ℬ(H) has a pure state that is not multiplicative on any masa

Abstract

Assuming the continuum hypothesis, we prove that ℬ(H) has a pure state whose restriction to any masa is not pure. This resolves negatively old conjectures of Kadison and Singer and of Anderson.

Let H be a separable infinite-dimensional Hilbert space and let ℬ(H) be the algebra of bounded operators on H. Kadison and Singer (1) suggested that every pure state on ℬ(H) would restrict to a pure state on some maximal abelian self-adjoint subalgebra (masa). Anderson (2) formulated the stronger conjecture that every pure state on ℬ(H) is diagonalizable, that is, of the form f(A) = lim_𝒰 〈Ae_n, e_n〉 for some orthonormal basis (e_n) and some ultrafilter 𝒰 over N.

An atomic masa is the set of all operators that are diagonalized with respect to some given orthonormal basis of H. Anderson's conjecture is related to a fundamental problem in C*-algebras also raised in ref. 1 and now known as the Kadison–Singer problem, which asks whether every pure state on an atomic masa of ℬ(H) has a unique extension to a pure state on ℬ(H). If (e_n) is an orthonormal basis of H, then every pure state f₀ on the corresponding atomic masa ℳ has the form f₀(A) = lim_𝒰〈Ae_n, e_n〉 for some ultrafilter 𝒰 over N and all A ∈ ℳ, and Anderson (3) showed that the same formula, now for A ∈ ℬ(H), defines a pure state f on ℬ(H). Thus, a positive solution to the Kadison–Singer problem would say that f is the only pure state on ℬ(H) that extends f₀.

In the presence of a positive solution to the Kadison–Singer problem, Anderson's conjecture is equivalent to the weaker statement that every pure state on ℬ(H) restricts to a pure state on some atomic masa. However, assuming the continuum hypothesis, we show that this weaker statement is false; in fact, there exist pure states on ℬ(H) whose restriction to any masa is not pure. It follows that there are pure states on ℬ(H) that are not diagonalizable. It seems likely that the statement “every pure state on ℬ(H) restricts to a pure state on some atomic masa” is also consistent with standard set theory. This, together with a positive solution to the Kadison–Singer problem, would imply the consistency of a positive answer to Anderson's conjecture.

The key lemma we need is the following. Let 𝒦(H) be the algebra of compact operators on H, let 𝒞(H) = ℬ(H)/𝒦(H) be the Calkin algebra, and let π : ℬ(H) → 𝒞(H) be the natural quotient map. We also write ȧ for π(a), for any a ∈ ℬ(H).

Lemma 0.1.

Let 𝒜 be a separable C*-subalgebra of ℬ(H) which contains 𝒦(H), let f be a pure state on 𝒜 that annihilates 𝒦(H), and let ℳ be a masa of ℬ(H). Then there is a pure state g on ℬ(H) that extends f and whose restriction to ℳ is not pure.

Proof:

By Proposition 6 of ref. 4 we can find an infinite-rank projection p ∈ ℬ(H) such that

for all a ∈ 𝒜.

Lemma 1.4 and Theorem 2.1 of ref. 5 imply that π(ℳ) is a masa of 𝒞(H). It follows that there is a projection q ∈ ℳ such that q̇ neither contains nor is orthogonal to ṗ. Otherwise ṗ would be in the commutant of π(ℳ), and hence would belong to π(ℳ) by maximality. But this would mean ṗ is minimal in π(ℳ) because any nonzero projection below ṗ neither contains nor is orthogonal to ṗ, and π(ℳ) has no minimal projections.

Let φ : 𝒞(H) → ℬ(K) be an irreducible representation of the Calkin algebra. It is faithful because 𝒞(H) is simple. Therefore, φ(q̇) neither contains nor is orthogonal to φ(ṗ), so we can find a unit vector υ ∈ K in the range of φ(ṗ) which is neither contained in nor orthogonal to the range of φ(q̇). Finally, define g(a) = 〈φ(ȧ)υ, υ〉 for all a ∈ ℬ(H). This is a pure state on ℬ(H) because φ ○ π is an irreducible representation of ℬ(H). It extends f because, using Eq. 1,

for all a ∈ 𝒜. Finally, its restriction to ℳ is not pure because the projection q ∈ ℳ has the property that

is strictly between 0 and 1, since υ is neither contained in nor orthogonal to the range of φ(q̇). □

Theorem 0.2.

Assume the continuum hypothesis. Then there is a pure state on ℬ(H) whose restriction to any masa is not pure.

Proof:

Let (a_α), α < ℵ ₁, enumerate the elements of ℬ(H). Since every von Neumann subalgebra of ℬ(H) is countably generated, a simple cardinality argument shows that there are only ℵ₁ such subalgebras. Hence, ℬ(H) has only ℵ₁ masas. Let (ℳ_α), α < ℵ₁, enumerate the masas of ℬ(H).

We now inductively construct a nested transfinite sequence of unital separable C*-subalgebras 𝒜_α of ℬ(H) together with pure states f_α on 𝒜_α such that for all α < ℵ₁

1. a_α ∈ 𝒜_{α + 1}.

2. if β < α then f_α restricted to 𝒜_β equals f_β.

3. 𝒜_{α + 1} contains a projection q_α ∈ ℳ_α such that 0 < f_{α + 1} (q_α) < 1.

Begin by letting 𝒜₀ be any separable C*-subalgebra of ℬ(H) that is unital and contains 𝒦(H) and let f₀ be any pure state on 𝒜₀ that annihilates 𝒦(H). At successor stages, use the lemma to find a projection q_α ∈ ℳ_α and a pure state g on ℬ(H) such that g|𝒜_α = f_α and 0 < g(q_α) < 1. By Lemma 4 of ref. 6 there is a separable C*-algebra 𝒜_{α + 1} ⊆ ℬ(H) that contains 𝒜_α, a_α, and q_α, and such that the restriction f_{α + 1} of g to 𝒜_{α + 1} is pure. To see this, write ℬ(H) as the union of a continuous nested transfinite sequence of separable C*-algebras ℬ_γ such that ℬ₀ is the C*-algebra generated by 𝒜_α, a_α, and q_α. The cited lemma guarantees that the restriction of g to some ℬ_γ will be pure. Thus, the construction may proceed. At limit ordinals α, let 𝒜_α be the closure of ∪_β<α 𝒜_β. The state f_α is determined by the condition f_α|𝒜_β = f_β, and it is easy to see that f_α must be pure. [If g₁ and g₂ are states on 𝒜_α such that f_α = (g₁ + g₂)/2, then for all β < α purity of f_β implies that g₁ and g₂ agree when restricted to 𝒜_β; thus g₁ = g₂.] This completes the description of the construction.

Now define a state f on ℬ(H) by letting f|_𝒜α = f_α. By the reasoning used immediately above, f is pure, and since 0 < f(q_α) < 1 for all α, the restriction of f to any masa is not pure.

It is interesting to contrast Theorem 0.2 with Theorem 9 of ref. 4, which states that (assuming the continuum hypothesis) any state on 𝒞(H) restricts to a pure state on some masa of 𝒞(H). This does not conflict with our result because there are many masas of 𝒞(H) that do not come from masas of ℬ(H) (regardless of the truth of the continuum hypothesis). Indeed, ℬ(H) has 2^ℵ₀ masas but 𝒞(H) has 2^2ℵ₀ masas. This can be seen by first finding 2^ℵ₀ mutually orthogonal nonzero projections p_α in 𝒞(H), then finding projections q_α¹, q_α² < p_α such that q_α¹ q_α² ≠ q_α² q_α¹ for each α, and finally for each set S ⊆ 2^ℵ₀ choosing a masa of 𝒞(H) that contains {q_α¹ : α ∈ S} and {q_α² : α ⊆ S}. It is easy to see that this produces 2^2ℵ₀ distinct masas.