39.05.02 · operator-algebras / nuclearity-exactness

Operator Systems, Arveson's Extension Theorem, and the Choi-Effros Theorem

shipped3 tiersLean: none

Anchor (Master): Paulsen *Completely Bounded Maps and Operator Algebras* (Cambridge UP, 2002) Ch. 6, 13-15; Brown-Ozawa *C*-Algebras and Finite-Dimensional Approximations* (AMS, 2008) Ch. 1; Effros-Ruan *Operator Spaces* (Oxford UP, 2000) Ch. 2-5

Intuition Beginner

A C*-algebra is a closed world of operators: you can add them, multiply them, star them, and the positive ones form a tidy cone. Sometimes you only care about a piece of that world that is closed under fewer operations. Keep the ability to add, to scale, to star, and to recognise which elements are positive, but give up multiplication. What remains is an operator system: a window onto the algebra that still respects positivity but no longer multiplies.

Why bother with a window instead of the whole house? Because the natural maps in this subject, the completely positive maps from the previous unit, often start life only on such a window. You measure part of a quantum apparatus and get a positivity-respecting rule defined on the part you can see. The first basic question is whether you can always extend that partial reading to the full apparatus without breaking positivity. Arveson's theorem says yes: any such partial reading finishes to a full one. This is the operator version of the rule that a sensible partial measurement never has to be thrown away.

The second question is the reverse. Suppose you are handed an abstract gadget with a notion of "positive at every matrix size" and a special element playing the role of the identity. Is it secretly a window onto some algebra of operators? The Choi-Effros theorem says yes again: the abstract positivity data is always realised concretely.

Visual Beginner

An operator system is a window $S$ inside a C*-algebra $A$ ; a positivity-respecting reading on the window always extends to the whole algebra.

The dictionary reads: the window $S$ is the operator system, the solid arrow $φ$ is the positivity-respecting reading defined only on the window, and the dashed arrow $\tilde{φ}$ is its extension to the full algebra. The lower panel is the converse: an abstract space carrying matrix-level positivity and an identity-like element is realised as a concrete window.

Worked example Beginner

Take the algebra of two-by-two complex matrices. Inside it, keep only the diagonal matrices: the ones whose two off-diagonal corners are zero. This collection is closed under addition, under scaling, and under the star (the conjugate transpose of a diagonal matrix is again diagonal). It contains the identity matrix. And we can recognise its positive members: a diagonal matrix is positive exactly when both diagonal entries are at least zero. So the diagonal matrices form a window: an operator system sitting inside the full matrix algebra.

Now define a reading on this window. Send each diagonal matrix to the single number obtained by averaging its two diagonal entries, then place that number on the diagonal of a new one-by-one matrix. This sends positive to positive, so it is a sensible reading on the window. Arveson's promise is that we can extend it to all two-by-two matrices, not just the diagonal ones. One extension keeps the same recipe: average the two diagonal entries of any matrix, ignore the corners. It still sends positive matrices to non-negative numbers, because the diagonal of a positive matrix is non-negative.

What this tells us: the window carried just enough structure (addition, star, an identity, a positivity test) to make the reading meaningful, and the reading extended to the whole algebra without inventing anything. The corners we ignored were exactly the multiplication data the window had dropped.

Check your understanding Beginner

Exercise (easy, multiple choice).

Which list of features does an operator system keep from its surrounding C*-algebra?

A. Addition, scaling, the star, and a notion of positivity — but not multiplication B. Only multiplication C. Addition and multiplication, but not the star D. Nothing; it is just an unstructured set

Hint

The point of the window is that it is closed under fewer operations than the full algebra; one familiar operation is dropped.

Answer

A. Addition, scaling, the star, and positivity, but not multiplication.

Feedback-correct: correct; an operator system is a self-adjoint subspace containing the identity, with the inherited positivity, and it need not be closed under products. Feedback-wrong: multiplication is exactly the operation a general operator system may lack; the star and the identity are kept.

Formal definition Intermediate+

Let $A$ be a unital C*-algebra 39.01.01. An operator system is a closed subspace $S \subseteq A$ that is self-adjoint ( $S = S^{*}$ ) and unital ( $1 \in S$ ). Self-adjointness makes $S$ a $*$ -vector space; together with $1 \in S$ it inherits not only the order on $S$ itself but the matrix order: for each $n \geq 1$ the matrix space $M_{n} (S) \subseteq M_{n} (A)$ carries the cone $$ M_n(S)+ = M_n(S) \cap M_n(A)+ , $$ where $M_{n} (A)_{+}$ is the positive cone of the C*-algebra $M_{n} (A)$ 39.05.01. The whole family $(M_{n} (S)_{+})_{n \geq 1}$ , together with the unit $1 \in S = M_{1} (S)$ , is the data that makes $S$ an operator system; the element $1$ is an order unit in the sense that for every self-adjoint $s \in S$ there is $r > 0$ with $r 1 + s \geq 0$ and $r 1 - s \geq 0$ .

A linear map $φ : S \to T$ between operator systems is unital if $φ (1) = 1$ , and completely positive (CP) if every amplification $φ_{n} = φ \otimes id_{n} : M_{n} (S) \to M_{n} (T)$ sends $M_{n} (S)_{+}$ into $M_{n} (T)_{+}$ ^{[Paulsen Ch. 6]}. A unital complete order isomorphism is a unital bijection $φ$ with $φ$ and $φ^{- 1}$ both CP; two operator systems are then identified.

An abstract operator system is a complex $*$ -vector space $V$ with a distinguished family of cones $C_{n} \subseteq M_{n} (V)_{s a}$ (the self-adjoint part of $M_{n} (V)$ ) that is a matrix ordering: each $C_{n}$ is a proper cone, and compatible, meaning $α^{*} C_{n} α \subseteq C_{m}$ for every scalar matrix $α \in M_{n, m} (C)$ . A self-adjoint element $e \in V$ is a matrix order unit if for every self-adjoint $v \in M_{n} (V)$ there is $r > 0$ with $r (1_{n} \otimes e) + v \in C_{n}$ ; it is Archimedean if whenever $r (1_{n} \otimes e) + v \in C_{n}$ for all $r > 0$ , then $v \in C_{n}$ ^{[Choi-Effros]}. The Archimedean condition is the abstract trace of norm-closedness.

For operator spaces the analogous abstract data is a family of norms. An operator space is a subspace $X \subseteq B (H)$ with the inherited matrix norms $∥ \cdot ∥_{n}$ on $M_{n} (X) \subseteq M_{n} (B (H)) = B (H^{\oplus n})$ . Abstractly, a matrix-normed space $(X, (∥ \cdot ∥_{n}))$ satisfies Ruan's axioms if $$ |\alpha x \beta|m \le |\alpha|,|x|n,|\beta| \quad (\alpha \in M{m,n}, \beta \in M{n,m}), \qquad \left| \begin{smallmatrix} x & 0 \ 0 & y \end{smallmatrix} \right|_{n+m} = \max{|x|_n, |y|_m} $$ for $x \in M_{n} (X)$ , $y \in M_{m} (X)$ ^[Ruan].

Counterexamples to common slips

An operator system is not a subalgebra. The diagonal-plus-antidiagonal subspace of $M_{2} (C)$ , or the span of $1, u, u^{*}$ for a unitary $u$ , is self-adjoint and unital but not closed under products. Operator systems drop multiplication; only the order and the unit survive.
The matrix order, not the order at level $1$ , is the right datum. Two operator systems can be order isomorphic at the scalar level $M_{1}$ yet fail to be completely order isomorphic. The transpose map on $M_{2} (C)$ is unital and positive (an order isomorphism at level $1$ ) but not $2$ -positive 39.05.01, so it is not a complete order isomorphism. Positivity must be tested at every matrix level.
Archimedean is not automatic for an abstract order unit. An order unit can fail the Archimedean condition; one must impose it (equivalently, pass to the Archimedeanisation) before the Choi-Effros realisation applies. Without it the abstract cone need not be the cone of a norm-closed concrete operator system.

Key theorem with proof Intermediate+

Theorem (Arveson's extension theorem). Let $A$ be a unital C-algebra, $S \subseteq A$ an operator system, and $φ : S \to B (H)$ a completely positive map. Then there exists a completely positive map $\tilde{φ} : A \to B (H)$ with $\tilde{φ} ∣_{S} = φ$ .* ^{[Arveson; Paulsen Ch. 6]}

Proof. Reduce first to finite-dimensional $H$ . If the theorem holds for every finite-dimensional compression, let $(P_{λ})$ be the net of finite-rank projections on $H$ ordered by range inclusion, set $φ_{λ} = P_{λ} φ (\cdot) P_{λ} : S \to B (P_{λ} H)$ , extend each to $\tilde{φ}_{λ} : A \to B (P_{λ} H) \subseteq B (H)$ , and note $∥ \tilde{φ}_{λ} (1) ∥ = ∥ φ_{λ} (1) ∥ \leq ∥ φ (1) ∥$ , so the net $(\tilde{φ}_{λ})$ is bounded. By Banach-Alaoglu in the point-weak- $*$ topology on the bounded set of maps $A \to B (H)$ it has a cluster point $\tilde{φ}$ , which is CP (each $\tilde{φ}_{λ}$ is and the cones are weak- $*$ closed) and restricts to $φ$ on $S$ .

So take $H = C^{m}$ , hence $B (H) = M_{m} (C)$ . Completely positive maps $S \to M_{m} (C)$ correspond to positive linear functionals on $M_{m} (S) = M_{m} (C) \otimes S$ by the canonical pairing: given $φ$ , define $s_{φ}$ on $M_{m} (S)$ by $$ s_\varphi\big([t_{ij}]\big) = \frac{1}{m}\sum_{i,j=1}^m \big\langle \varphi(t_{ij}) e_j, e_i\big\rangle , $$ where $(e_{i})$ is the standard basis of $C^{m}$ . Complete positivity of $φ$ is equivalent to positivity of $s_{φ}$ on the cone $M_{m} (S)_{+}$ : the matrix $[t_{ij}] \geq 0$ is paired against the rank-one positive functional encoded by the vector $\sum_{i} e_{i} \otimes e_{i}$ , and $φ_{m} ([t_{ij}]) \geq 0$ unwinds to $s_{φ} ([t_{ij}]) \geq 0$ . The unital pairing identity $s_{φ} (1_{M_{m} (S)}) = \frac{1}{m} tr φ (1)$ records the norm.

Now $M_{m} (S)$ is an operator system inside $M_{m} (A)$ with order unit $1_{m} = 1_{M_{m}} \otimes 1_{A}$ , and $s_{φ}$ is a positive linear functional on it. A positive functional on an operator system is bounded with $∥ s_{φ} ∥ = s_{φ} (1_{m})$ . By the Hahn-Banach theorem applied to the real ordered Banach space $M_{m} (A)_{s a}$ with order unit $1_{m}$ — extend the positive functional from the order-unit subspace $M_{m} (S)_{s a}$ to $M_{m} (A)_{s a}$ preserving the norm $∥ \cdot ∥ =$ the order-unit norm, so that positivity is preserved — there is a positive functional $\tilde{s}$ on $M_{m} (A)$ with $\tilde{s} ∣_{M_{m} (S)} = s_{φ}$ and $\tilde{s} (1_{m}) = s_{φ} (1_{m})$ . Reverse the pairing: $\tilde{s}$ defines a completely positive $\tilde{φ} : A \to M_{m} (C)$ by $⟨ \tilde{φ} (a) e_{j}, e_{i} ⟩ = m \tilde{s} (e_{ij} \otimes a)$ , and $\tilde{φ} ∣_{S} = φ$ because $\tilde{s}$ extends $s_{φ}$ . $□$

Bridge. Arveson's extension theorem builds toward the entire approximation theory of nuclear and exact C*-algebras, and it appears again in 39.05.03 where the identity map of a nuclear algebra is approximated by CP maps that factor through matrix algebras, each factoring map produced by extending off a finite-dimensional operator system. The foundational reason it holds is exactly the Stinespring dilation 39.05.01 reorganised as a positivity statement: a CP map into $B (H)$ is a matrix-level positive functional, and extending it is the order-theoretic Hahn-Banach theorem for the order unit $1$ . This is exactly the completely positive lift of the scalar Hahn-Banach extension of states used in the Gelfand-Naimark theory, and it generalises that scalar fact in one direction while it is dual to the injectivity of $B (H)$ in another — $B (H)$ receives every extension precisely because it is injective in the category of operator systems. Putting these together, the central insight is that $B (H)$ is the operator-system analogue of an injective module, and the bridge is that injectivity is what makes the partial-to-full extension automatic, which is the structural fact the whole nuclearity chapter rests on.

Exercises Intermediate+

Exercise 5 (medium, short-answer).

Show that the transpose $t : M_{2} (C) \to M_{2} (C)$ , viewed as a unital map of operator systems, is positive but not a complete order isomorphism.

Hint

Transposition preserves eigenvalues (so is positive at level $1$ ), but test $t_{2}$ on the positive matrix $[e_{ij}] \in M_{2} (M_{2} (C))$ .

Answer

Transposition is unital and sends a self-adjoint matrix to one with the same eigenvalues, hence positive: it is an order isomorphism at level $1$ . But the amplification $t_{2} = t \otimes id_{2}$ applied to the positive rank-one matrix $[e_{ij}] = (\sum_{i} e_{i 1}) (\sum_{j} e_{1 j})^{*} \in M_{2} (M_{2} (C))$ returns the swap operator, which has eigenvalue $- 1$ . So $t_{2}$ is not positive, $t$ is not $2$ -positive, and therefore not completely positive; a complete order isomorphism requires $t$ and $t^{- 1}$ both completely positive. Thus $t$ is a level- $1$ order isomorphism that fails completely, exhibiting the matrix order as strictly finer than the order at level $1$ .

Rubric: full credit for the level- $1$ positivity, the $[e_{ij}]$ amplification, and the eigenvalue $- 1$ certifying failure of $2$ -positivity.

Exercise 6 (hard, short-answer).

Prove that $B (H)$ is injective in the category of operator systems with completely positive maps: for any operator systems $S \subseteq T$ and any CP map $φ : S \to B (H)$ , there is a CP extension $\tilde{φ} : T \to B (H)$ .

Hint

Realise $T$ inside its containing C*-algebra and apply Arveson's extension theorem to the inclusion $S \subseteq T \subseteq A$ .

Answer

Let $T \subseteq A$ for a unital C*-algebra $A$ (every operator system sits in its generated C*-algebra), so $S \subseteq T \subseteq A$ is a chain of operator systems. Given CP $φ : S \to B (H)$ , Arveson's extension theorem extends $φ$ to a CP map $Φ : A \to B (H)$ . Restrict: $\tilde{φ} = Φ ∣_{T} : T \to B (H)$ is CP (restriction of a CP map to a sub-operator-system is CP, since each amplification restricts) and $\tilde{φ} ∣_{S} = Φ ∣_{S} = φ$ . Hence the CP extension across $S \subseteq T$ exists, which is the defining injectivity (lifting) property of $B (H)$ .

Rubric: full credit for embedding $T$ in a C*-algebra, applying Arveson, and restricting to $T$ .

Exercise 7 (hard, short-answer).

Let $Φ : B (H) \to B (H)$ be a unital completely positive idempotent ( $Φ^{2} = Φ$ ) onto an operator system $S = Φ (B (H))$ . Define the Choi-Effros product $a \circ b = Φ (ab)$ on $S$ and show $Φ (a Φ (b)) = Φ (Φ (a) b) = Φ (Φ (a) Φ (b))$ for all $a, b$ , so the product is well-defined on $S$ .

Hint

Use the multiplicative-domain identity from Stinespring: for a UCP idempotent, every $s \in S$ lies in the multiplicative domain of $Φ$ , so $Φ (s a) = Φ (s) Φ (a) = s Φ (a)$ .

Answer

Since $Φ$ is a UCP idempotent, $Φ ∣_{S} = id_{S}$ . For $s = Φ (b) \in S$ apply the multiplicative-domain analysis of $Φ$ 39.05.01: a UCP map satisfies $Φ (s a) = Φ (s) Φ (a)$ and $Φ (a s) = Φ (a) Φ (s)$ whenever $s$ is in the multiplicative domain ${x : Φ (x^{*} x) = Φ (x)^{*} Φ (x), Φ (x x^{*}) = Φ (x) Φ (x)^{*}}$ . Every $s \in S$ lies there because $Φ$ fixes $S$ and Kadison-Schwarz becomes equality on the fixed-point set of a UCP idempotent: $Φ (s^{*} s) \geq Φ (s)^{*} Φ (s) = s^{*} s$ while also $Φ (s^{*} s) \leq Φ (Φ (s^{*} s))$ forces $Φ (s^{*} s) = s^{*} s = Φ (s)^{*} Φ (s)$ . Therefore $$ \Phi(a,\Phi(b)) = \Phi(a),\Phi(b) \cdot \text{?} $$ more precisely, with $s = Φ (b)$ in the multiplicative domain, $Φ (a Φ (b)) = Φ (a) Φ (Φ (b)) = Φ (a) Φ (b)$ , and symmetrically $Φ (Φ (a) b) = Φ (Φ (a)) Φ (b) = Φ (a) Φ (b)$ and $Φ (Φ (a) Φ (b)) = Φ (a) Φ (b)$ . All three equal $Φ (a) Φ (b)$ , so $a \circ b := Φ (ab)$ depends only on $Φ (a), Φ (b)$ , hence descends to a well-defined bilinear product on $S$ . Associativity and the C*-identity for $∥ \cdot ∥$ on $S$ then make $(S, \circ)$ a C*-algebra with the inherited norm and involution.

Rubric: full credit for placing $S$ in the multiplicative domain (equality in Kadison-Schwarz for the idempotent) and deducing the three equalities.

Lean formalization Intermediate+

lean_status: none — Mathlib carries the C*-algebra positivity layer (CStarAlgebra, cfc, matrices over a C*-algebra and their order) but neither operator systems as a named structure (a self-adjoint unital subspace with the inherited family of matrix cones $M_{n} (S)_{+}$ ), the matrix-ordered $*$ -vector space with an Archimedean order unit, completely positive maps between abstract ordered spaces, nor the extension and realisation theorems. The intended statement reads schematically:

import Mathlib.Analysis.CStarAlgebra.Basic
import Mathlib.Analysis.CStarAlgebra.Matrix

variable {A : Type*} [CStarAlgebra A] {H : Type*}
  [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]

/-- Arveson extension: a completely positive map from an operator system
S ⊆ A into B(H) extends to a completely positive map on all of A. -/
theorem arveson_extension
    (S : Submodule ℂ A) (hS_unit : (1 : A) ∈ S) (hS_star : ∀ s ∈ S, star s ∈ S)
    (φ : S →ₗ[ℂ] (H →L[ℂ] H))
    (hcp : ∀ n (x : Matrix (Fin n) (Fin n) S), 0 ≤ x.map (Submodule.subtype S) →
            0 ≤ (x.map (fun s => φ s)))            -- complete positivity on S
    :
    ∃ ψ : A →ₗ[ℂ] (H →L[ℂ] H),
      (∀ s : S, ψ (s : A) = φ s) ∧
      (∀ n (x : Matrix (Fin n) (Fin n) A), 0 ≤ x → 0 ≤ (x.map ψ)) :=
  sorry  -- reduce to finite m, pair with positive functional on Mₘ(S),
         -- order-unit Hahn-Banach extension to Mₘ(A), pair back

Advanced results Master

The extension theorem and the abstract characterisation organise the operator-system theory into a self-dual package.

The Choi-Effros theorem. Let $V$ be a complex $*$ -vector space with a matrix ordering $(C_{n})$ and an Archimedean matrix order unit $e$ . Then there is a Hilbert space $K$ and a unital complete order isomorphism of $V$ onto a concrete operator system $S \subseteq B (K)$ ^{[Choi-Effros]}. The realisation is built from the matrix states: the set of UCP maps $V \to M_{n} (C)$ separates points (Archimedean order unit) and assembling a faithful family $⨁_{α} φ_{α}$ embeds $V$ completely order isomorphically. Thus abstract matrix-order data and concrete operator systems are the same theory; the matrix order, not the order at level $1$ , is the complete invariant, and the transpose example shows level $1$ alone is insufficient.

Injectivity and the Choi-Effros product. An operator system $S$ is injective if it has the extension property of the previous exercise: every CP map into $S$ extends across any operator-system inclusion. Equivalently $S$ is the image of a UCP idempotent $Φ : B (H) \to B (H)$ with $Φ (B (H)) = S$ . The Choi-Effros theorem on injective operator systems endows $S$ with the product $a \circ b = Φ (ab)$ , the involution and norm inherited from $B (H)$ , making $(S, \circ)$ a C*-algebra — indeed a monotone-complete (AW $^{*}$ , and for $H$ separable a von Neumann) C*-algebra — even though $S$ need not be a subalgebra of $B (H)$ . Injective von Neumann algebras are precisely the ranges of normal conditional expectations, and the chain of equivalences injective $=$ semidiscrete $=$ hyperfinite $=$ amenable for factors is the Connes classification, of which this construction is the operator-system germ.

Wittstock's completely bounded extension. The order-theoretic Hahn-Banach of Arveson has a norm-theoretic twin. A linear map $φ : S \to B (H)$ from an operator space is completely bounded with $∥ φ ∥_{c b} = sup_{n} ∥ φ_{n} ∥$ , and Wittstock's theorem extends a completely bounded map off a subspace of a C*-algebra to a completely bounded map on the whole algebra with the same cb-norm; for unital self-adjoint $φ$ on an operator system this specialises to Arveson via the identity $∥ φ ∥_{c b} = ∥ φ (1) ∥$ for CP maps. Ruan's theorem closes the abstraction: a matrix-normed space satisfying Ruan's two axioms is completely isometric to a concrete operator space $X \subseteq B (H)$ , the operator-space analogue of the Choi-Effros realisation ^[Ruan].

Paulsen's $2 \times 2$ trick. Completely bounded maps and completely positive maps are linked by the operator system $S_{φ} = {λ 1 ψ (y)^{*} φ (x) μ 1}$ : a map $φ$ on an operator space $X$ is completely contractive iff the associated map on the Paulsen system $S_{X} = (C 1 X^{*} X C 1)$ is completely positive. This reduces the completely bounded theory to the completely positive theory, so Wittstock's theorem follows from Arveson's by passing to the Paulsen system, and the two Hahn-Banach theorems are one theorem viewed through the $2 \times 2$ construction.

Synthesis. The operator-system framework is the foundational reason the maps of this subject extend: a CP map into $B (H)$ is a matrix-level positive functional, and this is exactly the order-theoretic Hahn-Banach datum, so Arveson's extension theorem generalises the scalar extension of states while it is dual to the injectivity of $B (H)$ . Putting these together, the central insight is the self-duality of the package — the Choi-Effros theorem says abstract matrix order is always concretely realised, and the Choi-Effros product says an injective such system is secretly a C*-algebra, so the category of operator systems contains its own algebra objects as its injective objects. This is exactly the operator-space mirror of Ruan's axioms, and the bridge to the rest of the chapter is that nuclearity 39.05.03 is the statement that the identity map factors approximately through matrix algebras by UCP maps, each factor built by the Arveson extension whose existence rests on $B (H)$ being injective. The whole nuclearity-exactness theory is the study of which algebras the injective-operator-system machinery can finitely approximate.

Full proof set Master

Proposition (CP maps into $M_{m}$ are positive functionals on $M_{m} (S)$ ). For an operator system $S$ and $m \geq 1$ , the pairing $φ \mapsto s_{φ}$ , $s_{φ} ([t_{ij}]) = \frac{1}{m} \sum_{i, j} ⟨ φ (t_{ij}) e_{j}, e_{i} ⟩$ , is a bijection between completely positive maps $φ : S \to M_{m} (C)$ and positive linear functionals on the operator system $M_{m} (S)$ . Indeed $M_{m} (S)_{+} = {[t_{ij}] : φ_{m} ([t_{ij}]) \geq 0 -detected}$ , and the matrix $[t_{ij}] \geq 0$ pairs against the vector $ζ = \sum_{i} e_{i} \otimes e_{i}$ as $⟨ φ_{m} ([t_{ij}]) ζ, ζ ⟩ = \sum_{i, j} ⟨ φ (t_{ij}) e_{j}, e_{i} ⟩ = m s_{φ} ([t_{ij}])$ , so positivity of $φ_{m}$ on the cone is positivity of $s_{φ}$ ; the inverse $⟨ φ (s) e_{j}, e_{i} ⟩ = m s_{φ} (e_{ij} \otimes s)$ recovers $φ$ , and complete positivity follows because every positive matrix in $M_{n} (M_{m} (S))$ rearranges to a positive element of $M_{nm} (S)$ paired against the corresponding $ζ$ .

Proposition (order-unit Hahn-Banach). Let $W$ be a real ordered vector space with Archimedean order unit $e$ , $W_{0} \subseteq W$ an order-unit subspace ( $e \in W_{0}$ ), and $f : W_{0} \to R$ positive with $∥ f ∥ = f (e)$ . Then $f$ extends to a positive $\tilde{f} : W \to R$ with $\tilde{f} (e) = f (e)$ . The order-unit norm $∥ w ∥ = in f {r > 0 : - r e \leq w \leq r e}$ makes $f$ a bounded functional of norm $f (e)$ on $(W_{0}, ∥ \cdot ∥)$ ; the classical Hahn-Banach theorem extends it to $\tilde{f}$ of the same norm; and $∥ \tilde{f} ∥ = \tilde{f} (e)$ forces positivity, since $w \leq r e$ gives $\tilde{f} (r e - w) = r f (e) - \tilde{f} (w) \geq 0$ by the norm bound $∣ \tilde{f} (w) ∣ \leq ∥ \tilde{f} ∥ ∥ w ∥$ . Applied to $W = M_{m} (A)_{s a}$ , $W_{0} = M_{m} (S)_{s a}$ , $e = 1_{m}$ , this is the engine of Arveson's theorem.

Proposition (Arveson extension). A CP map $φ : S \to B (H)$ on an operator system $S \subseteq A$ extends to a CP map $A \to B (H)$ . Reduce to $H = C^{m}$ by the point-weak- $*$ compactness argument on finite-rank compressions; pass to the positive functional $s_{φ}$ on $M_{m} (S)$ ; extend it by the order-unit Hahn-Banach proposition to a positive functional $\tilde{s}$ on $M_{m} (A)$ with $\tilde{s} (1_{m}) = s_{φ} (1_{m})$ ; pair back to obtain the CP extension $\tilde{φ}$ with $\tilde{φ} ∣_{S} = φ$ . The three propositions chain to give the theorem.

Proposition (the matrix states separate points of an Archimedean operator system). If $(V, (C_{n}), e)$ is a matrix-ordered $*$ -vector space with Archimedean matrix order unit, then for every nonzero self-adjoint $v \in V$ there is $n$ and a UCP map $φ : V \to M_{n} (C)$ with $φ (v) \neq = 0$ . By the Archimedean property the state space ${f : V \to C positive, f (e) = 1}$ is weak- $*$ compact and separates self-adjoint elements through their order-unit norm; amplifying to matrix states ( $V \to M_{n}$ ) only enriches the separating family. The direct sum $⨁_{α} φ_{α}$ over a separating family of matrix states is a unital complete order injection of $V$ into $B (⨁_{α} C^{n_{α}})$ , whose image is a concrete operator system — the Choi-Effros realisation.

Proposition (Choi-Effros product makes an injective operator system a C-algebra).* Let $Φ : B (H) \to B (H)$ be a UCP idempotent with range $S$ . The fixed-point set $S$ lies in the multiplicative domain of $Φ$ : for $s \in S$ , Kadison-Schwarz 39.05.01 gives $Φ (s^{*} s) \geq Φ (s)^{*} Φ (s) = s^{*} s$ , while $Φ$ unital idempotent and $∥Φ∥ = 1$ give $Φ (s^{*} s) \leq s^{*} s$ on the fixed set, so $Φ (s^{*} s) = Φ (s)^{*} Φ (s)$ and likewise for $s s^{*}$ . Hence $Φ (s a) = Φ (s) Φ (a) = s Φ (a)$ and $Φ (a s) = Φ (a) s$ for $s \in S$ . Define $a \circ b = Φ (ab)$ on $S$ ; the multiplicative-domain identities show it depends only on $a, b \in S$ and is associative: $(a \circ b) \circ c = Φ (Φ (ab) c) = Φ (ab c) = Φ (a Φ (b c)) = a \circ (b \circ c)$ . With the inherited involution and norm, $∥ a \circ a^{*} \dots ∥$ satisfies the C*-identity because the norm is the operator norm restricted to the fixed set, so $(S, \circ, *, ∥ \cdot ∥)$ is a C*-algebra. When $Φ$ is normal and $H$ separable, $S$ is a von Neumann algebra.

Connections Master

Completely positive maps and the Stinespring dilation theorem 39.05.01 — Stinespring is the engine of Arveson's theorem: a CP map into $B (H)$ is a compressed representation, and the extension recompresses an extended one. The multiplicative-domain analysis there is exactly what equips an injective operator system with the Choi-Effros product, so this unit is the operator-system completion of the dilation theory.
C-algebras: axioms, spectrum, and the continuous functional calculus 39.01.01* — the matrix order $M_{n} (S)_{+} = M_{n} (S) \cap M_{n} (A)_{+}$ that defines an operator system is inherited from the C*-positivity of $M_{n} (A)$ built on the functional calculus, and the order-unit norm that drives the Hahn-Banach extension is the C*-norm restricted to $S$ ; the whole framework runs on that order.
Nuclear and exact C-algebras 39.05.03* — nuclearity is the approximate factorisation of the identity through matrix algebras by UCP maps $A \to M_{n} \to A$ , each UCP map produced by an Arveson extension off a finite-dimensional operator system; the injectivity of $B (H)$ established here is precisely what makes those approximating maps exist.
The predual, normal states, and the $σ$ -weak topology 39.03.02 — the normal conditional expectations onto a von Neumann subalgebra are the normal UCP idempotents whose ranges carry the Choi-Effros product, linking injective operator systems to the predual and normality theory of von Neumann algebras.
Amenable groups, Følner sequences, and invariant means 39.05.06 — injectivity of a group von Neumann algebra is equivalent to amenability of the group, and an invariant mean is the prototypical UCP idempotent (a conditional expectation onto the scalars); this unit supplies the operator-system meaning of that equivalence.

Historical & philosophical context Master

The extension theorem is due to William Arveson, who in his 1969 Acta Mathematica paper "Subalgebras of C*-algebras" proved that a completely positive map defined on an operator system extends to the ambient C*-algebra, and isolated the operator system and the completely bounded map as the basic objects of a noncommutative function theory ^{[Arveson 1969]}. Arveson's motivation was the noncommutative analogue of the disc algebra and the boundary-representation theory generalising the Šilov boundary; the extension theorem played the role of the Hahn-Banach theorem in that program. The abstract characterisation came from Man-Duen Choi and Edward Effros, whose 1977 Journal of Functional Analysis paper "Injectivity and operator spaces" identified the matrix order with an Archimedean order unit as the exact abstract data of an operator system and constructed the multiplication making an injective operator system a C*-algebra ^{[Choi-Effros 1977]}. Zhong-Jin Ruan's 1988 thesis work supplied the parallel matrix-norm axioms characterising operator spaces, and the Effros-Ruan monograph organised the resulting theory.

The conceptual shift is from algebras to ordered (or normed) matrix spaces as the primary objects: an operator system records positivity at every matrix level and forgets multiplication, yet the Choi-Effros theorem recovers multiplication exactly when the system is injective. The injective objects are the conditional-expectation ranges, and through Connes's work the injective factors are the amenable ones, so the operator-system extension theory is one source of the deep equivalence between approximation, amenability, and injectivity that governs the classification of von Neumann algebras and the nuclearity of C*-algebras.

Bibliography Master

Arveson, W. B., "Subalgebras of C*-algebras", Acta Mathematica 123 (1969), 141-224.
Choi, M.-D. and Effros, E. G., "Injectivity and operator spaces", Journal of Functional Analysis 24 (1977), 156-209.
Ruan, Z.-J., "Subspaces of C*-algebras", Journal of Functional Analysis 76 (1988), 217-230.
Wittstock, G., "Ein operatorwertiger Hahn-Banach Satz", Journal of Functional Analysis 40 (1981), 127-150.
Paulsen, V., Completely Bounded Maps and Operator Algebras, Cambridge Studies in Advanced Mathematics 78, Cambridge University Press, 2002. Ch. 6, 13-15.
Effros, E. G. and Ruan, Z.-J., Operator Spaces, London Mathematical Society Monographs 23, Oxford University Press, 2000. Ch. 2-5.
Brown, N. P. and Ozawa, N., C-Algebras and Finite-Dimensional Approximations*, Graduate Studies in Mathematics 88, American Mathematical Society, 2008. Ch. 1.

Operator-algebras spine, second unit of the nuclearity-exactness chapter. Produced as the operator-systems / Arveson-extension / Choi-Effros anchor: operator systems as self-adjoint unital matrix-ordered subspaces, the abstract matrix-order axioms with an Archimedean order unit, Arveson's extension theorem with its order-unit-Hahn-Banach proof, the Choi-Effros realisation of abstract operator systems and the Choi-Effros product on injective systems, Ruan's operator-space axioms, and Wittstock's completely bounded version via the Paulsen system. Builds on the Stinespring dilation and CP-map theory of 39.05.01.

Prerequisites

39.05.01
39.01.01

Tier anchors

beginner: Strogatz-style 'a partial measuring instrument that still respects positivity, plus the rule that you can always finish reading it on the full apparatus'; the partial-information picture from quantum measurement
intermediate: Paulsen *Completely Bounded Maps and Operator Algebras* (Cambridge UP, 2002) Ch. 6, 13; Brown-Ozawa *C*-Algebras and Finite-Dimensional Approximations* (AMS, 2008) Ch. 1
master: Paulsen *Completely Bounded Maps and Operator Algebras* (Cambridge UP, 2002) Ch. 6, 13-15; Brown-Ozawa *C*-Algebras and Finite-Dimensional Approximations* (AMS, 2008) Ch. 1; Effros-Ruan *Operator Spaces* (Oxford UP, 2000) Ch. 2-5

References

Arveson, W. B. — Subalgebras of C*-algebras · Acta Math. 123 (1969), 141-224 — the extension theorem for operator systems and the injectivity of B(H)
Choi, M.-D. and Effros, E. G. — Injectivity and operator spaces · J. Funct. Anal. 24 (1977), 156-209 — abstract operator systems, the Archimedean order unit, the Choi-Effros product
Paulsen, V. — Completely Bounded Maps and Operator Algebras · Cambridge Studies in Advanced Mathematics 78 (2002), Ch. 6 Arveson extension, Ch. 13 abstract operator systems, Ch. 15 injectivity
Ruan, Z.-J. — Subspaces of C*-algebras · J. Funct. Anal. 76 (1988), 217-230 — the abstract matrix-norm characterisation of operator spaces
Effros, E. G. and Ruan, Z.-J. — Operator Spaces · London Math. Soc. Monographs 23, Oxford UP (2000), Ch. 2-5 operator spaces, abstract operator systems, the matrix-norm axioms

Estimated time

beginner: 18m
intermediate: 52m
master: 94m