Closed two-sided ideal of a C*-algebra

Let $A$ be a $C^*$ -algebra. A closed two-sided ideal of $A$ is a norm-closed linear subspace $I\subseteq A$ satisfying

a x\in I\quad\text{and}\quad x a\in I \qquad(a\in A,\ x\in I).

Every norm-closed two-sided ideal in a $C^*$ -algebra is automatically self-adjoint: $x\in I$ implies $x^*\in I$ . Consequently $I$ , with the operations and norm inherited from $A$ , is itself a $C^*$ -algebra. The terms $C^*$ -ideal and closed ideal normally refer to this notion.

Quotients and kernels

The algebraic quotient $A/I$ , with

\lVert a+I\rVert=\inf_{x\in I}\lVert a+x\rVert \quad\text{and}\quad (a+I)^*=a^*+I,

is again a quotient $C^*$ -algebra. Thus closedness is essential: an arbitrary algebraic ideal need not yield a complete Hausdorff quotient. Conversely, the kernel of every $*$ -homomorphism between $C^*$ -algebras is a closed two-sided ideal. These two operations organize ideals as the subobjects that support $C^*$ -quotients.

Positivity and approximate identities

Each closed ideal $I$ has a positive contractive approximate identity. Moreover, ideals are hereditary for positive elements: if $0\leq b\leq a$ with $a\in I$ , then $b\in I$ . Equivalently, $I_+=I\cap A_+$ , and $I$ is linearly spanned by its positive elements. These analytic properties go beyond the defining absorption conditions and distinguish $C^*$ -ideals from general ring ideals.

Examples and distinctions

For a locally compact Hausdorff space $X$ and an open subset $U$ , extension by zero identifies $C_0(U)$ with the ideal of functions in $C_0(X)$ vanishing on $X\setminus U$ . In $K(\mathcal H)$ , the only closed two-sided ideals are $0$ and $K(\mathcal H)$ . A $C^*$ -subalgebra need not be an ideal, because it may fail absorption by elements of the ambient algebra.

References

Gerard J. Murphy, $C^*$ -Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: §3.1 on ideals, quotients, and approximate identities.
Gert K. Pedersen, $C^*$ -Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: §§1.4 and 1.8 on ideals and their positive structure.