closed ideals in $C^{*}$-algebras are self-adjoint

Theorem - Every closed (http://planetmath.org/ClosedSet) two-sided ideal (http://planetmath.org/IdealOfAnAlgebra) $ℐ$ of a $C^{*}$-algebra (http://planetmath.org/CAlgebra) $𝒜$ is self-adjoint (http://planetmath.org/InvolutaryRing), i.e.

if $x\in ℐ$ then $x^{*}\in ℐ$.

Proof : Let ${ℐ}^{*}:=\{a^{*}:a\in ℐ\}$.

Since $ℐ$ is closed and the involution mapping is continuous, it follows that ${ℐ}^{*}$ is also closed.

We claim that ${ℐ}^{*}$ is also a of $𝒜$. To see this let $a,b\in ℐ$, $x\in 𝒜$ and $\lambda \in ℂ$. Then

• $a^{*}+\lambda b^{*}={(a+\overline{\lambda }b)}^{*}\in {ℐ}^{*}$ since $a+\overline{\lambda }b\in ℐ$

• $x{a}^{*}={(a{x}^{*})}^{*}\in {ℐ}^{*}$ since $a{x}^{*}\in ℐ$.

• $a^{*}x={(x^{*}a)}^{*}\in {ℐ}^{*}$ since $x^{*}a\in ℐ$

Let $ℬ:=ℐ\cap {ℐ}^{*}$.

$ℬ$ is a $C^{*}$-subalgebra of $𝒜$ (it is a norm-closed, involution-closed, subalgebra of $𝒜$).

It is known that every $C^{*}$-algebra has an approximate identity consisting of positive elements with norm less than $1$ (see this entry (http://planetmath.org/CAlgebrasHaveApproximateIdentities)).

Let ${(e_{\lambda })}_{\lambda \in \mathrm{\Lambda }}$ be an approximate identity for $ℬ$ with the above :

1. each $e_{\lambda }$ is positive (hence self-adjoint) and

2. $\parallel e_{\lambda }\parallel \le 1{\forall }_{\lambda \in \mathrm{\Lambda }}$

We now prove $ℐ$ is self-adjoint:

Let $a\in ℐ$. We have that

${\parallel a^{*}-a^{*}{e}_{\lambda }\parallel }^2$$=$$\parallel {(a^{*}-a^{*}{e}_{\lambda })}^{*}\cdot (a^{*}-a^{*}{e}_{\lambda })\parallel$

$=$$\parallel (a-e_{\lambda }a)\cdot (a^{*}-a^{*}{e}_{\lambda })\parallel$

$=$$\parallel (a{a}^{*}-a{a}^{*}{e}_{\lambda })-e_{\lambda }(a{a}^{*}-a{a}^{*}{e}_{\lambda })\parallel$

$\le$$\parallel a{a}^{*}-a{a}^{*}{e}_{\lambda }\parallel +\parallel e_{\lambda }\parallel \cdot \parallel a{a}^{*}-a{a}^{*}{e}_{\lambda }\parallel$

$\le$$\parallel a{a}^{*}-a{a}^{*}{e}_{\lambda }\parallel +\parallel a{a}^{*}-a{a}^{*}{e}_{\lambda }\parallel$

$=$$2\parallel a{a}^{*}-a{a}^{*}{e}_{\lambda }\parallel$

Taking limits in both we obtain

$$\underset{\lambda }{lim}{\parallel a^{*}-a^{*}{e}_{\lambda }\parallel }^2\le \underset{\lambda }{lim}\mathrm{\hspace{0.33em}2}\parallel a{a}^{*}-a{a}^{*}{e}_{\lambda }\parallel =0$$

since $a{a}^{*}\in ℐ\cap {ℐ}^{*}=ℬ$ and ${(e_{\lambda })}_{\lambda \in \mathrm{\Lambda }}$ is an approximate identity for $ℬ$.

As $e_{\lambda }\in ℐ$ we see that $a^{*}{e}_{\lambda }\in ℐ$.

We conclude from the limit above that $a^{*}$ is in the closure of $ℐ$. Therefore $a^{*}\in ℐ$.

Hence, $ℐ$ is self-adjoint. $\mathrm{\square }$