Faithful representation of a C*-algebra - Imported Knowlpedia Content

Definition

Let $A$ be a $C^*$ -algebra and let $\pi:A\to\mathcal B(H)$ be a representation on a complex Hilbert space. The representation is faithful when $\pi$ is injective, equivalently

\ker\pi=\{0\}.

Faithfulness means that the operator realization loses no algebra elements or relations. It does not require $\pi$ to be irreducible, surjective onto $\mathcal B(H)$ , unital, or nondegenerate. Thus faithfulness is the specialization to representations of the general notion of a faithful $*$ -homomorphism.

Equivalent characterizations

Every $*$ -homomorphism between $C^*$ -algebras is contractive, and such a map is injective exactly when it is isometric. Consequently,

\pi\text{ is faithful} \quad\Longleftrightarrow\quad \lVert\pi(a)\rVert=\lVert a\rVert\text{ for every }a\in A.

Its image is then norm closed, and $\pi$ is a $*$ -isomorphism from $A$ onto the concrete $C^*$ -algebra $\pi(A)$ Murphy, §2.1.

Existence and use

The Gelfand–Naimark theorem states that every $C^*$ -algebra has a faithful representation on some Hilbert space Murphy, Theorem 3.4.1. This permits abstract $C^*$ -algebras to be studied as norm-closed self-adjoint operator algebras without changing their norms.

Examples and distinctions

The defining action of $K(H)$ on $H$ is faithful. If $\pi:A\to\mathcal B(H)$ is faithful, then $a\mapsto\pi(a)\oplus0$ on $H\oplus K$ remains faithful but is degenerate when $K\neq0$ . By contrast, a quotient representation $A\to A/I$ with $I\neq0$ is not faithful because its kernel is $I$ .

References

Gerard J. Murphy, C-Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: §2.1 on injective $*$ -homomorphisms and Theorem 3.4.1 on faithful representations.
Gert K. Pedersen, C-Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: Chapter 3 on represented $C^*$ -algebras.