Primitive ideal of a C*-algebra

Let $A$ be a $C^*$ -algebra. A primitive ideal of $A$ is a closed two-sided ideal $P\subsetneq A$ for which there is a nonzero irreducible representation

\pi:A\longrightarrow\mathcal B(H)

with $P=\ker\pi$ . Equivalently, $A/P$ has a faithful irreducible representation. The representation $\pi$ need not itself be faithful unless $P=\{0\}$ , and distinct irreducible representations may have the same primitive kernel. The set of all primitive ideals is denoted $\operatorname{Prim}(A)$ .

Quotients, pure states, and maximal ideals

If $\omega$ is a pure state of $A$ , its GNS representation is irreducible, so its kernel is primitive. Conversely, every primitive ideal is the kernel of an irreducible GNS representation associated with a suitable pure state after the standard treatment of nonunital algebras.

Every maximal proper closed two-sided ideal is primitive: its simple quotient has a faithful irreducible representation. The converse fails in general; a primitive ideal need not be maximal. Primitive ideals therefore remember irreducible representation theory more finely than maximal ideals alone.

Prime versus primitive

Every primitive ideal of a $C^*$ -algebra is prime: if closed two-sided ideals $I,J$ satisfy $IJ\subseteq P$ , then $I\subseteq P$ or $J\subseteq P$ . For separable $C^*$ -algebras, every prime ideal is primitive, but without separability the converse can fail. This is a theorem with a hypothesis, not an alternative unconditional definition.

Examples and interpretation

For $A=C_0(X)$ with $X$ locally compact Hausdorff, the primitive ideals are

P_x=\{f\in C_0(X):f(x)=0\},\qquad x\in X.

Thus primitive ideals recover ordinary points in the commutative case. The zero ideal of $\mathcal K(H)$ , for $H\neq0$ , is primitive because the defining representation on $H$ is faithful and irreducible. By contrast, the zero ideal of a direct sum $A_1\oplus A_2$ , with both summands nonzero, is not primitive: every irreducible representation annihilates one summand.

This point-like role motivates equipping the primitive ideals with the hull–kernel topology to form the primitive ideal space.

References

Jacques Dixmier, C-Algebras, North-Holland Mathematical Library 15, North-Holland, 1977. Publisher record. Relevant: §§2.5 and 3.1 on irreducible representations, primitive ideals, and hull–kernel constructions.
Gert K. Pedersen, C-Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: Chapter 4 on prime and primitive ideals and their representation-theoretic structure.