Kernel of a *-homomorphism - Imported Knowlpedia Content

Definition

For a $*$ -homomorphism $\phi:A\to B$ of $C^*$ -algebras, its kernel is

\ker\phi=\{a\in A:\phi(a)=0\}.

It is a closed two-sided ideal: linearity gives a closed linear subspace, multiplicativity gives absorption from both sides, and preservation of the involution gives self-adjointness. The homomorphism is injective exactly when $\ker\phi=\{0\}$ . Thus the kernel records precisely which elements of the domain become indistinguishable from zero. Because $*$ -homomorphisms are automatically continuous, closedness of the kernel requires no separate continuity hypothesis.

Factorization through the quotient

Let $q:A\to A/\ker\phi$ be the quotient map. There is a unique $*$ -homomorphism

\widetilde\phi:A/\ker\phi\longrightarrow B,\qquad \widetilde\phi(a+\ker\phi)=\phi(a),

such that $\phi=\widetilde\phi\circ q$ . It is injective and therefore isometric. Its range is $\phi(A)$ , so

A/\ker\phi\cong\phi(A)

as $C^*$ -algebras. In particular, the range of every $*$ -homomorphism between $C^*$ -algebras is norm closed Murphy, §3.1.

Universal property

More generally, if $I$ is a closed two-sided ideal of $A$ , then a $*$ -homomorphism $\psi:A\to C$ factors uniquely through the quotient $C^*$ -algebra $A/I$ exactly when $I\subseteq\ker\psi$ . This is the universal property of the quotient and is often the efficient way to construct homomorphisms out of $A/I$ .

Examples

Evaluation at $x$ on $C_0(X)$ has kernel $\{f\in C_0(X):f(x)=0\}$ . A faithful representation $\pi:A\to B(H)$ has zero kernel, whereas the quotient map $A\to A/I$ has kernel exactly $I$ . These examples connect points, representations, and ideals through the same construction.

References

Gerard J. Murphy, $C^*$ -Algebras and Operator Theory, Academic Press, 1990. Publisher DOI record. Relevant: §3.1 on ideals, quotient algebras, and the $C^*$ -algebraic isomorphism theorem.
Gert K. Pedersen, $C^*$ -Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. Publisher DOI record. Relevant: §§1.2 and 1.8 on morphisms, kernels, and quotients.