Gelfand–Naimark theorem

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.

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

One standard proof combines positive functionals with the GNS construction.

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.