NOTES ON THE UNIQUE EXTENSION PROPERTY

WILLIAM ARVESON

Abstract. In a recent paper, Dritschel and McCullough established the existence of completely positive maps of operator algebras that have a unique extension property. In this expository note we give a more explicit rendering of that result geared to operator systems, and discuss consequences.

1. Maximality

An operator system is a self-adjoint linear subspace S of a unital C^{∗}-algebra that contains the unit; we usually require that the C^{∗}-algebra be generated by S, and express that by writing S ⊆ C^{∗}(S). We consider unital completely positive (UCP) maps ϕ : S → B(H), that is, completely positive maps that carry the unit of S to the identity operator of B(H). Such maps satisfy ϕ(x^{∗}) = ϕ(x)^{∗}, x ∈ S. A linear map ϕ : S → B(H) that preserves the unit is completely positive iff it is completely contractive. If S is a linear subspace of C^{∗}(S) containing 1, then every completely contractive unital map of S extends uniquely to a UCP map of S + S^{∗} (see [Arv69]).

Let S ⊆ C^{∗}(S) be an operator system. Given two UCP maps ϕ_{k} : S → B(H_{k}), k = 1, 2, we write ϕ_{1} ≤ ϕ_{2} if H_{1} ⊆ H_{2} and P_{H_{1}}ϕ_{2}(x) ↾_{H_{1}}= ϕ_{1}(x), x ∈ S; ϕ_{2} is called a dilation of ϕ_{1} and ϕ_{1} is called a compression of ϕ_{2}. The relation ≤ is transitive, and one has ϕ_{1} ≤ ϕ_{2} and ϕ_{2} ≤ ϕ_{1} iff H_{1} = H_{2} and ϕ_{1} = ϕ_{2}. Thus, ≤ defines a partial ordering of UCP maps of S. Every UCP map ϕ : S → B(H) can be dilated in a trivial way by forming a direct sum ϕ⊕ψ where ψ : S → B(K) is another UCP map.

Definition 1.1. A UCP map ϕ : S → B(H) is said to be maximal if it has no nontrivial dilations: ϕ′ ≥ ϕ ⟹ ϕ′ = ϕ ⊕ ψ for some UCP map ψ.

A dilation ϕ_{2} of ϕ_{1} need not satisfy H_{2} = [C^{∗}(ϕ_{2}(S))H_{1}], C^{∗}(ϕ_{2}(S)) denoting the C^{∗}-algebra generated by ϕ_{2}(S) ⊆ B(H_{2}), but it can always be replaced with a smaller dilation of ϕ_{1} that has this property. Notice that this reduction imposes an upper bound on the dimension of H_{2} in terms of the dimension of H_{1} and the cardinality of S. ϕ_{1} is maximal iff the only dilation ϕ_{2} ≥ ϕ_{1} that satisfies H_{2} = [C^{∗}(ϕ_{2}(S))H_{1}] is ϕ_{2} = ϕ_{1} itself.

Let ϕ : S → B(H) be a UCP map and let F be a (possibly empty) subset of S × H. We will say that ϕ is maximal on F if for every dilation ψ of ϕ acting on K ⊇ H, we have

ψ(x)ξ = ϕ(x)ξ, (x, ξ) ∈ F.

A UCP map ϕ : S → B(H) is maximal iff it is maximal on S × H. If ϕ is maximal on F ⊆ S ×H and ψ ≥ ϕ, then ψ is maximal on F. We require the following result, inspired by an observation of N. Ozawa.

Lemma 1.2. For every UCP map ϕ : S → B(H) and every (x, ξ) ∈ S × H, there is a dilation of ϕ that is maximal on (x, ξ).

Proof. Since for every dilation ψ ≥ ϕ we have ∥ψ(x)ξ∥ ≤ ∥x∥ · ∥ξ∥ < ∞, we can find a dilation ϕ_{1} of ϕ for which ∥ϕ_{1}(x)ξ∥ is as close to sup{∥ψ(x)ξ∥ : ψ ≥ ϕ} as we wish. Continuing inductively, we find a sequence of UCP maps ϕ ≤ ϕ_{1} ≤ ϕ_{2} ≤ ⋯ such that ϕ_{n} : S → B(H_{n}), H ⊆ H_{1} ⊆ H_{2} ⊆ ⋯, and

∥ϕ_{n+1}(x)ξ∥ ≥ sup_{ψ≥ϕ_{n}} ∥ψ(x)ξ∥ − 1/n.

Let H_{∞} be the closure of the union ∪_{n}H_{n} and let ϕ_{∞} : S → B(H_{∞}) be the unique UCP map that compresses to ϕ_{n} on H_{n} for every n. Note that ϕ_{∞} is maximal on (x, ξ). Indeed, if ψ ≥ ϕ_{∞} then ψ ≥ ϕ_{n} for every n ≥ 1, and

∥ϕ_{∞}(x)ξ∥ ≥ ∥P_{H_{n+1}}ϕ_{∞}(x)ξ∥ = ∥ϕ_{n+1}(x)ξ∥ ≥ ∥ψ(x)ξ∥ − 1/n.

Hence ∥ϕ_{∞}(x)ξ∥ ≥ ∥ψ(x)ξ∥. It follows that

∥ψ(x)ξ − ϕ_{∞}(x)ξ∥^{2} = ∥ψ(x)ξ − P_{H_{∞}}ψ(x)ξ∥^{2} = ∥ψ(x)ξ∥^{2} − ∥ϕ_{∞}(x)ξ∥^{2} ≤ 0

so that ψ(x)ξ = ϕ_{∞}(x)ξ as asserted. □

Theorem 1.3. Every UCP map ϕ_{0} : S → B(H_{0}) can be dilated to a maximal UCP map ϕ : S → B(H).

Proof. We show first that ϕ_{0} can be dilated to a UCP map ϕ_{1} : S → B(H_{1}) that is maximal on S × H_{0}. To that end, let λ be an ordinal sufficiently large that there is a surjection α ∈ λ ↦ x_{α} ∈ S × H_{0}; hence S × H_{0} = {x_{α} : α < λ}. We claim that there is a family of UCP maps ϕ_{α} : S → B(H_{α}), indexed by the ordinals α ≤ λ, which satisfy ϕ_{α} ≥ ϕ_{0} together with

(i) ϕ_{α} is maximal on {x_{β} : β < α}, and

(ii) α ≤ β ⟹ ϕ_{α} ≤ ϕ_{β}.

Once the existence of this family is established, one can set ϕ_{1} = ϕ_{λ}.

Proceeding inductively, for α = 0 we set ϕ_{α} = ϕ_{0}, noting that (i) is vacuous for α = 0. Assuming that α ≤ λ is an ordinal for which {ϕ_{β} : β < α} has been defined and satisfies (i) and (ii) on the initial segment {β < α}, define ϕ_{α} as follows. If α has an immediate predecessor α − 1, Lemma 1.2 implies that ϕ_{α−1} can be dilated to a UCP map ϕ_{α} : S → B(H_{α}) that is maximal on x_{α−1}. If α is a limit ordinal then the Hilbert spaces H_{β}, β < α, are linearly ordered by inclusion; we take H_{α} to be the closure of their union and ϕ_{α} : S → B(H_{α}) to be the unique UCP map that compresses to ϕ_{β} on H_{β} for every β < α. In either case properties (i) and (ii) persist for the augmented family {ϕ_{β} : β ≤ α}. This defines {ϕ_{α} : α ≤ λ}.

Now one can use ordinary induction on the preceding result to find an increasing sequence of Hilbert spaces H_{0} ⊆ H_{1} ⊆ H_{2} ⊆ ⋯ and UCP maps ϕ_{n} : S → B(H_{n}) such that ϕ_{n+1} is a dilation of ϕ_{n} that is maximal on S × H_{n}, n = 0, 1, 2, . . .. Let H_{∞} be the closure of ∪_{n}H_{n} and let ϕ_{∞} : S → B(H_{∞}) the unique UCP map that compresses to ϕ_{n} on H_{n} for every n ≥ 1. Note that for every dilation ψ : S → B(K) of ϕ_{∞} and every n ≥ 1, both ψ and ϕ_{∞} are dilations of ϕ_{n+1}, so by maximality of ϕ_{n+1} on S × H_{n} we have

ψ(x)ξ = ϕ_{n+1}(x)ξ = ϕ_{∞}(x)ξ, (x, ξ) ∈ S × H_{n}.

It follows that ϕ_{∞} is maximal on S × ∪_{n}H_{n}, hence on its closure S × H_{∞}. □

2. The unique extension property

Dritschel and McCullough have used the term boundary representation for UCP maps ϕ : S → B(H) that have unique competely positive extensions to representations of C^{∗}(S) [DM03]. We avoid conflict with the terminology of [Arv69] and [Arv72] by making use of the following:

Definition 2.1. A UCP map π : S → B(H) is said to have the unique extension property if

(i) π has a unique completely positive extension π̃ : C^{∗}(S) → B(H), and (ii) π̃ is a representation of C^{∗}(S) on H.

The unique extension property for π : S → B(H) is equivalent to the assertion that every extension of π to a completely positive map ϕ : C^{∗}(S) → B(H) should be multiplicative on C^{∗}(S). If the extension π̃ of such a map π to C^{∗}(S) is an irreducible representation then the extension is a boundary representation in the sense of [Arv69]; otherwise it is not.

In the following result we adapt an observation of Muhly and Solel [MS98] so as to relate maximality to the unique extension property.

Proposition 2.2. A UCP map π : S → B(H) is maximal iff it has the unique extension property.

Proof. Assume first that π is maximal and let ϕ : C^{∗}(S) → B(H) be a completely positive extension of it. We have to show that ϕ is multiplicative. By Stinespring’s theorem, there is a representation σ : C^{∗}(S) → B(K) on a Hilbert space K ⊇ H such that ϕ(x) = P_{H}σ(x) ↾_{H}, x ∈ C^{∗}(S). We can assume that the dilation is minimal in that K = [σ(C^{∗}(S))H] = [C^{∗}(σ(S))H]. By maximality of π, K = H and ϕ = σ is multiplicative.

Conversely, suppose that π has the unique extension property and let ϕ : S → B(K) be a dilation of π acting on K ⊇ H with K = [C^{∗}(ϕ(S))H]. We show that K = H and ϕ = π. By Theorem 1.2.9 of [Arv69] ϕ can be extended to a completely positive linear map ϕ̃ : C^{∗}(S) → B(K). Since the compression of ϕ̃ to H defines a completely positive map of C^{∗}(S) to B(H) that restricts to π on S, the unique extension property implies that P_{H} ϕ̃P_{H} is multiplicative on C^{∗}(S). So for x ∈ C^{∗}(S),

P_{H} ϕ̃(x)^{∗}P_{H} ϕ̃(x)P_{H} = P_{H} ϕ̃(x^{∗}x)P_{H} ≥ P_{H} ϕ̃(x)^{∗}ϕ̃(x)P_{H},

since ϕ̃(x^{∗}x) ≥ ϕ̃(x)^{∗}ϕ̃(x); hence |(1−P_{H})ϕ̃(x)P_{H}|^{2} ≤ 0. This implies that H is invariant under the set of operators ϕ̃(C^{∗}(S)) ⊇ ϕ(S), and therefore under C^{∗}(ϕ(S)). Thus K = [C^{∗}(ϕ(S))H] = H, and ϕ = π follows. □

Combining Theorem 1.3 with Proposition 2.2, we obtain the main result of [DM03]. Note that while the discussion of [DM03] is limited to operator algebras, the arguments there carry over to operator systems as well.

Corollary 2.3 (Dritschel-McCullough). Every UCP map ϕ_{0} : S → B(H_{0}) can be dilated to a UCP map π : S → B(H) with the unique extension property.

3. Silov ideals and C^{∗}-envelopes

We have seen that a UCP map is maximal iff it has the unique extension property. This allows one to strengthen the invariance principle of Theorem 2.1.2 of [Arv69] by a simpler argument than the original.

Proposition 3.1 (Invariance Principle). Let S_{k} ⊆ C^{∗}(S_{k}), k = 1, 2, be two operator systems and let θ : S_{1} → S_{2} be a unital completely isometric linear map of S_{1} on S_{2}. For every UCP map π_{1} : S_{1} → B(H) with the unique extension property, the UCP map π_{2} : S_{2} → B(H) defined by π_{2} ∘ θ = π_{1} has the unique extension property.

Proof. Consider the UCP map π_{2} : S_{2} → B(H) defined by π_{2} = π_{1} ∘ θ^{−1}. By Proposition 2.2, it suffices to show that π_{2} is maximal, given that π_{1} is maximal. Let ϕ : S_{2} → B(K) be a UCP map acting on K ⊇ H that compresses to π_{2} and satisfies K = [C^{∗}(ϕ(S_{2}))H]. Then ϕ∘θ is a UCP map of S_{1} to B(K) that compresses to π_{1} and satisfies K = [C^{∗}(π(S_{1}))H], so by maximality of π_{1} we have ϕ ∘ θ = π_{1}, hence ϕ = π_{1} ∘ θ^{−1} = π_{2}. □

We also make use of some terminology from [Arv69] and [Arv72].

Definition 3.2. Let S ⊆ C^{∗}(S) be an operator system. A boundary ideal for S is an ideal J ⊆ C^{∗}(S) with the property that the natural projection of C^{∗}(S) on C^{∗}(S)/J restricts to a completely isometric map on S. If there exists a boundary ideal J̄ that contains every other boundary ideal for S, then J̄ is called the Silov ideal for S.

The Silov ideal of an operator system was introduced in [Arv69] and [Arv72], where its existence was established for many examples, and applications to concrete problems in operator theory were described in detail. But the problem of the existence of the Silov ideal for general operator systems was left open. It was finally settled in the affirmative by Hamana, as a consequence of his analysis of injective operator systems [Ham79]. As pointed out by Dritschel and McCullough [DM03], Corollary 2.3 implies the existence of the Silov ideal and the C^{∗}-envelope following an argument of [Arv69]. This has the advantage of sidestepping issues of injectivity that were essential in Hamana’s proof of the existence of the C^{∗}-envelope for operator systems. We now reiterate this point in some detail.

Corollary 3.3 (Existence of the Silov ideal). For every operator system S ⊆ C^{∗}(S), the set of all boundary ideals for S has a largest element J̄.

Assuming that J̄ = {0}, as we may after passing to a quotient, then C^{∗}(S) is the smallest C^{∗}-algebra that can be generated by S in the following sense: for every unital completely isometric linear map θ of S into some other unital C^{∗}-algebra B with B = C^{∗}(θ(S)) there is a unique ∗-homomorphism σ of B onto C^{∗}(S) such that σ ∘ θ is the identity map of S.

Proof. There is a maximal UCP map π : S → B(H) whose restriction to S is completely isometric; indeed, starting with any completely isometric UCP map ϕ_{0} : S → B(H_{0}), Theorem 1.3 implies that ϕ_{0} can be dilated to such a map π. Proposition 2.2 implies that we may extend π uniquely to a representation of C^{∗}(S) on H, which we denote by the same latter π. Let J̄ = ker π. J̄ is a boundary ideal since π restricts to a completely isometric map of S.

Let J be any other boundary ideal for S, and consider the natural projection x ∈ C^{∗}(S) ↦ ẋ ∈ C^{∗}(S)/J. the map ψ : Ṡ → B(H) defined by ψ(ṡ) = π(s), s ∈ S is unit preserving and completely contractive, hence completely positive, and therefore has a completely positive extension ψ̃ : C^{∗}(S)/J → B(H). Since ψ̃(ṡ) = π(s) for s ∈ S, ψ̃(ẋ) = π(x) for x ∈ C^{∗}(S) by the unique extension property for π. Hence π vanishes on J and J ⊆ J̄ follows.

The last paragraph follows from the invariance principle Proposition 3.1. Indeed, since J̄ = {0} the extension of the map π : S → B(H) of the first paragraph to C^{∗}(S) is a faithful representation. Consider the UCP map π ∘ θ^{−1} : θ(S) → B(H). Proposition 3.1 implies that π ∘ θ^{−1} has the unique extension property. Letting σ_{0} : B → B(H) be its extension to a representation of B, we find that σ_{0}(θ(s)) = π(s) for s ∈ S, and therefore σ_{0}(θ(s_{1}) ⋯ θ(s_{n})) = π(s_{1} ⋯ s_{n}) for s_{1}, …, s_{n} ∈ S, n = 1, 2, . . .. It follows that σ_{0}(B) = π(C^{∗}(S)), hence the map defined on B by σ = π^{−1} ∘ σ_{0} is a homomorphism of B onto C^{∗}(S) that carries θ(s) to s, s ∈ S. □

Remark 3.4 (Existence of the C^{∗}-envelope). The second paragraph of Corollary 3.3 implies that in general, the Silov ideal J̄ ⊆ C^{∗}(S) has the property that the natural projection S → Ṡ ⊆ C^{∗}(S)/J̄ exhibits C^{∗}(S)/J̄ as the C^{∗}-envelope of S.

References

[Arv69] W. Arveson. Subalgebras of C^{∗}-algebras. Acta Math., 123:141–224, (1969).

[Arv72] W. Arveson. Subalgebras of C^{∗}-algebras II. Acta Math., 128:271–308, (1972).

[DM03] M. Dritschel and S. McCullough. Boundary representations for operator algebras. J. Op. Th.(to appear), pages 1–7, (2003). arXiv:math.OA/0107217.

[Ham79] M. Hamana. Injective envelopes of operator systems. Publ. Res. Inst. Math. Sci. (Kyoto), 15:773–785, (1979).

[MS98] P. Muhly and B. Solel. An algebraic characterization of boundary representations. In Nonselfadjoint operator algebras, operator theory, and related topics, pages 189–196, Basel, (1998). Birkhäuser.

University of California, Berkeley, CA, 94720

   E-mail address: arveson@math.berkeley.edu