A BOOLEAN ACTION OF C(M, U(1)) WITHOUT A SPATIAL MODEL AND A RE-EXAMINATION OF

THE CAMERON–MARTIN THEOREM

JUSTIN TATCH MOORE AND SŁAWOMIR SOLECKI

Abstract. We will demonstrate that if M is an uncountable compact metric space, then there is an action of the Polish group of all continuous functions from M to U(1) on a separable probability algebra which preserves the measure and yet does not admit a point realization in the sense of Mackey. This is achieved by exhibiting a strong form of ergodicity of the Boolean action known as whirliness. This is in contrast with Mackey’s point realization theorem, which asserts that any measure preserving Boolean action of a locally compact second countable group on a separable probability algebra can be realized as an action on the points of the associated probability space. In the course of proving the main theorem, we will prove a result concerning the infinite dimensional Gaussian measure space (ℝ^{ℕ}, γ_{∞}) which is in contrast with the Cameron–Martin Theorem.

1. Introduction

Suppose that G is a Polish group acting invariantly on a probability space (X, μ) where X is equipped with a Polish topology.^{1} Let B_{μ} denote the quotient of the Borel subsets of X modulo the μ-null sets. The original action induces a continuous action of G on B_{μ}, where B_{μ} is topologized by the metric d_{μ}(A, B) = μ(A△B). An action of G on B_{μ} induced by a continuous homomorphism mapping G into Aut(B_{μ}, μ)

2000 Mathematics Subject Classification. 22F10, 22A05, 28C20, 28D15. Key words and phrases. Boolean action, Cameron-Martin, group action, infinite-

dimensional, point realization, Polish group, spatial model, whirly.

We would like to thank Nathaniel Eldredge, Leonard Gross, and Alexander Kechris for reading an early draft of this article and offering a number of useful suggestions. The research of the first author was supported by NSF grant DMS– 0757507. The research of the second author was supported by NSF grant DMS– 1001623. Any opinions, findings, and conclusions or recommendations expressed in this article are those of the authors and do not necessarily reflect the views of the NSF .

1Recall that Polish is a synonym for separable and metrizable by a complete metric.

is commonly referred to as a Boolean action. In the situation above, the action of G on (X, μ) is commonly referred to as a spatial model (or point realization) of this Boolean action. In this paper, we will focus exclusively on Boolean actions on separable probability algebras; the underlying set in a spatial model is by definition required to be a standard Borel space (i.e., the Borel sets arise from a Polish topology).

One can ask under what circumstances a Boolean action has a spatial model. Mackey proved the first general result of this sort, demonstrating that if G is a locally compact Polish group, then every Boolean action of G has a spatial model [7]. Mackey’s proof relied heavily on the existence of Haar measure on G and therefore the situation for non-locally compact groups remained completely open.

Recently, Glasner, Tsirelson, and Weiss [4] introduced the notion of a whirly Boolean action as a sufficient criterion for the non existence of a spatial model for a Boolean action. A Boolean action of a Polish group G on B_{μ} is whirly if whenever V is a neighborhood of 1_{G} and A, B ∈ B_{μ} are positive, there is a g in V such that μ((g·A)∩B) > 0. It was proved in [4] that such Boolean actions not only do not have spatial models, but in fact do not have non trivial factors with spatial models. They also showed that all ergodic Boolean actions of Polish Lévy groups are whirly, and exhibited such an action for the Lévy group L^{0}(ν, U(1)), where U(1) is the circle group and ν is a atomless Borel probability measure on a Polish space. The construction of their Boolean action was generalized by Pestov [8] to L^{0}(ν, S) for all non-trivial compact Polish groups S. Here L^{0}(ν, S) is the group of all (equivalence classes) of ν-measurable functions with values in S taken with the convergence in measure topology.

In the other direction, Glasner and Weiss proved that Boolean actions of closed subgroups of the Polish group of all permutations of ℕ always have spatial models [5]. Kwiatkowska and the second author extended this and Mackey’s results, proving that Boolean actions of isometry groups of locally compact separable metric spaces have spatial models [6].

It remained an open problem whether Mackey’s theorem could be extended to groups of the form C(M, S) whenever M is a compact metric space and S is a compact Polish group. Here C(M, S) is the Polish group of continuous functions from M into S with the pointwise operation and the topology of uniform convergence. Part of the interest in such groups comes from their position on the border between the two classes of groups discussed above: the L^{0} groups and the isometry groups of locally compact separable metric spaces. On the one hand, C(M, S) maps continuously via inclusion onto a dense subgroup of L^{0}(ν, S) for each Borel probability measure ν on M. On the other hand, the closed normal subgroups H ◁C(M, S) for which C(M, S)/H is compact separate the identity from non-identity elements in C(M, S). This property places the groups C(M, S) close to isometry groups of locally compact separable metric spaces, since, by [6, Theorem 1.2], its strengthening “in each neighborhood of 1_{G} there is a closed normal subgroup H for which G/H is compact” guarantees that a Polish group G is the isometry group of a locally compact separable metric space.

Kwiatkowska and the second author noticed, using the second author’s spectral analysis of Boolean actions of C(M, U(1)), that if M is countable, then Boolean actions of C(M, U(1)) always admit spatial models. In the present article, we will construct an example of an action of C(2^{ℕ}, U(1)) on a separable atomless probability algebra which is whirly and hence has no spatial model. Notice that if M is any uncountable compact metric space, then M contains a homeomorphic copy P of the Cantor set. Standard arguments show that the restriction map f ↦ f ↾ P defines a continuous epimorphism from C(M, U(1)) onto C(P, U(1)) ≃ C(2^{ℕ}, U(1)). Furthermore, if f : P → U(1), then f has a continuous extension to M with the same distance from the identity. This yields the main result of the paper.

Theorem 1.1. If M is an uncountable compact metric space, then C(M, U(1)) admits a Boolean action on a separable atomless probability algebra which is whirly.

The construction itself — and Theorem 3.7 which concerns it and is at the root of the proof of Theorem 1.1 — may also be of independent interest from a probabilistic point of view. Also, this appears to be the first whirly action whose whirliness does not come from a form of concentration of measure on the acting group; see [4], [5], and [8].

In the process of proving the main result, we will also establish some results about the space (ℝ^{ℕ}, γ_{∞}), where γ_{∞} is the product measure arising from the standard Gaussian measure on ℝ.

Theorem 1.2. If a ∈ ℝ and K ⊆ ℝ^{ℕ} is a Borel set with γ_{∞}(K) > 0, then for γ_{∞}-a.e. y in ℝ^{ℕ}, γ_{∞}(\sqrt{1 + a^{2}}K + ay) > 0.

The result above is in contrast with the Cameron-Martin Theorem (see, e.g., [2, Example 2.3.5, Theorem 3.2.3]) which implies that the translation of γ_{∞} by x ∈ ℝ^{ℕ} is orthogonal to γ_{∞} unless x is an element of ℓ^{2} (a set with γ_{∞}-measure 0). In fact, it is not difficult to show, using an analysis similar to that in the proof of the Cameron-Martin Theorem, that if (a, b) ≠ (±1, 0), then the affine transformation x ↦ ax+by sends a set of γ_{∞}-measure 1 to a set of γ_{∞}-measure 0 for γ_{∞}-a.e. y. Also, Theorem 1.2 does not hold for transformations x ↦ ax + by unless a^{2} = b^{2} + 1. This will be discussed in the final section.

Our notation is mostly standard. We will take ℕ to include 0 and all indexing will begin with 0. If n is a natural number, then 2^{n} will be used to denote the collection of all binary sequences of length n. We will use 2^{<ω} to denote the collection of all binary sequences of finite length. If σ is such a sequence and n is not bigger than the length |σ| of σ, then σ ↾ n is the sequence consisting of the first n entries of σ. If σ and τ are sequences, σ^{⌢}τ will denote their concatenation. If x is a point in a metric space X and δ > 0, then N_{δ}(x) denotes the open δ-ball about x. The variables ϵ, δ, and s will always denote real numbers; i, j, k, m, and n will always denote integers. We will use [1] as our general reference for probability. Further information on measure theory and Polish group actions can be found in [9].

2. The construction

Let γ denote the standard Gaussian measure on ℂ viewed as ℝ^{2}; γ has density

\frac{1}{2π} e^{− \frac{x^{2}+y^{2}}{2}}

with respect to the Lebesgue measure on ℝ^{2}. Define I = 2^{<ω} ∪ {∗} where ∗ is some symbol not in 2^{<ω} and let γ^{I} denote the product measure on ℂ^{I}. Let U_{σ} (σ ∈ I) denote the coordinate functions on ℂ^{I}, regarded as complex random variables (i.e. these are i.i.d. standard complex Gaussian random variables). Define random variables Z_{σ} (σ ∈ 2^{<ω}) recursively by

Z_{∅} = U_{∗}

Z_{σ^{⌢}0} = \frac{Z_{σ} + U_{σ}}{\sqrt{2}}

Z_{σ^{⌢}1} = \frac{Z_{σ} − U_{σ}}{\sqrt{2}} .

Observe that, for a given σ, Z_{σ^{⌢}0} and Z_{σ^{⌢}1} are independent. It follows inductively that for each n, {Z_{σ} : σ ∈ 2^{n}} ∪ {U_{σ} : |σ| ≥ n} is a family of independent standard complex Gaussian random variables. Hence if we define, for each σ ∈ 2^{n} and k ∈ ℕ,

U_{σ,n+k} = \frac{1}{\sqrt{2}^{k}} ∑_{τ∈2^{k}} U_{σ^{⌢}τ},

then {Z_{σ} : σ ∈ 2^{n}} ∪ {U_{σ,k} : (σ ∈ 2^{n}) ∧ (k ≥ n)} is an independent family of standard complex Gaussian random variables.

Now we define the set underlying the measure space on which the Boolean action of C(2^{ℕ}, U(1)) will be performed. For each n ∈ ℕ, define X_{n} = ℂ^{2^{n}} and let γ_{n} denote the product measure on X_{n}. If n ∈ ℕ, define π_{n+1,n} : X_{n+1} → X_{n} by

π_{n+1,n}(x)(σ) = \frac{x(σ^{⌢}0) + x(σ^{⌢}1)}{\sqrt{2}}

and if n, k ∈ ℕ, let π_{n+k,n} : X_{n+k} → X_{n} be the composite projection

π_{n+k,n}(x)(σ) = \frac{1}{\sqrt{2}^{k}} ∑_{τ∈2^{k}} x(σ^{⌢}τ).

It is easily verified that γ_{n}(π_{n,m}^{−1}(A)) = γ_{m}(A) whenever A is a Borel subset of X_{m} and m < n. Define X_{ω} to be the collection of all functions x : 2^{<ω} → ℂ such that for all σ ∈ 2^{<ω}

x(σ) = \frac{x(σ^{⌢}0) + x(σ^{⌢}1)}{\sqrt{2}}

The projections π_{n} : X_{ω} → X_{n} are defined by restriction. Observe that X_{ω} is a closed linear subspace of ℂ^{2^{<ω}} with the product topology.

We define a measure γ_{ω} on X_{ω} as follows. Note that, for each σ,

Z_{σ} = \frac{Z_{σ^{⌢}0} + Z_{σ^{⌢}1}}{\sqrt{2}}

and, therefore, the random variables Z_{σ} (σ ∈ 2^{<ω}) induce a linear bijection from ℂ^{I} to X_{ω}, where bijectivity follows from the identity

U_{σ} = \frac{Z_{σ^{⌢}0} − Z_{σ^{⌢}1}}{\sqrt{2}} .

Let Φ be the inverse of this bijection and define γ_{ω} to be the pullback of γ^{I} under Φ: γ_{ω}(A) = γ^{I}(Φ(A)). It follows from the observations above that γ_{ω} satisfies γ_{ω}(π_{n}^{−1}(A)) = γ_{n}(A) whenever n ∈ ℕ and A ⊆ X_{n}. For each σ ∈ 2^{<ω} and k ≥ |σ|, we will also regard Z_{σ}, U_{σ}, and U_{σ,k} as random variables in the measure space (X_{ω}, γ_{ω}) by identifying this space with (ℂ^{I}, γ^{I}) via Φ.

Conditional probability will play a crucial role in our proof. In particular, we will need the following result often known as the Measure Disintegration Theorem.

Theorem 2.1. (see [3, 5.14]) Let (X, μ) and (Y, ν) be standard probability spaces and let h : X → Y be a Borel surjection such that μ(h^{−1}(A)) = ν(A) whenever A is a Borel subset of Y. Then there is a mapping y ↦ μ_{y} defined on Y which satisfies the following conditions:

• for each y in Y, μ_{y} is a Borel probability measure on X and μ_{y}(h^{−1}(y)) = 1;

• for each Borel A ⊆ X, y ↦ μ_{y}(A) is a Borel measurable function;

• for every Borel A ⊆ X, μ(A) = ∫ μ_{y}(A)dν(y).

For each n, fix a Borel measurable mapping x ↦ γ_{x} defined on X_{n} such that, for a given x ∈ X^{n}, γ_{x} is a Borel probability measure on X_{ω} with

γ_{x}({x ∈ X_{ω} : π_{n}(x) = x}) = 1,
γ_{ω}(A) = ∫ γ_{x}(A)dγ_{n}(x).

Next we define the Boolean action of C(2^{ℕ}, U(1)) on (X_{ω}, γ_{ω}). Let S_{n} denote all elements g of C(2^{ℕ}, U(1)) such that g(x) depends only on the first n coordinates of x ∈ 2^{ℕ}. Let S_{ω} denote the union of the sets S_{n} as n ranges over ℕ. Note that we can naturally identify S_{n} with U(1)^{2^{n}}. If m ≥ n, g is in S_{n}, and σ ∈ 2^{m}, we will use g(σ) to denote the common value of g(x) for those x ∈ 2^{ℕ} extending σ. Observe that if x is a function defined on finite binary sequences of length at least n into ℂ and for every σ of length at least n

x(σ) = \frac{x(σ^{⌢}0) + x(σ^{⌢}1)}{\sqrt{2}},

then there is a unique element of X_{ω} which extends x.

If x is in X_{ω} and g is in S_{n}, then g · x is defined by letting (g · x)(σ), for σ of length at least n, be g(σ)x(σ) ∈ ℂ. This is well defined for sequences σ of length at least n (notice that g is in S_{m} for all m ≥ n); the result extends uniquely to sequences of length less than n to yield an element of X_{ω}. Thus, we have defined an action of S_{ω} on X_{ω} and hence on B_{ω}. It should be clear that this action preserves γ_{ω}.

Lemma 2.2. The action of S_{ω} on B_{ω} extends to a continuous action of C(2^{ℕ}, U(1)) on B_{ω}.

Proof. The lemma is a consequence of the following statement.

For every W ⊆ X_{k} which is a product of open disks and for every ϵ > 0, there is a δ > 0 such that if n > k and g, h ∈ S_{n} with |g(σ)−h(σ)| < δ for all σ ∈ 2^{n}, then γ_{n}((g · π_{n,k}^{−1}(W))△(h · π_{n,k}^{−1}(W))) < ϵ.

To prove the above statement, first observe that it suffices to prove it for k = 0 (that is, W is an open disk in ℂ). Let z and r be such that W = N_{r}(z). Let s > 0 be sufficiently small that

γ(N_{r+s}(x) \ N_{r−s}(x)) < ϵ/3.

Define δ = s\sqrt{ϵ/3}. Now suppose that g, h ∈ S_{n} are given with |g(σ) − h(σ)| < δ for all σ ∈ 2^{n}. Notice that

x ↦ π_{n,0}(g^{−1} · x) − π_{n,0}(h^{−1} · x) ∈ ℂ

defines a complex Gaussian random variable on (X_{n}, γ_{n}) with mean 0 and variance less than δ^{2}. In particular, by Chebyshev’s inequality

γ_{n}(\{x ∈ X_{n} : |π_{n,0}(g^{−1} · x) − π_{n,0}(h^{−1} · x)| ≥ s\}) ≤ \frac{ϵ}{3}.

Finally, observe that x ∈ g · π_{n,0}^{−1}(W) is equivalent to π_{n,0}(g^{−1} · x) ∈ W (and similarly for h). Thus, if we let

E = (g · π_{n,0}^{−1}(W))△(h · π_{n,0}^{−1}(W))
E_{0} = {x ∈ X_{n} : π_{n,0}(x) ∈ N_{r+s}(x) \ N_{r−s}(x)}
E_{1} = {x ∈ X_{n} : |π_{n,0}(g^{−1} · x) − π_{n,0}(h^{−1} · x)| ≥ s},

then E ⊆ gE_{0} ∪ hE_{0} ∪ E_{1} and hence γ_{n}(E) < ϵ. The statement and the lemma follow. □

We finish this section by recalling some standard facts about the space (ℂ^{I}, γ^{I}). Let H denote the collection of all elements of ℂ^{I} with finite support, regarded as a group under addition.

Lemma 2.3. For every Borel A ⊆ ℂ^{I} and every h ∈ H, γ^{I}(A) > 0 if and only if γ^{I}(A + h) > 0.

Proof. This follows from the fact that translations are absolutely continuous in finite dimensional powers of (ℂ, γ) and the properties of product measures. (It is also a trivial consequence of the Cameron-Martin Theorem; see [2, Example 2.3.5, Theorem 3.2.3].) □

Lemma 2.4. If A ⊆ ℂ^{I} is Borel and H-invariant (i.e., A + H = A), then γ^{I}(A) is either 0 or 1.

Proof. This is a special case of Kolmogorov’s Zero-One Law. □

3. Proof of the main theorem

For each s ∈ ℝ, k ∈ ℕ define the following element of C(2^{ℕ}, U(1)):

g_{s,k}(x) = \frac{1 + is(−1)^{x(k)}}{\sqrt{1 + s^{2}}}

Lemma 3.1. For each n and γ_{n}-a.e. z ∈ X_{n},

{U_{σ,k} : (σ ∈ 2^{n}) ∧ (n ≤ k)}

is a family of mutually independent standard complex Gaussian random variables with respect to γ_{z}.

Proof. This follows immediately from the observation made above that

{Z_{σ} : σ ∈ 2^{n}} ∪ {U_{σ,k} : (σ ∈ 2^{n}) ∧ (n ≤ k)}

is an independent family of standard complex Gaussian random variables with respect to γ_{ω}. □

Lemma 3.2. If n ∈ ℕ and K ⊆ X_{n} is Borel with γ_{n}(K) > 0, then for each s > 0 and γ_{n}-a.e. z ∈ X_{n}

g_{s,k} · π_{n}^{−1}(K), for n ≤ k

are mutually independent with respect to γ_{z}, each having γ_{z}-measure γ_{n}(\frac{\sqrt{1+s^{2}}K−z}{s}).

Proof. Let U_{n,k} denote the vector valued random variable

(U_{σ,k} : σ ∈ 2^{n}).

Let y be an element of X_{ω} and σ be an element of 2^{n}. Observe that the value of π_{n}(g_{s,n} · y) at σ is

\frac{y(σ^{⌢}0) + isy(σ^{⌢}0) + y(σ^{⌢}1) − isy(σ^{⌢}1)}{\sqrt{2}\sqrt{1 + s^{2}}}

= \frac{1}{\sqrt{1 + s^{2}}} (\frac{y(σ^{⌢}0) + y(σ^{⌢}1)}{\sqrt{2}} + is\frac{y(σ^{⌢}0) − y(σ^{⌢}1)}{\sqrt{2}})

and thus

π_{n}(g_{s,n} · y) = \frac{π_{n}(y) + isU_{n,n}(y)}{\sqrt{1 + s^{2}}}

More generally, if n ≤ k, then

π_{n}(g_{s,k} · y) = \frac{π_{n}(y) + isU_{n,k}(y)}{\sqrt{1 + s^{2}}}.

Fix z ∈ X_{n}. It follows that if π_{n}(y) = z, then π_{n}(g_{s,k} · y) is in K if and only if

\frac{1}{\sqrt{1 + s^{2}}} (z + isU_{n,k}(y)) ∈ K

if and only if

U_{n,k}(y) ∈ −i\frac{\sqrt{1 + s^{2}}K − z}{s} .

The conclusion follows from Lemma 3.1 after observing that multiplication by −i preserves the standard Gaussian measure. □

Lemma 3.3. If K ⊆ X_{ω} is Borel and a ∈ ℝ, then

∫ γ_{ω}(\sqrt{1 + a^{2}}K + ay)dγ_{ω}(y) = γ_{ω}(K)

Remark 3.4. Note that, since there is a continuous linear measure preserving bijection between (X_{ω}, γ_{ω}) and (ℝ^{ℕ}, γ_{∞}), an analogous identity holds for the measure space (ℝ^{ℕ}, γ_{∞}). This can also be verified by direct computation.

Proof. We first show that if n ∈ ℕ, K ⊆ X_{n} is Borel, and a ∈ ℝ, then

∫ γ_{n}(\sqrt{1 + a^{2}}K + az)dγ_{n}(z) = γ_{n}(K)

If a = 0, the above equality clearly holds. If a ≠ 0, set s = −1/a. Applying Lemma 3.2 we have

γ_{n}(K) = γ_{ω}(π_{n}^{−1}(K)) = γ_{ω}(g_{s,n} · π_{n}^{−1}(K))
= ∫ γ_{z}(g_{s,n} · π_{n}^{−1}(K))dγ_{n}(z)
= ∫ γ_{n}(\sqrt{1 + a^{2}}K + az)dγ_{n}(z),

as required.

Observe now that ν(K) = ∫ γ_{ω}(\sqrt{1 + a^{2}}K + ay)dγ_{ω}(y) defines a Borel probability measure on X_{ω}. By the equality proved above, this measure coincides with γ_{ω} on sets of the form π_{n}^{−1}(K) for K a Borel subset of X_{n}. Therefore ν equals γ_{ω}. □

Lemma 3.5. If a ∈ ℝ and K ⊆ X_{ω} is Borel with γ_{ω}(K) > 0, then

γ_{ω}(\{x ∈ X_{ω} : γ_{ω}(\sqrt{1 + a^{2}}K + ax) > 0\}) = 1

Remark 3.6. Theorem 1.2 follows immediately by again making the observation that there is a continuous linear measure preserving bijection between (X_{ω}, γ_{ω}) and (ℝ^{ℕ}, γ_{∞}).

Proof. Let K and a be given as in the statement of the lemma and define

A = \{x ∈ X_{ω} : γ_{ω}(\sqrt{1 + a^{2}}K + ax) > 0\}

and let B be the image of A under Φ. First observe that by Lemma 3.3, γ_{ω}(A) > 0. If x is in A, then the linearity of Φ implies that

γ^{I}(\sqrt{1 + a^{2}}Φ(K) + aΦ(x)) > 0.

By Lemma 2.3, if h is in H, then

γ^{I}(\sqrt{1 + a^{2}}Φ(K) + aΦ(x) + ah) > 0

and therefore x+Φ^{−1}(h) is also in A. This means that B is H-invariant and therefore must have measure 1 by Lemma 2.4. Since γ_{ω}(A) = γ^{I}(B), this completes the proof. □

We are now ready to establish that our near action is whirly.

Theorem 3.7. For every Borel K ⊆ X_{ω} with 0 < γ_{ω}(K) and for every ϵ > 0 there are m and n with

γ_{ω}( ⋃_{k=n}^{n+m} g_{ϵ,k} · K) > 1 − ϵ.

Remark 3.8. The functions g_{ϵ,k} are in fact Lipschitz. Notice, however, that while the uniform distance from g_{ϵ,k} to 1 does not depend on k (and tends to 0 as ϵ approaches 0), the Lipschitz norm of g_{ϵ,k} tends to infinity with respect to k for any fixed ϵ > 0.

Proof. Let K and ϵ be given as in the statement of the theorem. We can, and do, assume that K is compact. Set a = −1/ϵ and fix a δ > 0 such that

γ_{ω}(\{y ∈ X_{ω} : γ_{ω}(\sqrt{1 + a^{2}}K + ay) ≥ δ\}) > \sqrt{1 − ϵ/2}.

This is possible by Lemma 3.5. Let m be sufficiently large such that 1 − (1 − δ)^{m} ≥ \sqrt{1 − ϵ/2}. Next by applying compactness of K, fix an n which is sufficiently large such that

γ_{ω}(K) > γ_{n}(π_{n}(K)) − \frac{ϵ}{2m}.

Define K′ = π_{n}(K) and set

A = \{z ∈ X_{n} : γ_{n}(\sqrt{1 + a^{2}}K′ + az) ≥ δ\}

noting that γ_{n}(A) > \sqrt{1 − ϵ/2}. By Lemma 3.2, for each z in A, we have that the sets g_{ϵ,k} · π_{n}^{−1}(K′), n ≤ k < n + m, are γ_{z}-mutually independent with γ_{z}-measure at least δ. Their union, therefore, has γ_{z}-measure at least 1 − (1 − δ)^{m} ≥ \sqrt{1 − ϵ/2}. Integrating, we obtain

γ_{ω}( ⋃_{k=n}^{n+m−1} g_{ϵ,k} · π_{n}^{−1}(K′)) = ∫_{X_{n}} γ_{z}( ⋃_{k=n}^{n+m−1} g_{ϵ,k} · π_{n}^{−1}(K′))dγ_{n}(z)

≥ ∫_{A} γ_{z}( ⋃_{k=n}^{n+m−1} g_{ϵ,k} · π_{n}^{−1}(K′))dγ_{n}(z) > (\sqrt{1 − \frac{ϵ}{2}})(\sqrt{1 − \frac{ϵ}{2}}) = 1 − \frac{ϵ}{2}

Finally,

γ_{ω}( ⋃_{k=n}^{n+m−1} g_{ϵ,k} · K) > γ_{ω}( ⋃_{k=n}^{n+m−1} g_{ϵ,k} · π_{n}^{−1}(K′)) − m\frac{ϵ}{2m} > 1 − ϵ.

□

4. Concluding remarks

It is natural to ask whether the main construction of this paper is possible if we replace C(M, U(1)) by C^{∞}(M, U(1)), where M is now some smooth manifold and C^{∞}(M, U(1)) is equipped with the metric arising from the C^{∞}-norm. Suppose that M is given and P ⊆ M is homeomorphic to 2^{ℕ}. The Boolean action itself can be constructed exactly as above. The key difference, however, is that for a fixed ϵ > 0, the extensions of the functions g_{ϵ,k} to M will necessarily have norms which diverge to infinity as k → ∞. The role of the functions g_{ϵ,k} seems so essential to our argument that it is tempting to conjecture that if M is a smooth manifold and S is a compact Lie group, then every Boolean action of C^{∞}(M, S) on a separable probability algebra has a spatial model.

We will finish by mentioning that Theorem 1.2 is sharp in the sense that it only applies to the affine transformations x ↦ ax + by when a^{2} = b^{2} + 1. To see this define, for x ∈ ℝ^{ℕ},

s(x) = lim_{n→∞} (\frac{1}{n} ∑_{k<n} |x(k)|^{2})^{\frac{1}{2}} .

While this limit may not exists, s(x) = 1 for γ_{∞}-a.e. x; set

K = {x ∈ ℝ^{ℕ} : s(x) = 1}.

Furthermore, if x and y are in ℝ^{ℕ} and a is in ℝ, then s(ax) = as(x). Next observe that if a, b ∈ ℝ, then by Remark 3.4,

γ_{∞}(\frac{a}{\sqrt{1 + b^{2}}} K) = ∫ γ_{∞}(\sqrt{1 + b^{2}} \frac{a}{\sqrt{1 + b^{2}}} K + by)dγ_{∞}(y)

= ∫ γ_{∞}(aK + by)dγ_{∞}(y).

If a^{2} ≠ b^{2} + 1, then (\frac{a}{\sqrt{1+b^{2}}} K) ∩ K = ∅. In particular, γ_{∞}(\frac{a}{\sqrt{1+b^{2}}} K) = 0 and hence γ_{∞}(aK + by) = 0 for γ_{∞}-a.e. y.

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Justin Tatch Moore: Department of Mathematics, Cornell University, Ithaca, NY 14853–4201, USA

  E-mail address: justin@math.cornell.edu

Sławomir Solecki: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, Illinois 61801-2975, USA

  E-mail address: ssolecki@math.uiuc.edu