Gelfand–Naimark theorem - Imported Knowlpedia Content

Statement

Gelfand–Naimark theorem. For every abstract complex $C^*$ -algebra $A$ , there are a Hilbert space $H$ and a faithful representation

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

The map $\pi$ is automatically isometric, so its image is a norm-closed $*$ -subalgebra of $\mathcal B(H)$ . It may be chosen nondegenerate; when $A$ is unital, it may be chosen unital. Thus the abstract axioms for a $C^*$ -algebra describe exactly the algebras that can be realized concretely as norm-closed operator algebras closed under adjoints.

GNS proof architecture

For each state of $A$ , the GNS construction produces a cyclic representation. Taking their universal direct sum gives a representation $\pi_u$ . Since states separate positive elements, $\pi_u(a)\neq0$ whenever $a\neq0$ , hence $\pi_u$ is faithful. The $C^*$ -identity then makes every injective $*$ -homomorphism isometric Murphy, Theorem 3.4.1.

Consequences and scope

The theorem justifies moving freely between abstract and concrete $C^*$ -algebras. It does not say that a representation is unique: one algebra usually has many inequivalent faithful representations on different Hilbert spaces. The commutative Gelfand representation theorem, which realizes a commutative $C^*$ -algebra as functions on its spectrum, is a stronger specialized statement and should not be conflated with this operator representation theorem.

Historical formulation

Gelfand and Naimark established the representation result in 1943 in the language of normed rings with involution. Modern presentations usually prove it by combining positive functionals with the GNS construction. The original article is Gelfand–Naimark, 1943.

References

Gerard J. Murphy, C-Algebras and Operator Theory*, Academic Press, 1990. DOI record. Relevant: Theorem 3.4.1 and its GNS proof.
I. Gelfand and M. Neumark, “On the imbedding of normed rings into the ring of operators in Hilbert space,” Matematicheskii Sbornik 12(54), no. 2 (1943), 197–217. Stable journal record. Relevant: the original faithful-representation theorem.