Faithful *-homomorphism

Let $A$ and $B$ be $C^*$ -algebras. A $*$ -homomorphism $\phi:A\to B$ is faithful if it is injective:

\ker\phi=\{0\}.

Equivalently, $\phi(a)=0$ implies $a=0$ . Faithfulness says that no nonzero algebra element, and hence no nonzero closed two-sided ideal, is lost under $\phi$ . It does not require $\phi$ to be surjective, unital, or nondegenerate. When $B=\mathcal B(H)$ , this is the usual faithfulness condition for a representation of $A$ on $H$ .

Isometry and concrete realization

A $*$ -homomorphism of $C^*$ -algebras is faithful exactly when it is isometric:

\lVert\phi(a)\rVert=\lVert a\rVert\qquad(a\in A).

Thus its range is closed, and $\phi$ is a $*$ -isomorphism from $A$ onto the $C^*$ -subalgebra $\phi(A)$ . This automatic isometry is a specifically $C^*$ -algebraic consequence of the $C^*$ -identity and spectral theory.

The Gelfand–Naimark theorem therefore says that every abstract $C^*$ -algebra admits a faithful $*$ -homomorphism into some bounded-operator algebra $\mathcal B(H)$ .

Kernels and quotients

For an arbitrary $*$ -homomorphism $\phi:A\to B$ , the closed two-sided ideal $\ker\phi$ measures precisely the failure of faithfulness. The induced map

A/\ker\phi\longrightarrow\phi(A),\qquad a+\ker\phi\longmapsto\phi(a),

is always a faithful $*$ -homomorphism and hence an isometric $*$ -isomorphism. In particular, faithfulness is a property of the map, not of either algebra in isolation.

Examples and distinctions

The inclusion of a $C^*$ -subalgebra $A\subseteq B$ is faithful. A quotient map $A\to A/I$ is faithful exactly when $I=\{0\}$ . The zero map from a nonzero algebra is never faithful.

References

Gerard J. Murphy, C-Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: §§2.1 and 3.4 on injective $*$ -homomorphisms, isometry, and faithful representations.
Gert K. Pedersen, C-Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: §1.2 on morphisms, kernels, and represented images.