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

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

if $x\in\mathcal{I}$ then $x^{*}\in\mathcal{I}$.

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

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

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

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

• $xa^{*}=(ax^{*})^{*}\in\mathcal{I}^{*}$ since $ax^{*}\in\mathcal{I}$.

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

Let $\mathcal{B}:=\mathcal{I}\cap\mathcal{I}^{*}$.

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

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

1. 1.

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

2. 2.

$\|e_{\lambda}\|\leq 1\;\;\;\forall_{\lambda\in\Lambda}$

We now prove $\mathcal{I}$ is self-adjoint:

Let $a\in\mathcal{I}$. We have that

 $\displaystyle\|a^{*}-a^{*}e_{\lambda}\|^{2}$ $\displaystyle=$ $\displaystyle\|(a^{*}-a^{*}e_{\lambda})^{*}\cdot(a^{*}-a^{*}e_{\lambda})\|$ $\displaystyle=$ $\displaystyle\|(a-e_{\lambda}a)\cdot(a^{*}-a^{*}e_{\lambda})\|$ $\displaystyle=$ $\displaystyle\|(aa^{*}-aa^{*}e_{\lambda})-e_{\lambda}(aa^{*}-aa^{*}e_{\lambda})\|$ $\displaystyle\leq$ $\displaystyle\|aa^{*}-aa^{*}e_{\lambda}\|+\|e_{\lambda}\|\cdot\|aa^{*}-aa^{*}e% _{\lambda}\|$ $\displaystyle\leq$ $\displaystyle\|aa^{*}-aa^{*}e_{\lambda}\|+\|aa^{*}-aa^{*}e_{\lambda}\|$ $\displaystyle=$ $\displaystyle 2\|aa^{*}-aa^{*}e_{\lambda}\|$

Taking limits in both we obtain

 $\lim_{\lambda}\|a^{*}-a^{*}e_{\lambda}\|^{2}\leq\lim_{\lambda}\;2\|aa^{*}-aa^{% *}e_{\lambda}\|=0$

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

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

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

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

Title closed ideals in $C^{*}$-algebras are self-adjoint ClosedIdealsInCalgebrasAreSelfadjoint 2013-03-22 17:30:42 2013-03-22 17:30:42 asteroid (17536) asteroid (17536) 9 asteroid (17536) Theorem msc 46L05 msc 46H10