can we say that $a=a^*$?

Thank you in advance!

Answer 1

In $C^*$algebra, we have $(a_1+a_2)^* = a_1^* + a_2^*$ which is $a_1+a_2$. We don't need commutativity.