COMMUTATIVE C^{∗}-ALGEBRAS AND SEQUENTIALLY NORMAL MORPHISMS

MARCEL DE JEU

Abstract. We show that the image of a commutative monotone sequentially complete C^{∗}-algebra, under a sequentially normal morphism, is again a monotone sequentially complete C^{∗}-algebra, and also a monotone sequentially closed C^{∗}-subalgebra. As a consequence, the image of an algebra of this type, under a sequentially normal representation in a separable Hilbert space, is strongly closed. In the case of a unital representation of C(X) in a separable Hilbert space, where X is a compact Hausdorff space, this implies that the von Neumann algebra generated by the image of C(X) is the image of the Baire functions on X under the extension of the representation to the bounded Borel functions.

1. Main result and application

It is well-known that the image of a von Neumann algebra under a normal unital representation is again a von Neumann algebra [3, Theorem 2.5.3]. In this note, we prove a theorem in the same vein, but now in the category of C^{∗}-algebras. The domain C^{∗}-algebra is in our case a commutative algebra, possessing a sequential order completeness property to be defined below. The result is subsequently applied in the context of a unital separable representation of C(X), where X is a compact Hausdorff space.

We start by recalling the relevant definitions. For the sake of clarity, let us mention explicitly that the C^{∗}-algebras in this note are not necessarily unital; neither are (if applicable) the morphisms.

Definition 1.1. Cf. [3, 3.9.2].

(1) A C^{∗}-algebra A is monotone sequentially complete if every bounded increasing sequence of self-adjoint elements of A has a supremum in A.

(2) A C^{∗}-subalgebra A of a C^{∗}-algebra B is a monotone sequentially closed C^{∗}-subalgebra of B if sup_{n≥1}^{B} a_{n} ∈ A, whenever a_{1} ≤ a_{2} ≤ … is a bounded increasing sequence of self-adjoint elements of A which has a supremum sup_{n≥1}^{B} a_{n} in B.

2000 Mathematics Subject Classification. Primary 46L05; Secondary 46L10.

Key words and phrases. C^{∗}-algebra, von Neumann algebra, monotone sequential complete, monotone sequentially closed, sequentially normal morphism.

During the preparation of this paper the author was supported by a PIONIER grant of the Netherlands Organisation for Scientific Research (NWO).

(3) A morphism ϕ : A ↦ B between two C^{∗}-algebras is sequentially normal if ϕ(sup_{n≥1} a_{n}) = sup_{n≥1} ϕ(a_{n}), for each bounded increasing sequence a_{1} ≤ a_{2} ≤ … of self-adjoint elements of A which has a supremum in A.

Our main result then reads as follows.

Theorem 1.2. Let A be a commutative monotone sequentially complete C^{∗}-algebra. Suppose ϕ : A ↦ B is a sequentially normal morphism into a C^{∗}-algebra B. Then ϕ(A) is a commutative monotone sequentially complete C^{∗}-algebra, and a monotone sequentially closed C^{∗}-subalgebra of B.

The proof is given in Section 2.

Corollary 1.3. Let A be a commutative monotone sequentially complete C^{∗}-algebra. Suppose π : A ↦ B(H) is a sequentially normal representation in a separable Hilbert space H. Then π(A) is strongly closed.

Indeed, π(A) is a monotone sequentially closed C^{∗}-subalgebra of B(H) by Theorem 1.2. Then Pedersen’s Up–Down theorem for separable Hilbert spaces [3, Theorem 2.4.3] implies that it is strongly closed.

As an application of Corollary 1.3, let X be a compact Hausdorff space, with associated C^{∗}-algebras C(X) of continuous functions and B_{b}(X) of bounded Borel measurable functions. Suppose that π : C(X) ↦ B(H) is a unital representation. Then, as is well known (see, e.g., [1]), π extends uniquely to a unital representation of B_{b}(X), again denoted by π, such that

(1.1)  (π(f)ξ, ξ) = ∫_{X} f(x) dμ_{ξ}(x) (f ∈ B_{b}(X), ξ ∈ H).

Here, for ξ ∈ H, μ_{ξ} denotes the positive and bounded unique regular Borel measure on X, which is provided by the Riesz representation theorem when one requires (1.1) to hold for all f ∈ C(X).

The constructions involved in extending π from C(X) to B_{b}(X) make it obvious that π maps B_{b}(X) into the strong closure of π(C(X)), implying that π(B_{b}(X)) and π(C(X)) have the same strong closure. If H is separable, then π(B_{b}(X)) is actually equal to the strong closure of π(C(X)) [1, Proposition 9.5.3]. This description of the von Neumann algebra generated by π(C(X)), as being equal to π(B_{b}(X)), is then a starting point for further investigation of the unital separable representation π of C(X).

The proof of the equality of π(B_{b}(X)) and the strong closure of π(C(X)) in the separable case is usually based on measure-theoretical arguments, and tends to be somewhat more involved than others in this circle of ideas, cf. [1, proof of Proposition 9.5.3]. Corollary 1.3, which is based on order properties, provides an

SEQUENTIALLY COMPLETE C^{∗}-ALGEBRAS 3

alternative approach. Indeed, we note that B_{b}(X) is monotone sequentially complete and that, as a consequence of (1.1) and the monotone convergence theorem, π is a sequentially normal representation of B_{b}(X). By Corollary 1.3, π(B_{b}(X)) is strongly closed, hence equal to the strong closure of π(C(X)).

In fact, we can now—still for a unital separable representation—also give a more economic description of the von Neumann algebra generated by π(C(X)). Namely, Corollary 1.3 also implies that the image of the monotone sequential completion of C(X) in B_{b}(X) is strongly closed, and is therefore, by the same reasoning, also equal to the von Neumann algebra generated by π(C(X)).

The monotone sequential completion of C(X) in B_{b}(X) is known as the Baire algebra. If X is second countable, then the Baire algebra coincides with B_{b}(X), but in other cases it may be a proper C^{∗}-subalgebra of B_{b}(X) [2, 6.2.10]. The Baire C^{∗}-algebra is perhaps the natural pre-image of the von Neumann algebra, generated by the image of C(X) under the unital separable representation π.

2. Proofs

The proof of Theorem 1.2 consists mainly of the combination of an elementary lifting result for functions and the Gelfand–Naimark isomorphism. The resulting Proposition 2.2 is somewhat stronger than needed for our purposes.

Lemma 2.1. Let X be a topological space, and let C_{0}(X) denote the continuous functions on X vanishing at infinity. Suppose that Y ⊂ X is a non-empty subset.

(1) If 0 ≤ f_{1} ≤ f_{2} ≤ … is a sequence of functions on Y, such that each f_{n} is the restriction of some element of C_{0}(X) to Y, then there exists a sequence 0 ≤ g_{1} ≤ g_{2} ≤ … in C_{0}(X) such that, for each n, g_{n} restricts to f_{n} and ∥g_{n}∥_{∞} = ∥f_{n}∥_{∞}.

(2) If f_{1} ≥ f_{2} ≥ … ≥ 0 is a sequence of functions on Y, such that each f_{n} is the restriction of some element of C_{0}(X) to Y, then there exists a sequence g_{1} ≥ g_{2} ≥ … ≥ 0 in C_{0}(X) such that, for each n, g_{n} restricts to f_{n} and ∥g_{n}∥_{∞} = ∥f_{n}∥_{∞}.

Proof. Suppose that h_{n} ∈ C_{0}(X) restricts to f_{n}. Replacing h_{n} with |h_{n}|, we may assume that h_{n} ≥ 0. After a subsequent replacement of h_{n} with min(h_{n}, ∥f_{n}∥_{∞}) ∈ C_{0}(X), we may assume that h_{n} ≥ 0 and that ∥h_{n}∥_{∞} = ∥f_{n}∥_{∞}. In the case of an increasing sequence, define g_{n} = max_{1≤i≤n} h_{i}. In the case of a decreasing sequence, define g_{n} = min_{1≤i≤n} h_{i}. Then the g_{n} have the required properties. □

More generally, we have the following result on isometric monotone lifts of positive monotone sequences.

Proposition 2.2. Let A and B be commutative C^{∗}-algebras. Suppose that ϕ : A ↦ B is a surjective morphism.

(1) If 0 ≤ b_{1} ≤ b_{2} ≤ … is a sequence in B, then there exists a sequence 0 ≤ a_{1} ≤ a_{2} ≤ … in A, such that, for each n, ϕ(a_{n}) = b_{n} and ∥a_{n}∥ = ∥b_{n}∥.

(2) If b_{1} ≥ b_{2} ≥ … ≥ 0 is a sequence in B, then there exists a sequence a_{1} ≥ a_{2} ≥ … ≥ 0 in A, such that, for each n, ϕ(a_{n}) = b_{n} and ∥a_{n}∥ = ∥b_{n}∥.

Proof. We may assume that B ≠ 0. In that case, there is a canonical map ϕ^{∗} : B̂ ↦ Â between the maximal ideal spaces, defined by ϕ^{∗}(β) = β ∘ ϕ for β ∈ B̂. Then ϕ^{∗} is injective, as a consequence of the surjectivity of ϕ. Using ϕ^{∗}, we identify B̂ as a set with its image in Â, and we view C_{0}(B̂) as functions on B̂ ⊂ Â. (Actually, ϕ^{∗} is a homeomorphic embedding of B̂ into Â, but this fact is not needed.) Using these two identifications, the following diagram is then commutative:

\xymatrix{A \ar[r]^{ϕ} \ar[d]_{Γ_{A}} & B \ar[d]^{Γ_{B}} \\ C_{0}(Â) \ar[r]_{Res} & C_{0}(B̂)}

Here Γ_{A} and Γ_{B} denote the Gelfand–Naimark isomorphisms, and Res is the restriction mapping. The proposition now follows from an application of Lemma 2.1 at the bottom part of the diagram. □

Proposition 2.3. Let A be a commutative monotone sequentially complete C^{∗}- algebra. Suppose ϕ : A ↦ B is a sequentially normal morphism into a C^{∗}-algebra B. Let b_{1} ≤ b_{2} ≤ … be a bounded sequence of self-adjoint elements of ϕ(A). Then this sequence has a supremum in B, and this supremum in B is an element of ϕ(A).

Proof. We may assume that b_{1} ≥ 0. Using Proposition 2.2 for the morphism ϕ : A ↦ ϕ(A), we find a bounded sequence a_{1} ≤ a_{2} ≤ … of self-adjoint elements of A, such that ϕ(a_{n}) = b_{n} for all n. Since A is monotone sequentially complete, sup_{n≥1} a_{n} exists in A. By the sequential normality of ϕ, sup_{n≥1} b_{n} exists in B, since it is equal to ϕ(sup_{n≥1} a_{n}). This also shows that this supremum in B is an element of ϕ(A). □

Proposition 2.3 implies Theorem 1.2, as the reader will easily verify.

References

[1] R.V. Kadison and J.R. Ringrose, Fundamentals of the Theory of Operator Algebras, Volume II, Academic Press, London, 1986.

[2] G.K. Pedersen, Analysis Now, Springer-Verlag, New York, 1989.

[3] G.K. Pedersen, C^{∗}-Algebras and their Automorphism Groups, Academic Press, London, 1979.

M.F.E. de Jeu, Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands

   E-mail address: mdejeu@math.leidenuniv.nl