Every closed ideal of $C^{\star}-$ algebra is self adjoint.

Question

I am looking for a proof of the following well known result.

Every closed ideal $I$ of $C^{\star}-$ algebra is self adjoint that is if $x \in I$ then $x^{\ast} \in I$

Any reference or ideas?

Answer 1

Hint: Let $J$ be a two sided ideal of the $C^{\star}$ algebra $\mathcal{A}$. Consider the $C^{\star}$ subalgebra $\mathcal{B}=J\cap J^{\star}$. Let $\{e_i \}_{i\in I}$ be an approximate identity of positive elements with $\|e_i\|\leq 1$ for $\mathcal{B}$. Show the net $a^{\star}e_{i}$ converges to $a^{\star}$.

Answer 2

Consult Theorem BA.4.1 (page 108) in

Barnes, Murphy, Smyth, West: Riesz and Fredholm theory in Banach algebras.