On uniform Eberlein compacta and c-algebras

Mirna Džamonja School of Mathematics University of East Anglia Norwich, NR4 7TJ, UK

M.Dzamonja@uea.ac.uk

November 16, 1998

Abstract

We investigate a question of Y. Benyamini, M. E. Rudin and M. Wage, on the existence of universal uniform Eberlein compacta of a given weight, more exactly the related question of the existence of a universal c-algebra of a given size. We show that for any regular λ > ℵ_{1} with 2^{ℵ_{0}} > λ, there is no c-algebra of size λ universal under c-embeddings. In fact, under these circumstances, for no μ < 2^{ℵ_{0}} are there μ c-algebras of size λ such that every c-algebra of size λ c-embeds into one of the μ given ones. ^{1}

1 Introduction.

We investigate the question of the existence of universal uniform Eberlein compacta. The question of the existence of universal uniform Eberlein compacta of a given size was asked by Y. Benyamini, M.E. Rudin and M. Wage as Problem 3 in their 1977 paper [BeRuWa]. In his recent paper [Be], M. Bell showed that this is equivalent to the existence of universal c-algebras under ordinary embeddings. He showed in [Be] that if λ = 2^{<λ}, then there is a c-algebra of size λ which is universal not just under ordinary embeddings, but also under a stronger notion of a c-embedding. We are interested in the question of the existence of such an algebra when the relevant instances of GCH fail. We show that for any regular cardinal λ > ℵ_{1} with 2^{ℵ_{0}} > λ, there can be no c-algebra of size λ into which every c-algebra of size λ c-embeds. In fact, under these circumstances, for no μ < 2^{ℵ_{0}} are there μ c-algebras of size λ such that every c-algebra of size λ embeds into one of the μ given ones. The niceness property of the c-algebras is not used in the above. We also show that ♣ + ¬CH implies that there is no c-algebra of size ℵ_{1} into which every c-algebra of size ℵ_{1} c-embeds, in fact, there are no μ < 2^{ℵ_{0}} c-algebras of size ℵ_{1} such that every c-algebra of size ℵ_{1} c-embeds into one of them. In fact, ♣_{club} suffices in the place of ♣. It is a result of M. Bell from [Be] that adding ℵ_{2} Cohen reals suffices to have a model in which there is no universal uniform Eberlein compact of weight ℵ_{1}, hence also no c-algebra of size ℵ_{1} universal even under ordinary embeddings.

1The author thanks Saharon Shelah for bringing M. Bells’ [Be] to her attention. This paper is dedicated to the memory of Amer Bešlagić.

A uniform Eberlein compact, abbreviated UEC, is a topological space homeomorphic to a weakly compact subspace of a Hilbert space. A UEC X^{∗} is said to be universal of weight λ iff every UEC of weight ≤ λ is a continuous image of it. The intuition suggesting that a universal object is the one into which all other objects of that kind embed is justified when one passes to the objects roughly dual to the UEC, so called c-algebras. Calling a c-algebra B^{∗} universal iff every other c-algebra of the same size embeds into B^{∗}, it turns out that there is a universal UEC of weight λ iff there is a universal c-algebra of size λ (M. Bell in [Be]). Note that this result does not follow immediately from the Stone duality theorem, as not every UEC is 0-dimensional.

For a history and further references on UEC, see papers [BeRuWa] and [Be]. The method of invariants used in this paper was developed in the context of linear orders by M. Kojman and S. Shelah in [KjSh 409], and they have used it in other contexts elsewhere.

2 Preliminaries.

Notation 2.1. For α > θ = cf(θ), we let

S_{θ}^{α} =^{def} {β < α : cf(β) = θ}.

Definition 2.2. (1) A boolean algebra B is a c-algebra iff there is a family {B_{n} : n < ω} of subsets of B such that

(i) n ≠ m ⟹ B_{n} ∩ B_{m} = ∅,

(ii) Each B_{n} consists of pairwise disjoint elements,

(iii) ⋃_{n<ω} B_{n} generates B,

(iv) ⋃_{n<ω} B_{n} has the nice property, meaning that for no finite F ⊆ ⋃_{n<ω} B_{n} do we have ⋁ F = 1.

We say that ⟨B_{n} : n < ω⟩ witnesses that B is a c-algebra. When discussing c-algebras, we always have in mind a fixed sequence witnessing this, although we may omit to mention it. We may refer to it as ⟨B_{n}(B) : n < ω⟩.

(2) If B_{l}^{∗} for l ∈ {0, 1} are c-algebras, then a 1-1 boolean homomorphism f : B_{0}^{∗} → B_{1}^{∗} is a c-embedding iff f“B_{n}(B_{0}^{∗}) ⊆ B_{n}(B_{1}^{∗}) for all n < ω. If there is such an embedding from B_{0}^{∗} to B_{1}^{∗}, we write

B_{0}^{∗} < B_{1}^{∗}.

(3) A c-algebra B^{∗} of size λ is universal λ iff for any c-algebra B of size ≤ λ, there is an embedding f : B → B^{∗}.

(4) A c-algebra B^{∗} of size λ is c-universal λ iff for any c-algebra B of size ≤ λ, there is a c-embedding f : B → B^{∗}.

(5) A boolean algebra is almost c iff it satisfies all the properties of c-algebras, except possibly for (iv) in (1) above.

Fact 2.3. (M. Bell, [Be]) (1) There is a universal UEC of size λ iff there is a universal c-algebra of size λ.

(2) If λ = 2^{<λ}, then there is a c-universal c-algebra of size λ.

Definition 2.4. Let λ = cf(λ) > ℵ_{0}.

(1) A sequence ⟨c_{δ} : δ ∈ S⟩ is called a club-sequence iff S is a stationary set of limit ordinals < λ, and for each δ ∈ S, the set c_{δ} is a club subset of δ.

(2) A club sequence ⟨c_{δ} : δ ∈ S⟩ is called a club-guessing sequence iff for every club E of λ there is a stationary set of δ ∈ S such that c_{δ} ⊆ E.

The following fact is a result of S. Shelah from [Sh e] [III 7.8], see also [Sh g], and a proof is also in M. Kojman and S. Shelah’s [KjSh 409].

Fact 2.5. (S. Shelah, [Sh e]) Suppose λ = cf(λ) > ℵ_{1}.

\underline{Then} there is a club guessing sequence ⟨c_{δ} : δ ∈ S ⊆ S_{ℵ_{0}}^{λ}⟩.

Definition 2.6. (1) ♣ is the statement that there is a sequence

⟨c_{δ} : δ limit < ω_{1}⟩

with c_{δ} ⊆ δ = sup(c_{δ}) such that for every A ⊆ ω_{1} unbounded,

{δ : c_{δ} ⊆ A}

is stationary.

(2) ♣_{club} is the statement obtained from ♣ by replacing above in “every A ⊆ ω_{1} unbounded”, the word “unbounded” by “club”.

3 Non-existence of c-universal c-algebras.

Theorem 3.1. Suppose that λ = cf(λ) > ℵ_{1} satisfies 2^{ℵ_{0}} > λ.

\underline{Then} there is no c-universal c-algebra of size λ, moreover for no μ < 2^{ℵ_{0}} are there μ c-algebras of size λ such that every c-algebra of size ≤ λ c-embeds into one of them.

Proof. Fix λ as in the hypothesis, and a club guessing sequence

c̄ = ⟨c_{δ} : δ ∈ S ⊆ S_{ℵ_{0}}^{λ}⟩

as guaranteed by Fact 2.5.

Definition 3.2. (1) Suppose that B is a boolean algebra of size λ. A sequence B̄ = ⟨B^{α} : α < λ⟩ is a filtration of B iff

(i) B^{α} ⊆ B^{α+1} and B^{0} = ∅.

(ii) B^{δ} = ⋃_{α<δ} B^{α} for δ limit,

(iii) |B^{α}| < λ,

(iv) ⋃_{α<λ} B^{α} = B.

(2) Suppose that B is a c-algebra of size λ with a filtration B̄, while δ ∈ S and b ∈ B \ B^{δ}. We define

InvB̄(b, c_{δ}) =^{def} {α ∈ c_{δ} : (∃m ≥ 1)(∃y ∈ B_{m}(B) ∩ B^{min(c_{δ}\(α+1))} \ B^{α}) [y ≥ b]} .

If α ∈ InvB̄(b, c_{δ}) because m ≥ 1 is such that for some y ≥ b we have y ∈ B_{m}(B) ∩ B^{min(c_{δ}\(α+1))} \ B^{α}, we say that α ∈ InvB̄(b, c_{δ}) by virtue of m.

Note 3.3. With the notation of Definition 3.2:

(1) For every m ≥ 1, there is at most one α ∈ InvB̄(b, c_{δ}) which is there by virtue of m (unless b = 0).

[Why? As elements of B_{m}(B) are pairwise disjoint.]

(2) |{A : (∃b ∈ B)(∃δ ∈ S)InvB̄(b, c_{δ}) = A}| ≤ λ.

Main Claim 3.4. Suppose that for δ ∈ S we are given A_{δ} ⊆ c_{δ} with sup(A_{δ}) = δ and otp(A_{δ}) = ω.

\underline{Then} there is a c-algebra B and filtration B̄ of B such that for all δ ∈ S there is b_{δ}^{0} ∈ B \ B^{δ} such that InvB̄(b_{δ}^{0}, c_{δ}) = A_{δ}.

Proof of the Main Claim. Let f_{δ} : ω → A_{δ} be the increasing enumeration of A_{δ}. We shall use α_{m}^{δ} to stand for f_{δ}(m).

For n < ω, let X_{n} be the boolean algebra generated by {a_{η} : η ∈ ^{≤n}λ} freely except for the equations

(1) a_{⟨⟩} = 1,

(2) [η ≠ ε & lg(η) = lg(ε)] ⟹ a_{η} ∧ a_{ε} = 0, (2) η⊴ε ⟹ a_{η} ≥ a_{ε}.

Let B^{∗} be the product algebra ⟨X_{n} : n < ω⟩. Let I be the ideal of eventually 0 elements of B^{∗}, and let B =^{def} B^{∗}/I.

For δ ∈ S let:

b_{δ}^{0} =^{def} [⟨a_{f_{δ}↾0}, a_{f_{δ}↾1}, … a_{f_{δ}↾n}, …⟩]

and for m ≥ 1 let b_{δ}^{m} be the class of the function in B^{∗} which is constantly equal to a_{f_{δ}↾m}. Note that b_{δ}^{m} ≥ b_{δ}^{0} for all m ≥ 1. Also note that b_{δ}^{0} are pairwise disjoint, as for δ_{1} ≠ δ_{2} both in S, we have that A_{δ_{1}} ∩ A_{δ_{2}} is finite.

For n < ω let B_{n} =^{def} {b_{δ}^{n} : δ ∈ S}. Our algebra B is given by

B =^{def} ⟨ ⋃_{n<ω} B_{n}⟩_{B}.

It is easily seen that B is a c-algebra of size λ.

Next we define the filtration B̄ = ⟨B^{α} : α < λ⟩ of B by letting B^{0} = ∅ and for α < λ

B^{α+1} =^{def} ⟨ ⋃_{n<ω} B_{n} ∩ {[b̄] : (∀n)(∃η ∈ ^{≤n}α) [(b(n) = a_{η} or b(n) = 0]}⟩_{B} ,

while B^{α} =^{def} ⋃_{β<α} B^{β} for α limit.

Clearly, B̄ is a filtration of B, and b_{δ}^{0} ∈ B \ B^{δ}. The proof will be finished once we prove:

Subclaim 3.5. For δ ∈ S we have InvB̄(b_{δ}^{0}, c_{δ}) = A_{δ}.

Proof of the Subclaim. Note that for δ ∈ S and m ≥ 1 we have b_{δ}^{m} ∈ B^{min(c_{δ}\(α_{m−1}^{δ}+1))} \ B^{α_{m−1}^{δ}}, hence α_{m−1}^{δ} ∈ InvB̄(b_{δ}^{0}) by virtue of m. Now use Note 3.3(1). ⋆_{3.5} ⋆_{3.4}

Claim 3.6. Suppose that Ā = ⟨A_{δ} : δ ∈ S⟩ is as in the statement of the Main Claim, and B = B[Ā] and B̄ are obtained as in the Main Claim. Further suppose that f : B → B^{∗} is a c-embedding,while B̄^{∗} = ⟨B_{∗}^{α} : α < λ⟩ is a filtration of B^{∗} (so |B^{∗}| = λ).

\underline{Then} there is a club E of λ such that for every δ ∈ S with the property c_{δ} ⊆ E we have

A_{δ} = InvB̄(b_{δ}^{0}, c_{δ}) = InvB̄^{∗}(f(b_{δ}^{0}), c_{δ}).

Proof of the Claim. Let

E =^{def} {α < λ : B_{∗}^{α} ∩ f“B = f“B^{α}}.

Hence E is a club of λ.

Suppose now that δ ∈ S and c_{δ} ⊆ E. Clearly

f(b_{δ}^{m}) ∈ B_{∗}^{min(c_{δ}\(α_{m−1}^{δ}+1))} \ B_{∗}^{α_{m−1}^{δ}},

by the definition of E, and the rest follows as in the proof of A_{δ} = InvB̄(b_{δ}^{0}, c_{δ}). ⋆_{3.6}

Proof of the Theorem finished. Suppose that B^{∗} is a c-algebra of size λ and B̄^{∗} any filtration of B^{∗}. As λ < 2^{ℵ_{0}}, we can choose for each δ ∈ S an unbounded A_{δ} ⊆ c_{δ} of order type ω, such that for no b ∈ B^{∗} \ B_{∗}^{δ} do we have InvB̄^{∗}(b, c_{δ}) = A_{δ}. Let Ā =^{def} ⟨A_{δ} : δ ∈ S⟩ and B =^{def} B[Ā].

Suppose f : B → B^{∗} is a c-embedding. Let E be a club of λ guaranteed by Claim 3.6, and let δ ∈ S be such that c_{δ} ⊆ E. Hence InvB̄^{∗}(f(b_{δ}^{0}), c_{δ}) = A_{δ}, a contradiction.

The part of the theorem involving μ < 2^{ℵ_{0}} is proved similarly. ⋆_{3.1}

Theorem 3.7. (1) Suppose that ♣_{club} + ¬CH holds.

\underline{Then} there is no c-universal c-algebra of size ℵ_{1}. Moreover for no μ < 2^{ℵ_{0}} are there μ c-algebras of size ℵ_{1} such that every c-algebra of size ≤ ℵ_{1} cembeds into one of them.

(2) Suppose that either λ = cf(λ) > ℵ_{1} satisfies 2^{ℵ_{0}} > λ, or λ = ℵ_{1} and ♣_{club} + ¬CH holds.

\underline{Then} there is no almost c-algebra of size λ which is c-universal for almost c-algebras of size ≤ λ, or even for c-algebras of size ≤ λ. Moreover no μ < 2^{ℵ_{0}} almost c-algebras of size λ taken together have that property. ⋆_{3.7}

Proof. The same proof as that of Theorem 3.1.

References

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[KjSh 409] M. Kojman and S. Shelah, Nonexistence of universal orders in

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[Sh g] S. Shelah, Cardinal Arithmetic, Oxford University Press 1994. [Sh e] S. Shelah, Universal classes, in preparation. [Sh 175] S. Shelah, Universal graphs without instances of CH, Annals of

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