39.01.03 · operator-algebras / c-star-algebras-basics

States, the GNS Construction, and the Gelfand-Naimark Representation Theorem

shipped3 tiersLean: none

Anchor (Master): Davidson *C*-Algebras by Example* Ch. I; Murphy Ch. 3; Dixmier *C*-Algebras* §2; Takesaki *Theory of Operator Algebras I* Ch. I §9

Intuition Beginner

A state is a way of averaging. Given a collection of operators you can add, multiply, scale, and reflect through the star, a state is a rule that assigns to each operator a single number — its average value — in a way that respects the structure: the average of a sum is the sum of averages, and the average of an operator that is a square-of-a-star is never negative. In quantum mechanics this is exactly the recipe for the expected outcome of a measurement once the system has been prepared.

The surprising fact is that every such averaging rule comes from a vector. Pick any state on your algebra, and you can build a space of arrows and a single special arrow inside it, so that averaging an operator means sandwiching it between that arrow and itself. The abstract rule becomes a concrete geometric measurement: point the arrow, apply the operator, and read off the overlap.

Once every averaging rule is realised this way, you can collect all of them at once. Doing so turns any algebra of this kind — no matter how abstractly it was defined — into a genuine algebra of operators acting on arrows. That is the payoff: the abstract becomes concrete.

Visual Beginner

A state turns into a Hilbert space with one marked vector; the operator acts on that vector and the average is the overlap with the marked vector.

The dictionary reads: the averaging rule becomes a marked vector; an algebra element becomes an operator on the space; and the average of that element becomes the overlap of the operator-moved vector with the marked vector. The marked vector is cyclic, meaning the operators sweep it around enough to fill the whole space.

Worked example Beginner

Take the algebra of two-by-two complex matrices. A simple averaging rule is "report the top-left entry": send a matrix to the number sitting in its first row and first column. Call this rule $φ$ . Check it averages sensibly: the top-left entry of a sum is the sum of top-left entries, and the constant matrix that scales everything by one reports the number $1$ .

Now check positivity on a star-times-itself. Take a matrix $a$ and form $a^{*} a$ . Its top-left entry is the sum of the squared sizes of the first column of $a$ , which is never negative. So $φ (a^{*} a) \geq 0$ , as a state requires.

Build the geometric picture. Use the first standard basis vector $e_{1}$ in two-dimensional space as the marked arrow. For any matrix $a$ , the overlap of $a e_{1}$ with $e_{1}$ is exactly the top-left entry of $a$ . So $φ (a)$ equals the overlap of $a e_{1}$ with $e_{1}$ , and the marked arrow is $e_{1}$ itself.

What this tells us: the abstract rule "report the top-left entry" is secretly the concrete measurement "apply the matrix to $e_{1}$ and read the overlap with $e_{1}$ ". The averaging rule and the marked vector carry the same information.

Check your understanding Beginner

Formal definition Intermediate+

Let $A$ be a C*-algebra 39.01.01. A linear functional $φ : A \to C$ is positive if $φ (a^{*} a) \geq 0$ for all $a \in A$ ; positivity forces the Cauchy-Schwarz inequality $∣ φ (b^{*} a) ∣^{2} \leq φ (a^{*} a) φ (b^{*} b)$ and the self-adjointness $φ (a^{*}) = \overline{φ (a)}$ . A positive functional is automatically bounded, and on a unital algebra $∥ φ ∥ = φ (1)$ . A state is a positive linear functional of norm $1$ ; in the unital case this is the normalisation $φ (1) = 1$ ^{[Murphy Ch. 3]}. The set of all states is the state space $S (A)$ .

The state space is a convex subset of the dual $A^{*}$ : a convex combination $tφ + (1 - t) ψ$ of states is again a state. With the weak- $*$ topology (pointwise convergence, $φ_{i} \to φ$ iff $φ_{i} (a) \to φ (a)$ for every $a$ ), the state space of a unital C*-algebra is weak- $*$ compact, being a weak- $*$ closed subset of the unit ball of $A^{*}$ , compact by Banach-Alaoglu 02.11.08. A pure state is an extreme point of $S (A)$ : a state that cannot be written as $tφ + (1 - t) ψ$ with $0 < t < 1$ and $φ \neq = ψ$ both states. By the Krein-Milman theorem the state space is the weak- $*$ closed convex hull of its pure states, so pure states exist in abundance.

A representation of $A$ is a $*$ -homomorphism $π : A \to B (H)$ into the bounded operators on a Hilbert space $H$ . A vector $ξ \in H$ is cyclic for $π$ if the orbit $π (A) ξ = {π (a) ξ : a \in A}$ is dense in $H$ . A representation is non-degenerate if $π (A) H$ is dense, and irreducible if $H$ has no closed $π (A)$ -invariant subspace other than $0$ and $H$ . Two representations $(π_{1}, H_{1})$ and $(π_{2}, H_{2})$ are unitarily equivalent if there is a unitary $U : H_{1} \to H_{2}$ with $U π_{1} (a) = π_{2} (a) U$ for all $a$ .

Counterexamples to common slips

A positive functional need not respect multiplication. On $M_{2} (C)$ the normalised trace $τ (a) = \frac{1}{2} tr (a)$ is a state, but $τ (ab) \neq = τ (a) τ (b)$ in general; states are not characters, and only in the commutative case do the pure states coincide with the characters.
Positivity needs the star, not the spectrum alone. A functional satisfying $φ (a) \geq 0$ merely for self-adjoint $a$ with positive spectrum is not the definition; positivity is the condition $φ (a^{*} a) \geq 0$ for every $a$ , which is what drives the Cauchy-Schwarz inequality.
A cyclic vector need not be a state vector for a faithful representation. The GNS vector $ξ_{φ}$ is cyclic but the representation $π_{φ}$ can have a kernel; faithfulness is recovered only by summing over enough states, as in the universal representation.

Key theorem with proof Intermediate+

Theorem (GNS construction). Let $A$ be a C-algebra and $φ$ a state on $A$ . There exist a Hilbert space $H_{φ}$ , a representation $π_{φ} : A \to B (H_{φ})$ , and a cyclic vector $ξ_{φ} \in H_{φ}$ such that* $$ \varphi(a) = \langle \pi_\varphi(a)\xi_\varphi, \xi_\varphi \rangle \qquad (a \in A). $$ The triple $(π_{φ}, H_{φ}, ξ_{φ})$ is unique up to unitary equivalence among cyclic triples implementing $φ$ . ^{[Murphy Ch. 3; Davidson Ch. I]}

Proof. Define a sesquilinear form on $A$ by $$ \langle a, b \rangle_\varphi = \varphi(b^* a). $$ It is linear in the first slot, conjugate-linear in the second (using $φ (a^{*}) = \overline{φ (a)}$ ), and positive semidefinite since $⟨ a, a ⟩_{φ} = φ (a^{*} a) \geq 0$ by positivity of $φ$ . By Cauchy-Schwarz the set $$ N_\varphi = { a \in A : \varphi(a^a) = 0 } $$ is exactly ${a : ⟨ a, b ⟩_{φ} = 0 for all b}$ , the null space of the form. It is a closed left ideal: if $a \in N_{φ}$ and $c \in A$ , then $\varphi((ca)^(ca)) = \varphi(a^* c^* c a) \le |c^*c|,\varphi(a^a) = 0 $, u s in g t ha t$ b \mapsto \varphi(a^ b a) $i s a p os i t i v e f u n c t i o na l d o mina t e d b y$ |b|\varphi(a^*a) $(t h e in e q u a l i t y$ c^*c \le |c|^2 1 $in t h e u ni t i s a t i o n t r an s f er s u n d er an y p os i t i v e f u n c t i o na l) . H e n ce$ ca \in N_\varphi$.

On the quotient vector space $A / N_{φ}$ the induced form $$ \langle a + N_\varphi, b + N_\varphi \rangle = \varphi(b^* a) $$ is a genuine inner product (positive definite, as the null directions were divided out). Let $H_{φ}$ be the completion of $A / N_{φ}$ to a Hilbert space, and write $[a] = a + N_{φ}$ for the image of $a$ .

Define the representation by left multiplication: $π_{φ} (a) [b] = [ab]$ . This is well-defined because $N_{φ}$ is a left ideal ( $b \in N_{φ} \Rightarrow ab \in N_{φ}$ ), and it is bounded: $$ |\pi_\varphi(a)[b]|^2 = \varphi(b^* a^* a b) \le |a^a|,\varphi(b^b) = |a|^2,|[b]|^2, $$ again using $a^{*} a \leq ∥ a ∥^{2} 1$ applied inside the positive functional $b \mapsto φ (b^{*} \cdot b)$ . So $π_{φ} (a)$ extends to a bounded operator on $H_{φ}$ with $∥ π_{φ} (a) ∥ \leq ∥ a ∥$ . The map $π_{φ}$ is linear, multiplicative ( $π_{φ} (a) π_{φ} (a^{'}) [b] = [a a^{'} b] = π_{φ} (a a^{'}) [b]$ ), and $$-preserving: $$ \langle \pi_\varphi(a)[b], [c]\rangle = \varphi(c^ a b) = \varphi((a^* c)^* b) = \langle [b], \pi_\varphi(a^)[c]\rangle, $$ so $\pi_\varphi(a)^ = \pi_\varphi(a^*) $. T h u s$ \pi_\varphi$ is a representation.

For the cyclic vector, assume first $A$ unital and set $ξ_{φ} = [1]$ . Then $$ \langle \pi_\varphi(a)\xi_\varphi, \xi_\varphi\rangle = \langle [a], [1]\rangle = \varphi(1^* a) = \varphi(a), $$ and $π_{φ} (A) ξ_{φ} = {[a] : a \in A} = A / N_{φ}$ is dense in $H_{φ}$ by construction, so $ξ_{φ}$ is cyclic and is a unit vector since $∥ ξ_{φ} ∥^{2} = φ (1) = 1$ . (In the non-unital case one uses an approximate unit $(u_{λ})$ : the net $[u_{λ}]$ is Cauchy and its limit $ξ_{φ}$ serves as the cyclic vector, with $φ (a) = lim_{λ} φ (u_{λ} a u_{λ}) = ⟨ π_{φ} (a) ξ_{φ}, ξ_{φ} ⟩$ .)

Uniqueness: suppose $(π, H, ξ)$ is another cyclic triple with $φ (a) = ⟨ π (a) ξ, ξ ⟩$ . Define $U [a] = π (a) ξ$ . This is isometric, $$ |\pi(a)\xi|^2 = \langle \pi(a^*a)\xi, \xi\rangle = \varphi(a^*a) = |[a]|^2, $$ so it is well-defined on $A / N_{φ}$ and extends to an isometry $U : H_{φ} \to H$ with dense range $π (A) ξ = H$ , hence a unitary. It intertwines, $U π_{φ} (a) = π (a) U$ , and sends $ξ_{φ}$ to $ξ$ . $□$

Bridge. The GNS construction builds toward the entire representation theory of C*-algebras, and it appears again in 39.03.01 where the von Neumann algebra $π_{φ} (A)^{''}$ generated by a representation is the next structural layer and the cyclic vector becomes a (separating, in the faithful case) trace or weight vector. The foundational reason it works is exactly the C*-positivity inequality $a^{*} a \leq ∥ a ∥^{2} 1$ pushed through the positive functional $φ$ : it both bounds the operators $π_{φ} (a)$ and forces the null ideal $N_{φ}$ to be a left ideal, so left multiplication descends to the quotient. This is exactly the abstract face of the quantum-mechanical statement that a prepared state is a vector and an observable's expectation is an inner product; the construction generalises the commutative Gelfand picture 39.01.02, where a character is a one-dimensional GNS representation and the state space degenerates to the character space. Putting these together, every averaging rule on an abstract algebra is realised as a concrete vector measurement, and the bridge is that summing these realisations over all states embeds the whole algebra into operators on a Hilbert space.

Exercises Intermediate+

Exercise 6 (hard, short-answer).

Prove that a state $φ$ is pure if and only if the GNS representation $π_{φ}$ is irreducible.

Hint

Relate positive functionals dominated by $φ$ to positive operators in the commutant $π_{φ} (A)^{'}$ via $T \mapsto ⟨ T π_{φ} (\cdot) ξ_{φ}, ξ_{φ} ⟩$ .

Answer

There is an order isomorphism between positive functionals $ψ \leq φ$ (meaning $φ - ψ$ is positive) and operators $T$ in the commutant $π_{φ} (A)^{'}$ with $0 \leq T \leq 1$ , given by $ψ (a) = ⟨ π_{φ} (a) T ξ_{φ}, ξ_{φ} ⟩$ . Indeed any such $ψ$ satisfies a Cauchy-Schwarz bound through $φ$ , so it factors through $H_{φ}$ as a bounded positive sesquilinear form, represented by a positive contraction $T$ ; the relation $ψ (ab)$ matching on both sides forces $T \in π_{φ} (A)^{'}$ . Now $φ$ is pure iff the only $ψ$ with $0 \leq ψ \leq φ$ are scalar multiples $tφ$ , $0 \leq t \leq 1$ ; under the isomorphism this says the only $T \in π_{φ} (A)^{'}$ with $0 \leq T \leq 1$ are scalars $t \cdot 1$ , i.e. $π_{φ} (A)^{'} = C 1$ . By Schur's lemma $π_{φ} (A)^{'} = C 1$ holds iff $π_{φ}$ is irreducible. Hence $φ$ pure $⟺$ $π_{φ}$ irreducible.

Rubric: full credit for the dominated-functional/commutant correspondence and the Schur's-lemma identification.

Exercise 7 (hard, short-answer).

Show that for every nonzero $a$ in a C*-algebra there is a state $φ$ with $φ (a^{*} a) = ∥ a ∥^{2}$ , and deduce that the universal representation is isometric.

Hint

Apply Hahn-Banach / the functional calculus to the positive element $a^{*} a$ inside the commutative subalgebra it generates, then extend to a state on all of $A$ .

Answer

The element $h = a^{*} a$ is positive with $∥ h ∥ = ∥ a ∥^{2}$ , and $∥ h ∥ \in σ (h)$ since $h \geq 0$ . In the commutative C*-algebra $C^{*} (h, 1) ≅ C (σ (h))$ the evaluation at the point $∥ h ∥ \in σ (h)$ is a character $τ$ with $τ (h) = ∥ h ∥$ ; it is a state on $C^{*} (h, 1)$ . By the Hahn-Banach extension for positive functionals (extend preserving positivity and norm), $τ$ extends to a state $φ$ on $A$ with $φ (h) = ∥ h ∥ = ∥ a ∥^{2}$ . Then in the universal representation $π_{u} = ⨁_{ψ \in S (A)} π_{ψ}$ , the summand $π_{φ}$ gives $∥ π_{φ} (a) ξ_{φ} ∥^{2} = φ (a^{*} a) = ∥ a ∥^{2} = ∥ a ∥^{2} ∥ ξ_{φ} ∥^{2}$ , so $∥ π_{φ} (a) ∥ = ∥ a ∥$ . Since each $π_{ψ}$ is norm-decreasing and one summand attains $∥ a ∥$ , $∥ π_{u} (a) ∥ = ∥ a ∥$ . Hence $π_{u}$ is isometric.

Rubric: full credit for the state attaining $∥ a ∥^{2}$ via functional calculus + Hahn-Banach and the isometry of $π_{u}$ .

Lean formalization Intermediate+

lean_status: none — Mathlib carries the C*-algebra layer (CStarAlgebra, positivity through cfc, the spectrum) and the dual-space / Banach-Alaoglu machinery, but the state space as a named weak-* compact convex set and the GNS construction as a stated theorem are not packaged: the form $φ (b^{*} a)$ , the null left ideal $N_{φ}$ , the completion $H_{φ}$ , the left-regular representation $π_{φ}$ with cyclic vector $ξ_{φ}$ , and the noncommutative Gelfand-Naimark embedding are absent as a consolidated development.

The intended statement reads schematically:

import Mathlib.Analysis.CStarAlgebra.Basic
import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic

variable {A : Type*} [CStarAlgebra A]

/-- GNS: from a state φ build a cyclic representation reproducing φ as a
vector functional. -/
theorem gns_construction (φ : A →ₗ[ℂ] ℂ)
    (hpos : ∀ a, 0 ≤ (φ (star a * a)).re) (hstate : φ 1 = 1) :
    ∃ (H : Type*) (_ : NormedAddCommGroup H) (_ : InnerProductSpace ℂ H)
      (π : A →⋆ₐ[ℂ] (H →L[ℂ] H)) (ξ : H),
      (∀ a, φ a = inner (π a ξ) ξ) ∧ (Dense (Set.range fun a => π a ξ)) :=
  sorry  -- quotient by N_φ, completion, left-regular representation

Advanced results Master

The representation theory assembles from the GNS construction by varying the state and collecting the pieces.

The universal representation and noncommutative Gelfand-Naimark. Form the universal representation $π_{u} = ⨁_{φ \in S (A)} π_{φ}$ on $H_{u} = ⨁_{φ} H_{φ}$ . For each nonzero $a$ , the functional calculus on $a^{*} a$ together with Hahn-Banach produces a state $φ$ with $φ (a^{*} a) = ∥ a ∥^{2}$ , so $∥ π_{φ} (a) ∥ = ∥ a ∥$ , whence $∥ π_{u} (a) ∥ = ∥ a ∥$ : the universal representation is isometric, in particular faithful. This is the Gelfand-Naimark theorem: every C*-algebra is isometrically $*$ -isomorphic to a norm-closed $*$ -subalgebra of $B (H_{u})$ . The abstract C*-axiom $∥ a^{*} a ∥ = ∥ a ∥^{2}$ is precisely the abstract characterisation — an involutive Banach algebra satisfying it is realisable as operators, and conversely any norm-closed $*$ -subalgebra of $B (H)$ satisfies it.

Pure states and irreducible representations. A state is pure iff its GNS representation is irreducible. The correspondence is an order isomorphism between functionals $ψ$ dominated by $φ$ and positive contractions in the commutant $π_{φ} (A)^{'}$ ; purity is extremality, which translates to $π_{φ} (A)^{'} = C 1$ , irreducibility by Schur's lemma. Two pure states give unitarily equivalent irreducible representations iff they are related by a unitary in the (multiplier) algebra; the pure state space modulo this equivalence is the spectrum $A$ of the C*-algebra, the noncommutative analogue of the character space, and for commutative $A$ it is exactly the Gelfand character space 39.01.02.

Existence of states and the order structure. States exist in profusion: the functional calculus and Hahn-Banach guarantee, for each self-adjoint $a$ and each $λ \in σ (a)$ , a state $φ$ with $φ (a) = λ$ . The state space recovers the order: $a \geq 0$ iff $φ (a) \geq 0$ for every state $φ$ , and $∥ a ∥ = sup_{φ} ∣ φ (a) ∣$ for self-adjoint $a$ . Thus the positive cone, the norm, and the spectrum are all readable off the state space, which is why $S (A)$ is a complete invariant of the order-unit structure.

Synthesis. The GNS construction is the foundational reason the abstract C*-axiom suffices to characterise operator algebras: it converts each state into a concrete cyclic representation, and summing over all states gives the central insight that the universal representation is isometric, so an abstract C*-algebra is exactly a norm-closed $*$ -subalgebra of some $B (H)$ . This is dual to the commutative Gelfand-Naimark theorem 39.01.02 — there characters embed a commutative algebra into $C (X)$ , here states embed an arbitrary algebra into $B (H)$ , and putting these together the character space is the pure-state space of a commutative algebra. The construction generalises in two directions that appear again across operator algebras: downward to the single normal element, where the GNS space of a state on $C^{*} (a, 1)$ is an $L^{2}$ space against a spectral measure and the calculus of 39.01.01 reappears, and upward to von Neumann algebras 39.03.01, where the bicommutant $π_{φ} (A)^{''}$ of a GNS representation is the weak closure and the cyclic vector becomes the separating trace or weight vector of Tomita-Takesaki theory. The bridge to the rest of the subject is that pure states are exactly the irreducible representations, so the noncommutative spectrum $A$ — the orbit space of pure states — is the object that replaces the character space once commutativity is dropped, and the entire classification program reads off this space.

Full proof set Master

Proposition (positive functionals are bounded, with $∥ φ ∥ = φ (1)$ in the unital case). Let $φ$ be positive on a unital C*-algebra $A$ . For self-adjoint $a$ with $∥ a ∥ \leq 1$ , $1 - a \geq 0$ and $1 + a \geq 0$ , so $φ (a) \leq φ (1)$ and $- φ (a) \leq φ (1)$ , giving $∣ φ (a) ∣ \leq φ (1)$ . For general $a$ with $∥ a ∥ \leq 1$ , Cauchy-Schwarz gives $∣ φ (a) ∣^{2} = ∣ φ (1^{*} a) ∣^{2} \leq φ (1) φ (a^{*} a) \leq φ (1)^{2} ∥ a ∥^{2} \leq φ (1)^{2}$ , where $φ (a^{*} a) \leq φ (1) ∥ a^{*} a ∥ \leq φ (1)$ uses $a^{*} a \leq ∥ a ∥^{2} 1 \leq 1$ . Hence $∥ φ ∥ \leq φ (1)$ , and the reverse is immediate from $φ (1) = ∣ φ (1) ∣ \leq ∥ φ ∥∥1∥ = ∥ φ ∥$ . So $∥ φ ∥ = φ (1)$ .

Proposition (Cauchy-Schwarz for positive functionals). For a positive functional $φ$ and $a, b \in A$ , the form $⟨ a, b ⟩ = φ (b^{*} a)$ is a positive semidefinite Hermitian form: $\overline{φ (b^{*} a)} = φ ((b^{*} a)^{*}) = φ (a^{*} b)$ and $φ (a^{*} a) \geq 0$ . The standard Cauchy-Schwarz argument for semidefinite forms — minimising $φ ((a - t b)^{*} (a - t b)) \geq 0$ over $t \in C$ — yields $∣ φ (b^{*} a) ∣^{2} \leq φ (a^{*} a) φ (b^{*} b)$ . Taking $b = 1$ (unital case) gives $∣ φ (a) ∣^{2} \leq φ (1) φ (a^{*} a)$ .

Proposition (the GNS form is well-defined and $π_{φ}$ is bounded). The null space $N_{φ} = {a : φ (a^{*} a) = 0}$ equals ${a : φ (b^{*} a) = 0 \forall b}$ by Cauchy-Schwarz, hence is a closed subspace; it is a left ideal by the domination $φ (a^{*} c^{*} c a) \leq ∥ c ∥^{2} φ (a^{*} a)$ . The induced inner product on $A / N_{φ}$ is positive definite. Left multiplication is bounded by $∥ π_{φ} (a) [b] ∥^{2} = φ (b^{*} a^{*} ab) \leq ∥ a ∥^{2} φ (b^{*} b)$ , again from $a^{*} a \leq ∥ a ∥^{2} 1$ pushed through the positive functional $x \mapsto φ (b^{*} x b)$ . So $π_{φ} (a)$ extends to $H_{φ}$ with $∥ π_{φ} (a) ∥ \leq ∥ a ∥$ , and $π_{φ}$ is a $*$ -homomorphism.

Proposition (Gelfand-Naimark: the universal representation is isometric). For nonzero $a$ , put $h = a^{*} a \geq 0$ with $∥ h ∥ = ∥ a ∥^{2}$ and $∥ h ∥ \in σ (h)$ . Evaluation at $∥ h ∥$ on $C^{*} (h, 1) ≅ C (σ (h))$ is a state $τ_{0}$ with $τ_{0} (h) = ∥ h ∥$ ; extend it by Hahn-Banach (preserving positivity and norm) to a state $φ$ on $A$ . In the GNS representation, $∥ π_{φ} (a) ξ_{φ} ∥^{2} = φ (a^{*} a) = ∥ a ∥^{2}$ , so $∥ π_{φ} (a) ∥ \geq ∥ a ∥$ ; with $∥ π_{φ} (a) ∥ \leq ∥ a ∥$ always, equality holds. Hence $∥ π_{u} (a) ∥ = sup_{φ} ∥ π_{φ} (a) ∥ = ∥ a ∥$ , and $π_{u}$ is an isometric $*$ -isomorphism onto its (norm-closed) image in $B (H_{u})$ .

Proposition (pure $⟺$ irreducible). Fix a state $φ$ . Positive functionals $ψ$ with $0 \leq ψ \leq φ$ correspond bijectively and order-isomorphically to operators $T \in π_{φ} (A)^{'}$ with $0 \leq T \leq 1$ via $ψ (a) = ⟨ π_{φ} (a) T ξ_{φ}, ξ_{φ} ⟩$ ; the cyclicity of $ξ_{φ}$ makes the correspondence injective and the domination bound makes it surjective. Extreme points of the order interval $[0, φ]$ are the $tφ$ on one side and the $0, 1$ -spectral projections of $T$ on the other. So $φ$ is extreme in the state space iff the only $T \in π_{φ} (A)^{'}$ in $[0, 1]$ are scalars, iff $π_{φ} (A)^{'} = C 1$ , iff (Schur) $π_{φ}$ is irreducible.

Proposition (existence of pure states separating points). For self-adjoint $a \neq = 0$ choose $λ \in σ (a)$ with $∣ λ ∣ = ∥ a ∥$ ; a state with $φ (a) = λ$ exists as above. The set of states $ψ$ with $ψ (a) = ∥ a ∥$ is a weak-* compact convex face of $S (A)$ , nonempty, so by Krein-Milman it has an extreme point, which is extreme in $S (A)$ as well (a face's extreme points are global extreme points), hence a pure state. Thus pure states separate self-adjoint elements, and by polarisation all elements: the pure states alone already give a faithful (atomic) representation.

Connections Master

Commutative C-algebras and Gelfand duality 39.01.02* — the commutative Gelfand-Naimark theorem is the GNS theory specialised: characters are exactly the pure states of a commutative C*-algebra, each is a one-dimensional GNS representation, and the noncommutative spectrum $A$ collapses to the character space; this unit is the noncommutative completion of that picture.
C-algebras: axioms, spectrum, and the continuous functional calculus 39.01.01* — the existence-of-states argument runs through the continuous functional calculus on the positive element $a^{*} a$ , and a state on the singly-generated subalgebra $C^{*} (a, 1)$ has a GNS space that is an $L^{2}$ space against a spectral measure, recovering the calculus from the representation side.
Hilbert space 02.11.08 — the GNS construction is a completion-to-a-Hilbert-space procedure: the state defines a semidefinite form, the null directions are quotiented, and the completion supplies the space on which the algebra acts; the cyclic vector is the geometric encoding of the state.
Von Neumann algebras and the bicommutant theorem 39.03.01 — the weak-operator closure $π_{φ} (A)^{''}$ of a GNS representation is the von Neumann algebra it generates, the bicommutant theorem identifies it with the bicommutant, and the cyclic vector becomes the separating/cyclic vector underlying Tomita-Takesaki modular theory.
Spectral triples and noncommutative geometry 39.06.01 — the noncommutative spectrum $A$ of pure states is the pointless space on which noncommutative geometry operates; the GNS representation supplies the Hilbert space a spectral triple acts on, and irreducibility singles out the points of the noncommutative space.

Historical & philosophical context Master

The construction is named for Israel Gelfand, Mark Naimark, and Irving Segal. Gelfand and Naimark proved in 1943 that an abstract C*-algebra (then a $B^{*}$ -algebra, defined by $∥ a^{*} a ∥ = ∥ a ∥^{2}$ ) is isometrically $*$ -isomorphic to a norm-closed $*$ -subalgebra of the bounded operators on a Hilbert space ^{[Gelfand-Naimark 1943]}. Their original argument carried a redundant extra axiom, later removed; the route through states and a single representation was given its definitive form by Segal in 1947, who introduced the systematic construction of a cyclic representation from a positive functional and connected pure states to irreducible representations ^{[Segal 1947]}.

The physical origin is quantum mechanics: a state in the operator-algebraic sense is the mathematical content of a quantum-mechanical state — a normalised positive expectation functional on the observables — and the GNS Hilbert space is the state's own representation space, the algebraic substitute for the wavefunction. The identification of pure states with irreducible representations is the algebraic form of the superposition principle, and the universal representation is the statement that the abstract algebra of observables is always realisable concretely. Von Neumann's earlier program of rings of operators supplied the ambient setting, and the GNS construction became the standard bridge from the abstract algebra of observables to a concrete Hilbert-space model used throughout algebraic quantum field theory and quantum statistical mechanics.

Bibliography Master

Davidson, K. R., C-Algebras by Example*, Fields Institute Monographs 6, American Mathematical Society, 1996. Ch. I.
Murphy, G. J., C-Algebras and Operator Theory*, Academic Press, 1990. Ch. 3.
Gelfand, I. and Naimark, M., "On the imbedding of normed rings into the ring of operators in Hilbert space", Mat. Sbornik 12 (1943), 197–213.
Segal, I. E., "Irreducible representations of operator algebras", Bulletin of the American Mathematical Society 53 (1947), 73–88.
Dixmier, J., C-Algebras*, North-Holland, 1977. §2.
Takesaki, M., Theory of Operator Algebras I, Springer, 1979. Ch. I §9.

Operator-algebras spine, representation-theoretic foundation of the C-basics chapter. Produced as the GNS / Gelfand-Naimark anchor: states and the state space, the cyclic representation built from a state, the universal representation and the noncommutative embedding into B(H), and the pure-state / irreducible-representation correspondence.*

Prerequisites

39.01.01
39.01.02
02.11.08

Tier anchors

beginner: Strogatz-style 'a state is an averaging rule, and every averaging rule comes from a vector in a Hilbert space'; the expectation-value picture from quantum mechanics
intermediate: Murphy *C*-Algebras and Operator Theory* Ch. 3 (states and the GNS construction); Davidson *C*-Algebras by Example* Ch. I
master: Davidson *C*-Algebras by Example* Ch. I; Murphy Ch. 3; Dixmier *C*-Algebras* §2; Takesaki *Theory of Operator Algebras I* Ch. I §9

References

Davidson, K. R. — C*-Algebras by Example · Ch. I, states, the GNS construction, the universal representation, Gelfand-Naimark
Murphy, G. J. — C*-Algebras and Operator Theory · Ch. 3, positive linear functionals, states, the GNS construction, irreducibility and pure states
Gelfand, I. and Naimark, M. — On the imbedding of normed rings into the ring of operators in Hilbert space · Mat. Sbornik 12 (1943), 197–213 — every C*-algebra embeds isometrically into B(H)
Segal, I. E. — Irreducible representations of operator algebras · Bull. Amer. Math. Soc. 53 (1947), 73–88 — the GNS construction and pure-state irreducibility

Estimated time

beginner: 18m
intermediate: 50m
master: 90m