Source
Gelfand & Neumark (1943), “On the imbedding of normed rings into the ring of operators in Hilbert space”, Matematicheskii Sbornik 12(54):197–217, LIT-313, read in full in NOTE-303 from the Math-Net.Ru text, with the theorem statements checked against the Russian résumé.
What was actually shown
The axioms. A normed -ring is a complex Banach algebra with unit e, ‖e‖ = 1, and an involution with x* = x, (xy)* = yx, ‖xx‖ = ‖x‖·‖x‖, ‖x*‖ = ‖x‖, and e + xx invertible. The authors note that the third and fourth may be replaced by ‖xx‖ = ‖x‖², and suspect the last two follow from the others (p. 198 footnote).
Theorem 1 (p. 198). Every normed *-ring maps isomorphically onto a closed subring of the bounded operators on a Hilbert space, isometrically and preserving *. The space is not assumed separable.
The construction (§3). For each maximal left ideal M, a positive functional f with f(e) = 1 vanishing on M is obtained by Banach’s separation lemma and Krein’s extension theorem. Setting (ξ, η) = f(y*x) on R/M gives a Hilbert space on which left multiplication acts by bounded operators. The direct sum over all M is faithful, because the maximal left ideals intersect in 0. Corollary 6, that a *-isomorphism of normed *-rings is isometric, makes the image closed. This is the GNS construction, under no name.
Lemma 1 (p. 199). A commutative normed ring with the involution axioms is isometrically -isomorphic to the ring of all continuous complex functions on its compact maximal-ideal space 𝔐, with x sent to the conjugate function.
§4. The simple weakly closed rings on separable Hilbert space are exactly the factors of types I_n, II₁ and III (Theorem 3). As a special case, the compact operators are the only non-trivial closed ideal of B(H) for separable H (Calkin).
What could have come out otherwise. Some abstract normed *-ring could have failed to be an operator algebra, or a commutative one could have been a proper subalgebra of C(𝔐). Neither happens.
What this does not say
- That Gelfand duality is a categorical duality. The paper gives the object-level isomorphism R ≅ C(𝔐), not functoriality or an equivalence of categories (NOTE-303).
- That the commutative machinery is proved here. It rests on Gelfand’s “Normierte Ringe” (1941) and Šilov’s boundary. For the maximal-ideal space itself, that 1941 paper is the primary source.
- Uniqueness or irreducibility of the representation. States versus pure states are not discussed. THEORY-004’s uniqueness up to unitary of a minimal realisation is not proved here.
- That Gelfand, Stone and Pontryagin are one duality. Lemma 1 is one leg. Neither this paper nor the other legs’ primary sources (LIT-353, LIT-333) state a common duality; that framing is later literature and the owner’s synthesis.
- That the extra axioms are needed. Later work showed them redundant, according to standard histories; the reading did not verify it.
Connections
- THEORY-020, Stone’s representation of Boolean algebras as clopen algebras, is the Boolean counterpart of Lemma 1; for a commutative algebra generated by projections the two coincide (NOTE-303).
- THEORY-645, Pontryagin duality, is the characters leg.
- THEORY-004 names the kernel-to-Hilbert-space construction as GNS. This paper is where it first appears.
- THEORY-042 uses §4’s factors: an algebra with superselection sectors has a non-trivial centre and is not a factor.
- THEORY-652: Theorem 3 here descends from Wedderburn’s trivial-centre corollary for simple algebras.