ON THE SET OF FINITE DIMENSIONAL SUBSPACES OF PREDUALS OF VON NEUMANN ALGEBRAS

NARUTAKA OZAWA

Abstract. We prove that the set of all d-dimensional subspaces of preduals of von Neumann algebras is compact in the topology given by the cb-distance.

1. Introduction and Preliminary Background.

Pisier introduced an operator space analogue of Banach-Mazur distance in [P1].

Definition 1. For completely isomorphic operator spaces E and F, we define d_{cb}(E, F) by

d_{cb}(E, F) = inf{∥φ∥_{cb}∥φ^{−1}∥_{cb} : φ a surjective complete isomorphism}

and define δ_{cb}(E, F) by δ_{cb}(E, F) = log d_{cb}(E, F).

For each d ∈ ℕ, the set of all d-dimensional operator spaces OS_{d} is a complete metric space with the metric δ_{cb}. This metric space OS_{d} is not compact [P1], and furthermore it was proved by Junge and Pisier [JP] that it is not separable for d > 2. Using this fact, they proved 𝔹(ℓ_{2}) ⊗_{min} 𝔹(ℓ_{2}) ≠ 𝔹(ℓ_{2}) ⊗_{max} 𝔹(ℓ_{2}).

In this article, we are interested in a subspace of OS_{d}.

Definition 2. For each d ∈ ℕ, let S_{d} be the subspace of the metric space OS_{d} consisting of all d-dimensional subspaces of preduals of von Neumann algebras.

It has been observed that S_{d} is a separable complete metric space. Indeed, it follows from Theorem 3 below and Corollary 4.3 in [JP], or Remark 2.13 in [O] that all E ∈ S_{d} can be embedded into one separable C^{∗}-algebra C^{∗}(𝔽_{∞}) completely isometrically and a fortiori that S_{d} is separable (cf. Proposition 2.6 in [JP]). In this article, we will prove that

Theorem 1. For each d ∈ ℕ, the metric space S_{d} is compact.

Effros, Junge and Ruan proved the following remarkable theorem [EJR].

Theorem 2 (Effros, Junge and Ruan). The predual of any von Neumann algebra is strongly locally reflexive as an operator space.

Later, Junge and Le Merdy found an alternate proof and a remarkable factorization theorem at the same time (Theorem 2.1 in [JLM]). We note that their theorem is actually stronger than the following.

Theorem 3 (Junge and Le Merdy). Let R_{∗} be the predual of a von Neumann algebra R. Then, for any finite dimensional subspace E in R_{∗} and any ε > 0, there are m ∈ ℕ, an operator space F and two maps β: E → F and α: F → R_{∗} such that F^{∗} ⊂ 𝕄_{m}, α ∘ β = id_{E} and ∥α∥_{cb}∥β∥_{cb} < 1 + ε.

A key ingredient of the proofs of these theorems was Junge’s ingenious application of the classical Kaplansky density theorem [J]. See also Lemma 2.1 in [EJR].

Lemma 4 (Junge). Let A and B be C^{∗}-algebras. Then, any complete contraction φ from A^{∗} into B can be approximated by weak^{∗}-continuous finite rank complete contractions in the point-norm topology.

Smith’s lemma [Sm] will also play an important rôle in the proof of Theorem 1.

Date: February 18, 2000.

The author is supported by JSPS.

Lemma 5 (Smith). For any map φ from any operator space X into 𝕄_{m}, we have ∥φ∥_{cb} = ∥id_{𝕄_{m}} ⊗ φ∥.

It follows that for any map ψ: 𝕄_{m}^{∗} → Y, we have

∥id_{𝕄_{m}} ⊗ ψ∥ = ∥id_{𝕄_{m}} ⊗ ψ^{∗}∥ = ∥ψ^{∗}∥_{cb} = ∥ψ∥_{cb}.

At the end of this section, we review some basic facts on ultraproduct for operator spaces. We assume that the reader is familiar with ultraproduct for Banach spaces [H] and operator spaces [P1, P2]. Let U be a free ultrafilter on ℕ. Let {X_{n}} be a sequence of operator spaces and let X̂ = ∏ X_{n}/U be the ultraproduct operator space. For x = (x_{n})_{n} ∈ ∏ X_{n}, we denote the corresponding element in X̂ by x̂ = [x_{n}]. For a sequence of completely bounded maps φ_{n}: X_{n} → Y_{n} with sup ∥φ_{n}∥_{cb} < ∞, we define the corresponding map φ̂: X̂ → Ŷ by φ̂([x_{n}]) = [φ(x_{n})] for x̂ = [x_{n}] ∈ X̂. By definition, we have

∥id_{𝕄_{m}} ⊗ φ̂∥ = lim_{n→U} ∥id_{𝕄_{m}} ⊗ φ_{n}∥.

We note that the cb-norm version of this equality need not to be true in general. However, by Lemma 5, the cb-norm version of this equality does hold when Y_{n} = 𝕄_{m} for all n. This implies that the standard embedding ∏ X_{n}^{∗}/U ⊂ (∏ X_{n})^{∗} is completely isometric, where the embedding is given by

⟨f̂, (x_{n})_{n}⟩ = lim_{n→U}⟨f_{n}, x_{n}⟩

for f̂ = [f_{n}] ∈ ∏ X_{n}^{∗}/U and (x_{n})_{n} ∈ ∏ X_{n}.

If {A_{n}} is a sequence of C^{∗}-algebras, then the ultraproduct operator space ∏ A_{n}^{∗}/U is the predual of some von Neumann algebra [G]. We include the argument for the convenience of the reader. Consider the standard embedding ∏ A_{n}^{∗}/U ⊂ (∏ A_{n})^{∗}. Obviously, ∏ A_{n}^{∗}/U is a closed subspace which is invariant under the action of ∏ A_{n}. Thus, by Theorem III.2.7 in [T], there is a central projection e ∈ (∏ A_{n})^{∗∗} such that ∏ A_{n}^{∗}/U = e(∏ A_{n})^{∗} and a fortiori (∏ A_{n}^{∗}/U)^{∗} = e(∏ A_{n})^{∗∗} is a von Neumann algebra. Acknowledgement. The author would like to thank Professor Gilles Pisier for stimulating conversations. This research was carried out while the author was visiting the Institut Henri Poincaré supported by Japanese Society for the Promotion of Science for the semester “Free Probability and Operator Spaces”. He gratefully acknowledges the kind hospitality from the Institut Henri Poincaré.

2. Proof of the Theorem

Proof of Theorem 1. We give ourselves d ∈ ℕ. Let A_{n} be C^{∗}-algebras and let E_{n} ⊂ A_{n}^{∗} be d-dimensional subspaces. Fix a free ultrafilter U on ℕ. Let Ê = ∏ E_{n}/U ⊂ ∏ A_{n}^{∗}/U. Since ∏ A_{n}^{∗}/U is the predual of some von Neumann algebra and all E_{n}’s are d-dimensional, we have Ê ∈ S_{d}. It suffices to show that Ê is a cluster point of the sequence {E_{n}}. Take 0 < ε < 1 arbitrary.

By Theorem 3, there are m ∈ ℕ, an operator space F and two maps β: Ê → F and α: F → ∏ A_{n}^{∗}/U such that F^{∗} ⊂ 𝕄_{m}, α ∘ β = idÊ and ∥α∥_{cb}∥β∥_{cb} < 1 + ε. Let G = α(F) ⊂ ∏ A_{n}^{∗}/U.

We have the standard completely isometric embedding ∏ A_{n}^{∗}/U ⊂ (∏ A_{n})^{∗}. Furthermore, we may assume that there is a completely isometric embedding (∏ A_{n})^{∗} ⊂ 𝔹(H) for some H. By Lemma 4, there is a weak^{∗}-continuous finite rank map φ: (∏ A_{n})^{∗} → 𝔹(H) with ∥φ∥_{cb} < 1 + ε such that φ|Ê = idÊ. Let

∑_{i} a^{i} ⊗ x^{i} ∈ (∏ A_{n}) ⊗_{min} 𝔹(H)

be a tensor product expression of φ, where a^{i} = (a_{n}^{i})_{n} ∈ ∏ A_{n}.

Let {g^{k} : 1 ≤ k ≤ K} be a basis for G such that {g^{k} : 1 ≤ k ≤ d} is an Auerbach basis for Ê. Let (g_{n}^{k})_{n} ∈ ∏ A_{n}^{∗} be a representative of g^{k} such that g_{n}^{k} ∈ E_{n} for 1 ≤ k ≤ d. Let ψ_{n}: G → A_{n}^{∗} be the map defined by ψ_{n}(g^{k}) = g_{n}^{k} for all 1 ≤ k ≤ K. By the definition of ultraproduct operator spaces, we have

lim_{n→U} ∥id_{𝕄_{m}} ⊗ ψ_{n}: 𝕄_{m}(G) → 𝕄_{m}(A_{n}^{∗})∥ = 1.

While, by the definition of the standard embedding, we have

lim_{n→U}⟨g_{n}^{k}, a_{n}^{i}⟩ = ⟨g^{k}, a^{i}⟩

for all k and i. Thus, given N_{0} ∈ ℕ, we may find N ≥ N_{0} such that

∥id_{𝕄_{m}} ⊗ ψ_{N}∥ < 1 + ε

and

∑_{i} |⟨g_{N}^{k}, a_{N}^{i}⟩ − ⟨g^{k}, a^{i}⟩|∥x^{i}∥ < ε/d

for all 1 ≤ k ≤ d. Now, consider θ := ψ_{N} ∘ α ∘ β: Ê → E_{N}. Since F^{∗} ⊂ 𝕄_{m}, it follows from the remark following Lemma 5 that

∥θ∥_{cb} ≤ ∥ψ_{N} ∘ α∥_{cb}∥β∥_{cb}
≤ ∥id_{𝕄_{m}} ⊗ (ψ_{N} ∘ α)∥∥β∥_{cb}
≤ ∥id_{𝕄_{m}} ⊗ ψ_{N}∥∥α∥_{cb}∥β∥_{cb} ≤ (1 + ε)^{2}.

Let σ: A_{N}^{∗} → 𝔹(H) be the map corresponding to ∑_{i} a_{N}^{i} ⊗ x^{i} ∈ A_{N}^{∗} ⊗_{min} 𝔹(H). Then, we have

∥σ∥_{cb} ≤ ∥φ∥_{cb} < 1 + ε

and for 1 ≤ k ≤ d, we have

∥σ ∘ θ(g^{k}) − g^{k}∥ ≤ ∑_{i} |⟨g_{N}^{k}, a_{N}^{i}⟩ − ⟨g^{k}, a^{i}⟩|∥x^{i}∥ < ε/d.

Therefore, by the small perturbation argument, we may find σ̃: E_{N} → Ê with ∥σ̃∥_{cb} < (1 + ε)/(1 − ε) such that σ̃ ∘ θ = idÊ. It follows that

d_{cb}(E_{N}, Ê) < (1 + ε)^{3}/(1 − ε)

and we are done.

Remark 6. Though we have used Theorem 3 and implicitly Theorem 6.6 in [EJR] in the proof, Theorem 1 implies the strong local reflexivity of the subspaces of preduals of von Neumann algebras.

A similar proof shows that there is m(d, ε) ∈ ℕ for each d ∈ ℕ and ε > 0 such that the conclusion of Theorem 3 holds for m = m(dim E, ε).

The proof of Theorem 1 would be useful to solve the problem raised in [O] negatively.

References

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Department of Mathematical Science, University of Tokyo, Komaba, 153-8914, Japan

   E-mail address: narutaka@ms.u-tokyo.ac.jp