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G-continuous functions and whirly actions

This paper continues the work Glasner-Tsirelson-Weiss, ArXiv math.DS/0311450. For a Polish group G the notions of G-continuous functions and whirly actions are further exploited to show that: (i) A G-action is whirly iff it admits no nontrivial spatial (= pointwise) factors. (ii) Every action of a Polish Levy group is whirly. (iii) There exists a Polish monothetic group which is not Levy but admits a whirly action. (iv) In the Polish group Aut(X,\mu), for the generic automorphism T, the action of the Polish group \Lambda(T) = closure {T^n: n \in Z} \subset Aut(X,\mu) on the Lebesgue space (X,\mu) is whirly. (v) The Polish additive group underlying a separable Hilbert space admits both spatial and whirly faithful actions. (vi) When G is a non-archimedean Polish group then every G-action is spatial.


Introduction
In this paper we will freely use notations and results from [6]. The reader is also referred to that paper for motivation and background. However for the convenience of the reader we will now briefly recall some definitions and results from [6], sometimes in a slightly modified way, that will be used here.
A Borel action of G on a Borel space (X, X) is a Borel map G × X → X satisfying the conditions, ex = x and g(hx) = (gh)x for all g, h ∈ G and all x ∈ X (e denotes the identity element of G). Such an object is called also a Borel G-space. When a G-space carries an invariant measure we define the notion of spatial G-action. Definition 1.1. Let G be a Polish group. By a spatial G-action we mean a Borel action of G on a standard Lebesgue space (X, X, µ) such that each g ∈ G preserves the measure µ. We say that two spatial actions are isomorphic, if there exists a measure preserving one to one map between two G-invariant subsets of full measure in the corresponding spaces which intertwines the G-actions (the same two sets for all g ∈ G).
Often however it is the case that one has to deal with near-actions (see the definition below and Zimmer [17,Def. 3.1]) or merely with an action of the group on a measure algebra (i.e. the Borel algebra modulo sets of measure zero) and it is then desirable to find a spatial model. Definition 1.2. Let G be a Polish group and (X, X, µ) a standard Borel space with a probability measure µ. By a near-action of G on (X, X, µ) we mean a Borel map G × X → X, (g, x) → gx with the following properties: (i) With e the identity element of G, ex = x for almost every x.
(ii) For each pair g, h ∈ G, g(hx) = (gh)x for almost every x (where the set of points x ∈ X of measure one where this equality holds may depend on the pair g, h). (iii) Each g ∈ G preserves the measure µ.
Let Aut (X) = Aut (X, X, µ) be the Polish group of all equivalence classes of invertible measure preserving transformations X → X, with the neighborhood basis at the identity formed by sets of the form N(A, ε) = {T ∈ Aut (X) : µ(A△T A) < ε}, for A ∈ X and ε > 0. The following proposition is from the introduction to [6]. (I) A near-action of G on (X, X, µ). (II) A continuous homomorphism from G to Aut (X). (III) A boolean action of G on (X, X, µ), that is, a continuous homomorphism from G to the automorphism group of the associated measure algebra.
Every spatial action is also a near-action and in that case the spatial action will be called a spatial model of the near-action.
Recall that a Polish G-space is a Polish space X together with a continuous action G × X → X of a Polish group G. Such an action will be called a Polish action. If in addition X is compact then it is a compact (Polish) G-space.
Every Polish action is also a Borel action. In that case the Polish action will be called a Polish model of the Borel action.
We have the following classical theorems, due to Mackey, Varadarajan    (a) Every Borel action of a Polish group admits a Polish model.
It was shown in [6] that many Polish groups have near-actions that do not admit spatial models. In fact it is shown there that for a Polish Lévy group any spatial action is necessarily trivial (Theorem 1.1 of [6]).
Lévy groups were introduced and studied by Gromov and Milman in [7]. We briefly recall the definitions of Lévy families and Lévy groups. See [11], [8], for detailed studies of the phenomenon of concentration of measure and a plethora of examples. Concise proofs of some of the outstanding instances of this phenomenon can be found in the appendix of [6]. Refer to [10] and [12] for further details on the history of the subject.
A family (X n , d n , µ n ), n = 1, 2, 3 . . . of metric spaces with probability measures µ n is called a Lévy family if the following condition is satisfied. Whenever A n ⊂ X n is a sequence of subsets such that lim inf µ n (A n ) > 0 then for any ε > 0, lim µ(B ε (A n )) = 1 (here B ε (A) is the ε neighborhood of A). A Polish group G is a Lévy group if there exits a family of compact subgroups K n ⊂ K n+1 such that the group F = ∪ n∈N K n is dense in G and the corresponding family (K n , d, m n ) is a Lévy family; here m n is the normalized Haar measure on K n , and d is a right-invariant compatible metric on G.
Since the group Aut (X) is a Lévy group it follows in particular that the natural near-action of Aut (X) on (X, X, µ) does not admit a spatial model.
Another approach used in [6] to the problem of finding spatial models to boolean actions is through the notions of G-continuous function and whirly action. Definition 1.6. Having a near-action (or boolean action) of G on (X, X, µ) we say The following theorem is a reformulation of Theorem 2.2 and Remark 2.3 of [6]. Theorem 1.7. Let (X, µ, G) be a boolean G-system and let D ⊂ X be the, G-invariant, sub σ-algebra generated by the G-continuous functions in L ∞ (µ). Then the factor boolean system (D, µ, G) is the maximum factor of (X, µ, G) which admits a spatial model. Thus A ∈ X is in D iff for every ε > 0 there is a G-continuous function f ∈ L ∞ such that f − 1 A 2 < ε. In particular (X, µ, G) admits a spatial model if and only if the collection of G-continuous functions is dense in L 2 (µ).
The following definition and proposition are from [6] (part (c) of Proposition 1.9 follows directly from the definition). Definition 1.8. A near-action of G on (X, X, µ) is whirly, if for all sets A, B ∈ X of positive measure, every neighborhood of e (the unit of G) contains g such that µ(A ∩ gB) > 0. Proposition 1.9.
(a) If a near-action is whirly, then all G-continuous functions are constants.
(b) A whirly action has no spatial model; moreover, such an action cannot have nontrivial spatial factor.
(c) A factor of a whirly action is whirly.
In Theorem 2.10 below we will show that conversely, a boolean action with no nontrivial spatial factor is whirly.
It was shown in [6] that the action of Aut (X) on (X, X, µ) is whirly, thereby providing a second proof to the fact that this action does not admit a spatial model. Theorem 2.11 below asserts that in fact every near-action of a Polish Lévy group is whirly.

Stable subsets
Let G be a Polish group and fix a sequence {U m } ∞ m=1 of neighborhoods of the identity element e ∈ G with the following properties.
(i) For every m, Let (X, µ, G) be a boolean G-dynamical system. We say that A ∈ X is positive when µ(A) > 0. Given a positive set A ∈ X and an open set U ⊂ G, recall that the set UA ∈ X was defined in [6] as Here U ′ ⊂ U is a countable dense subset of U. This defines UA uniquely as an element of the measure algebra X and the definition does not depend on the choice of U ′ . Note that for all u ∈ U we have uA ⊂ UA.
We say that A ∈ X is stable if it is either empty orÃ = A. Proof. 1. Clear.

For each
henceÃ ⊂ ∞ k=1 U k A =Ã. By 1, alsoÃ ⊂Ã, henceÃ =Ã. 3. Given g ∈ G and n ∈ N there exists an m ∈ N with g −1 U m g ⊂ U n . Hence hence also gÃ ⊂ gA. The claims 4 and 5 are easy to verify.
is a sequence of stable sets then so is A = ∞ k=1 A k . Proof. By Proposition 2.2 we can assume that A k+1 ⊂ A k for every k. Fix k 0 . Then for each n, A ⊂ U n A ⊂ U n A k 0 , hencẽ Lemma 2.5. Let A be a positive stable set and ε > 0, then there exists a stable set D ⊂ A c with µ(A c \ D) < ε and a k such that µ(U k A ∩ U k D) = 0.
We now set D =Ẽ so that D is stable and Proposition 2.6. If A =Ã ∈ X is stable then A c is a union of an increasing sequence of stable sets.
Proof. If A = ∅ the assertion is clear. Otherwise, given ε, we use Lemma 2.5 to find a stable subset D ε ⊂ A c with µ(A c \ D ε ) < ε. Next use an exhaustion argument to show that A c is a union of a sequence of stable sets. Finally we can arrange for this sequence to be an increasing one by means of Proposition 2.2.
Definition 2.7. Let A ⊂ X be the σ-algebra generated by the stable sets.
Theorem 2.8. Every A ∈ A is a union of an increasing sequence of stable sets.
Proof. 1. Let A 0 be the collection of sets A ∈ X which are a union of an increasing sequence of stable sets. We will show that A 0 = A. Since A 0 contains the stable sets it is enough to show that it is closed under the operations of taking countable union and taking complements. The closure under countable union is evident. Suppose that By an exhaustion argument and an application of Proposition 2.2 we conclude that B is an increasing union of stable sets.
The proof of the next theorem is a variation on the classical Urysohn lemma. Theorem 2.9. If (X, µ, G) is a boolean G-system then D = A; i.e. the sub σ-algebra generated by the G-continuous factors coincides with the sub σ-algebra generated by the stable sets. In particular (X, µ, G) admits a spatial model iff X = A.
Proof. If f is a G-continuous function and In proving the converse inclusion we can now restrict our attention to the boolean factor (A, µ, G). In other words we now assume, with no loss of generality, that X = A.
Next observe that, in view of Theorem 1.7, it suffices to prove the following claim. Given A ∈ X and ε > 0 there exists a G-continuous Of course we can assume that both A and A c are positive.
Applying Theorem 2.8 and Lemma 2.5 we choose stable subsets D 0 ⊂ A, D 1 ⊂ A c and an m 0 such that (We say that U m 0 D 0 and U m 0 D 1 are disjoint.) Define f to have the values 0 on D 0 and 1 on D 1 . Next define, by induction on the binary tree, a system of disjoint stable subsets {D r : r = j 2 n , n ∈ N, 0 ≤ j < 2 n } whose union will be X, with the following provision. Note that if UD = D, for a positive set D and a symmetric neighborhood U of e in G, then also UD c = D c and both sets are stable. In the following construction we will assume, for convenience, that at each stage the various sets of the form U m D r \ D r are positive sets. If this is not the case then, by the above remark D as well as D c are stable and the construction will terminate at that point. The corresponding interval of dyadic rationals will be missing from the range of f .
Step 1: hence also the sets U m 0 +2 D 0 , U m 0 +2 D 1/2 and U m 0 +2 D 1 are pairwise disjoint. We now choose m 1 > m 0 + 2 such that We define f to have the value 1/2 on the stable set D 1/2 and declare that f will have values in the Step 2: Next consider the two disjoint stable sets D 0 and D 1/2 as subsets of the set And, at the same time, the two disjoint stable sets D 1 and D 1/2 as subsets of the set Note that, e.g., For a suitable m 2 > m 1 + 2 we will have {µ(U m 2 D r \ D r ) : j = 0, 1/4, 1/2, 3/4, 1} < 1 2 2 , and the sets U m 2 D r , r ∈ {0, 1/4, 1/2, 3/4, 1} are pairwise disjoint.
The key point (in proving the G-continuity of f ) is to observe that, e.g., Once this construction is completed we note that It is now easy to conclude that f − 1 A 2 < ε and that f is G-continuous.
The next two results are direct corollaries of Theorem 2.9.
Theorem 2.10. A boolean G-system (X, µ, G) is whirly iff it admits no nontrivial spatial factors.
Proof. If there are no spatial factors then Theorem 1.7 implies that there are no nontrivial G-continuous functions, hence, by Theorem 2.9, no nontrivial stable sets. It follows that µ(UA) = 1 for every positive set A ∈ X and a neighborhood of the identity U in G. By [6] this condition is equivalent to (X, µ, G) being whirly. Conversely, if (X, µ, G) is whirly then every stable set is either null or the whole space and by Theorem 2.9, D = A is also trivial. Thus (X, µ, G) admits no nonconstant G-continuous functions and by Theorem 1.7 there are no nontrivial spatial factors.
Theorem 2.11. Every ergodic boolean action of a Polish Lévy group is whirly.
Proof. By Theorem 1.1 of [6] every ergodic spatial action of a Polish Lévy group is trivial; i.e. a one point system. Evidently our theorem is now a consequence of this fact together with Theorem 2.10.

Some structure theory
Let G be a Polish group and Y, Y, ν, G) a spatial G-system. Denoting by M(Y ) the space of Borel measures on the Lebesgue space Y , we observe that the spatial action of G on Y induces an action of G on M(Y ). By [1] we can choose (Y, G) to be a compact model where the action G×Y → Y is jointly continuous, so that the induced action (M(Y ), G) is jointly continuous as well. This observation implies that every quasifactor (hence in particular every factor) of the system Y, Y, ν, G) has a spatial model (see [5]). Theorem 8.4 of [5] asserts that two ergodic systems (X, X, µ, G) and Y, Y, ν, G) are not disjoint iff Y, Y, ν, G) admits a nontrivial quasifactor which is a factor of (X, X, µ, G). Thus part 2 of the following theorem is a direct consequence of Theorem 2.10 above. For completeness, and for those who are not familiar with the theory of quasifactors, we provide an alternative direct proof. 1. Every factor of a spatial G-system is a spatial system. 2. A G-system is whirly iff it is disjoint from every spatial G-system.
Since E Z is a contraction, both on L 2 and on L ∞ , it follows easily that if f ∈ L ∞ (Y, ν) is a G-continuous function then so is E Z f . An application of Theorem 1.7 yields an L 2 -dense subset of L 2 (Y, ν) consisting of G-continuous functions. Its image under E Z will provide an L 2 -dense subset of L 2 (Z, ν). An application of the other direction of Theorem 1.7 finishes the proof.
2. If a G-system is disjoint from every spatial system then in particular it admits no nontrivial spatial factors and by Theorem 2.10 it is whirly.
Conversely, assume that (X, X, µ, G) is a whirly near-action and that (Y, Y, ν, G) is spatial. By [1] we can assume that Y is a compact space and that the action G × Y → Y is jointly continuous. Let λ be a joining of the two systems. That is, λ is a probability measure on X × Y , with projections µ and ν on X and Y respectively.
be the disintegration of λ over µ. Let Γ ⊂ G be a countable dense subgroup. Then, by the G-invariance of λ, for µ-a.e. x we have γλ x = γλ x for all γ ∈ Γ. If λ = µ × ν then the map x → λ x is not a constant µ-a.e. and there exists a continuous function is not a constant µ-a.e. Choose a value c ∈ R and a δ > 0 such that 0 < µ{x : f (x) > c} = a < 1, and 0 < µ{x : f (x) > c + δ} < a.
Let U be a neighborhood of e in G such that sup y∈Y |F (γy)−F (y)| < δ 10 for all γ ∈ U.
for µ-a.e. x. Thus for µ-a.e. x. Denote then we have for all γ ∈ U, γA ⊂ B, i.e. UA ⊂ B. This implies that A ⊂Ã ⊂ B, so thatÃ is a nontrivial stable set in X. By Theorem 2.9 this contradicts our assumption that (X, X, µ, G) is whirly. Thus λ = µ × ν and we have shown that (X, X, µ, G) and (Y, Y, µ, G) are disjoint.
Theorem 3.1 suggests an analogy -whirly is analogous to weak mixing while spatial is analogous to Kronecker (a system (X, X, µ, G) is a Kronecker system if the finite dimensional G-subrepresentations are dense in L 2 (µ)). The following corollary enhances this analogy. Proof. Since every Kronecker action embeds in an action of a compact group we deduce that every Kronecker system admits a spatial model. Since weak mixing is characterized as the property of having no nontrivial Kronecker factor, the corollary follows from Theorem 2.10

Non-archimedean group actions are spatial
Recall that a topological group is called non-archimedean if there is a basis for the topology at the identity consisting of open subgroups. In this section we will apply our criterion for an action to admit a spatial model to show that for Polish non-archimedean groups every near-action admits a spatial model.
Although we will not have an occasion to use it, we remind the reader of the following interesting characterization of Polish non-archimedean groups (see [  A theorem that will be used in our proof is the celebrated Ryll-Nardzewski theorem [14] (see also [3,Theorem III.5.2]). Recall that a topological dynamical system (Q, T ), where a group T acts continuously on a compact space Q, is called affine if Q is a convex subset of a topological linear space E and each t ∈ T acts as an affine transformation (that is, t(αx + (1 − α)y) = αtx + (1 − α)ty for every x, y ∈ Q and 0 ≤ α ≤ 1). The action is called distal with respect to a norm · on E if for every x = y in Q we have inf t∈T tx − ty > 0.
Theorem 4.2. Let E be a separable Banach space, Q a weakly compact convex subset of E. Let (Q, T ) be an affine dynamical system such that the action of T is distal in the norm topology, then T has a fixed point in Q. Proof. Let (X, X, µ, G) be a near-action of the Polish non-archimedean group G and let g → U g be the associated Koopman representation on L 2 (µ) given by U g f (x) = f (g −1 x), g ∈ G, f ∈ L 2 (X, µ). Given f ∈ L 2 (µ) and ε > 0 we will next show that the ball B ε (f ) = {k ∈ L 2 (µ) : k − f ≤ ε} contains a G-continuous L ∞ (µ) function.
By the continuity of the representation g → U g , there exists an open subgroup H ⊂ G such that U h f ∈ B ε (f ) for every h ∈ H. Being a norm closed bounded convex subset of L 2 (µ), B ε (f ) is weakly closed as well hence weakly compact. We let then clearly each f M is H-fixed and for sufficiently large M we have f M ∈ B ε (f ). We then fix such an M and claim that f M is G-continuous.
In fact, the group G being second countable, we see that the homogeneous space G/H is a countable space and that consequently the G-orbit of f M in L ∞ (µ) is a countable set. By Proposition 2.6 of [6] we deduce that f M is indeed a G-continuous function.
Note that what we have shown so far clearly implies that the G-continuous, essentially bounded functions are dense in L 2 (µ). Thus, in order to complete the proof of the theorem it only remains to apply Theorem 1.7.
Remark 4.4. One can use a weaker fixed point theorem here. In fact, since the action of H on Q is weakly almost periodic, it follows from the general theory of such systems that every minimal subsystem of Q is equicontinuous (see e.g. [5,Chapter 1]). Now an affine dynamical system always contains a strongly proximal minimal subsystem (see [3,Chapter III]). Finally, a minimal system which is both proximal and equicontinuous is necessarily a fixed point.

The generic automorphism of a Lebesgue space is whirly
Let (X, X, µ) be a standard Lebesgue space and denote by G = Aut (X) the Polish group of its automorphisms, i.e. the equivalence classes of invertible measure preserving transformations X → X, equipped with the topology of convergence in measure. For elements T of G we have the following well known dichotomy. Either the subgroup {T n : n ∈ Z} is a discrete subset of G or Λ(T ) = cls {T n : n ∈ Z} is a non-discrete monothetic Polish subgroup of G. In the latter case we say that T is a rigid transformation. Thus T is rigid iff there exists a sequence n k ր ∞ with lim k→∞ T n k = Id. A rigid T is called whirly if the Λ(T )-action on (X, X, µ) is whirly. This property of T can be characterized directly, with no reference to Λ(T ), as follows.
Definition 5.1. An automorphism T ∈ G is called whirly if for every neighborhood U of the identity in G and any two sets A, B ∈ X of positive measure there is n ∈ Z such that T n ∈ U and µ(T n A ∩ B) > 0.
The following theorem shows that whirly automorphisms are not at all rare.
Theorem 5.2. The collection of whirly transformations is a dense G δ subset of G.
Proof. For convenience we let X = [0, 1] and µ be normalized Lebesgue measure on X. Choose a sequence of pairs {(A k , B k )} ∞ k=1 which is dense in X × X, the two-fold product of the measure algebra. For each m ∈ N and 0 ≤ j < 2 m let J For k, m ∈ N we now define The sequence of positive constants δ m will be determined later. Note however that δ m is independent of k. Clearly each V k,m is an open subset of G and we set Claim 1: Each S ∈ R is whirly. Proof: Given A, B positive sets in X and m 0 ∈ N we have to find an n ∈ N with S n ∈ U m and µ(S n A ∩ B) > 0. For an ε > 0, to be determined soon, we find a pair By assumption S ∈ V k 0 ,m 0 hence there exists n such that S n ∈ U m 0 and We now observe that for sufficiently small ε this will imply By Baire's category theorem our proof will be complete once we show that each V k,m is dense in G. Proof: Since the set of weakly mixing automorphisms is dense in G it suffices to show that, given a weakly mixing T 0 ∈ G and ε > 0, there exists an element S ∈ V with µ{x ∈ X : T 0 x = Sx} < ε.
By weak mixing of T 0 there exists n 0 such that 1 n 0 << ε and µ(T n 0 0 A ∩ B) > B} are pairwise disjoint and the set E = X \ ∪ N −1 j=1 T j 0 B has measure < ε. We split B into two parts B γ and B 1−γ of measures γµ(B) and (1 − γ)µ(B) respectively. The size of 0 < γ < 1 will be determined later.
On each of the floors of the γ-part of the tower, except for the ceiling T N −1 0 B γ , we set S = T 0 . On the ceiling we set S = T −N +1 0 , so that S becomes periodic of period N on the γ-part of the tower.
On each of the n n 0 -block of floors of the (γ − 1)-part of the tower, we define S similarly to be periodic on that block with period n 0 , so that again S = T 0 on all floors of the tower except for the n ceilings of the n 0 -blocks. Finally, on the remainder E, S is defined as the identity. Clearly µ{x ∈ X : T 0 x = Sx} < ε and we will next show that S is in V .
We first observe that if γ ≤ 2 −2m then S n 0 ∈ U m . Next note that, by the choice of n, on the γ-part of the tower, say τ γ , we have We are finally in a position to determine the required size of the constants δ m . If we set so that S is indeed in V .

Further examples
As was shown in [6] a Lévy group admits no nontrivial spatial actions. On the other hand every action of a locally compact group is spatial. In this section we will show that: (I) There exists a Polish monothetic group which is not Lévy but admits a whirly action. (II) The abelian Polish group which underlies an infinite dimensional separable Hilbert space H admits both spatial and whirly faithful actions.
(I) In the following discussion we will use some basic facts from the theory of IP -sequences and IP -convergence. We refer to H. Furstenberg's book [2] for the necessary background. Definition 6.1. Let T ∈ Aut (X); we say that the action of T on X is whirly of all orders if for each n ≥ 1 the action of T × T × · · · × T (n-times) on X n is whirly. Lemma 6.2. If for T ∈ Aut (X) the Polish group Λ(T ) is Lévy then the action of T on X is whirly of all orders.
Proof. By Corollary 3.2 the system (X, T ) is weakly mixing and therefore for each n ≥ 1 the system (X n , T × T × · · · × T ) is ergodic. By Theorem 2.11 the latter system is also whirly. Lemma 6.3. Let T ∈ Aut (X) be such that the T -action on X is whirly of all orders. Let U be a neighborhood of the identity in Λ(T ). Given positive sets A, B ∈ X there exists an IP -sequence {p ν } such that for every ν, T pν ∈ U and µ(T pν A ∩ B) > 0.
Proof. We denote by N e the filter of neighborhoods of the identity in Λ(T ). Let U 1 = U and choose p 1 such that T p 1 ∈ U 1 and µ(T p 1 A ∩ B) > 0. Let D = T p 1 A ∩ B and find U 2 ∈ N e such that U 2 ⊂ U 1 and T p 1 U 2 ⊂ U 1 . Since the action of T is whirly of all orders we can choose p 2 such that T p 2 ∈ U 2 and µ(T p 2 D∩B) > 0, µ(T p 2 A∩B) > 0. We now have, for m = p 1 , p 2 and p 1 + p 2 , Assume p 1 , p 2 , . . . , p n have been found so that (6.1) is satisfied for all m = p i 1 + p i 2 + · · · + p i k with i 1 < i 2 < · · · < i k ≤ n. For each such m set D m = T m A ∩ B and choose U n+1 ∈ N e such that U n+1 ⊂ U n and T m U n+1 ⊂ U for every such m. Since the action of T is whirly of all orders we can choose p n+1 with T p n+1 ∈ U n+1 and with the property that It now follows that (6.1) is satisfied for all m = p i 1 + p i 2 + · · · + p i k with i 1 < i 2 < · · · < i k ≤ n + 1. This concludes the inductive step of the construction of the required IP -sequence.
Proposition 6.4. Let T ∈ Aut (X) be such that the T -action on X is whirly of all orders and let α be an irrational number. Denote by R α the rotation by α on T: where m is Lebesgue measure on T. Then 1. This dynamical system is rigid and we denote by L = Λ(T ×R α ) ⊂ Aut (X ×T) the corresponding perfect Polish monothetic topological group. 2. The action of L on X via the projection on the first coordinate is whirly. 3. L admits a continuous character and a nontrivial spatial action. In particular L is not Lévy. 4. Let K = {β ∈ T : (Id, β) ∈ L} then K is a closed subgroup of T and K = {0} iff the natural projection L → Λ(T ) is a topological isomorphism.
Proof. 1. Since T is a rigid transformation there exists a sequence n i ր ∞ such that lim T n i = Id in Λ(T ). It then follows that for some IP sequence n ν we have IPlim T nν = Id (see [2]). This fact, in turn, implies that for some sub-IP -sequence n ν ′ we have IP -lim n ν ′ α = 0 in T and therefore also IP -lim(T n ν ′ , n ν ′ α) = (Id, 0) in L. Thus T × R α is a rigid transformation and L is a perfect Polish monothetic group.
2. Note first that an arbitrary neighborhood of the identity in L contains a neighborhood of the form V = L ∩ (U × J ε ), where U is a neighborhood of the identity in Λ(T ) and J ε = {β ∈ T : |β| < ε} ( | · | denotes the distance to the closest integer). Thus in order to show that the action of L on X is whirly it suffices to show that for every pair of positive sets A, B ∈ X and every V as above there exists g = (S, β) ∈ V with µ(SA ∩ B) > 0. An application of Lemma 6.1 yields an IP -sequence {p ν } such that for every ν, T pν ∈ U and µ(T pν A ∩ B) > 0. Since T is a compact topological group if follows that for some sub-IP -sequence {p ν ′ }, IP -lim n ν ′ α = 0, hence eventually (T n ν ′ , n ν ′ α) ∈ V = L ∩ (U × J ε ). This completes the proof of part 2.
Via this character L acts spatially on T. The image of χ contains the dense subgroup {nα : n ∈ Z} and in particular χ is nontrivial. Since by Theorem 1.1 of [6] a Lévy group does not admit nontrivial spatial actions we conclude that L is not Lévy.
Remark 6.5. It follows from part 4 of Proposition 6.4 that if in addition to the conditions of that proposition T satisfies also the condition K = {0} then the action of Λ(T ) on X is whirly yet Λ(T ) is not a Lévy group. Such a T will provide a negative answer to Problem 2 below (see also [4,Theorem 3.3]).
(II) We first define a spatial action of H represented as H = ℓ 2 (N). As usual let T = R/Z be the 1-torus and let λ be normalized Lebesgue measure on T. For x ∈ ℓ 2 (N) with coordinates x = (x 1 , x 2 , . . . ) define a translation Taking Y = T N × T N and µ = λ N × λ N , the map x → A 1 (x) clearly defines a spatial measure preserving system (Y, Y, µ, H). Moreover this action is a topological system on a compact space and the action is equicontinuous. In fact the image of H under A 1 : H → Homeo (Y) is dense in the compact subgroup of Homeo (Y) (isomorphic to T N × T N ) of translations of Y . Of course the second component is there to make the action faithful.
We will next describe a faithful whirly near-action of H. This time we choose to view H as the Hilbert space L 2 ([0, 1], λ). Start from the Polish monothetic Lévy group G = L 0 [0, 1], S 1 of all (equivalence classes of) measurable functions [0, 1] → S 1 , where S 1 = {z ∈ C : |z| = 1} and the topology on G is given by L 2 -convergence (see [4]). The map θ : H → G, θ(f ) = e if defines a continuous homomorphism of H onto G. Consider G as a subgroup of the unitary group U(H) (the elements of G acting as multiplication operators) and let (X, X, µ, G) be the associated Gaussian near-action. Define for f ∈ L 2 (λ), This defines a faithful near-action (X × X, µ × µ, H). Let L = cls (A 2 (H)), where the closure is taken in the Polish group Aut (X × X). Of course L ⊂ G × G ⊂ Aut (X × X), and we claim that actually L = G × G. In fact, if for each n we denote by H n the finite dimensional subspace of H which consists of all the square integrable functions f : [0, 1] → R which are measurable with respect to the partition J n = [ j 2 n , j + 1 2 n ] : 0 ≤ j < 2 n , and let G n = θ(H n ), then it is easily checked that cls A 2 (H n ) = G n × G n . Since the union of the increasing sequence of compact groups G n × G n is dense in G × G we conclude that indeed L = G × G.
Next observe that the Polish groups G and G × G are clearly isomorphic (for example via the map which sends g ∈ G to the pair (g 1 , g 2 ) of its restrictions to [0, 1/2] and [1/2, 1] respectively). In particular, we deduce that G × G is a Lévy group. By Theorem 2.11 its action on X × X is whirly.