Essential ideal of a C*-algebra - Imported Knowlpedia Content

Definition

Let $A$ be a $C^*$ -algebra. A closed two-sided ideal $I\triangleleft A$ is essential if

I\cap J\ne\{0\}

for every nonzero closed two-sided ideal $J\triangleleft A$ . Equivalently, the annihilator

I^\perp=\{a\in A:aI=\{0\}\}

is zero; in a $C^*$ -algebra this is also equivalent to $\{a\in A:Ia=\{0\}\}=\{0\}$ . Thus essentiality says that no nonzero part of $A$ is disjoint from $I$ . It strengthens merely being a nonzero two-sided ideal.

Equivalent characterizations

The following conditions are equivalent for a closed ideal $I$ :

$I$ is essential;
$I^\perp=\{0\}$ ;
the canonical $*$ -homomorphism $A\to M(I)$ , obtained from left and

right multiplication on $I$ , is injective.

The $C^*$ -identity is important here: it makes the left and right annihilator conditions agree and converts a nonzero annihilating element into a nonzero ideal disjoint from $I$ Pedersen, §3.12.

Examples

Every $C^*$ -algebra is an essential ideal in itself. The compact operators $K(\mathcal H)$ form an essential ideal in $B(\mathcal H)$ when $\mathcal H\ne0$ . More generally, if $X$ is locally compact Hausdorff and $U\subseteq X$ is open, the ideal $C_0(U)\subseteq C_0(X)$ is essential exactly when $U$ is dense. By contrast, either coordinate summand in $A_1\oplus A_2$ , with both summands nonzero, is not essential because it has zero intersection with the other summand.

Role in multiplier algebras

The multiplier algebra $M(I)$ is the largest unital $C^*$ -algebra in which $I$ sits as an essential ideal, in the sense that every $C^*$ -algebra containing $I$ essentially acts faithfully on $I$ by multipliers. The essentiality hypothesis is exactly what prevents ambient elements from disappearing under this action Lance, §2.

References

Gert K. Pedersen, $C^*$ -Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: §3.12 on essential ideals and multipliers.
E. Christopher Lance, Hilbert $C^*$ -Modules: A Toolkit for Operator Algebraists, Cambridge University Press, 1995. DOI record. Relevant: §2 on multiplier algebras and essential ideals.