Actions of discrete amenable groups into the normalizers of full groups of ergodic transformations

Abstract

We apply the Evans–Kishimoto intertwining argument to the classification of actions of discrete amenable groups into the normalizer of a full group of an ergodic transformation. Our proof does not depend on the types of ergodic transformations.

1 Introduction

The purpose of this article is the study actions of discrete amenable groups into the normalizer of a full group of an ergodic transformation on the Lebesgue space. The study of such objects has been motivated by the theory of operator algebras, in particular, the classification of group actions on von Neumann algebras.

The study of automorphism groups of operator algebras is one of the central subjects in the theory of operator algebras, and the classification of automorphisms and group actions has been developed since Connes’ seminal works [5, 6]. In particular, classification of actions of discrete amenable groups on injective factors has been completed by many hands [13–16, 18, 20, 21]. These works heavily depend on the type of factor involved. However, we present a unified approach in [17] based on the Evans–Kishimoto method [9], and gave a proof that does not depend on the type of the factor in question.

There are corresponding results in ergodic theory. The first result is due to Connes and Krieger [7]. They developed the technique of applying ultraproducts to measure spaces and their transformations, and classified transformations (that is, actions of $\mathbb {Z}$ ) in the normalizer of a full group of type II. The result of Connes and Krieger has been generalized in [2] in the case of type II transformation and general discrete amenable groups, in [1] in the case of type III $_{\unicode{x3bb} }$ transformations ( $\unicode{x3bb} \ne 0$ ) and general discrete amenable groups, and finally in [3] in the case of type III $_0$ transformation and general discrete amenable groups. These results mentioned above depend on the type of the transformation, and it is natural to expect that our unified approach [17] is valid for the classification of actions of discrete amenable groups into the normalizers of full groups on Lebesgue spaces. In fact, the answer is affirmative and this is the main result of this article.

This classification result is very similar to that of the classification of actions of discrete amenable groups on injective factors. We will explain it in detail. Let $(X, \mathcal {B},\mu )$ be a Lebesgue space, T an ergodic transformation, and $N[T]$ the normalizer of a full group $[T]$ . Let $\alpha :G\rightarrow N[T]$ be a homomorphism, which we call an action of G into $N[T]$ . The invariant for $\alpha $ is a pair $(N_{\alpha },\operatorname {\mathrm {mod}}(\alpha ))$ , where $N_{\alpha }=\{g\in G\mid \alpha _g\in N[T]\}$ is a normal subgroup of G and $\operatorname {\mathrm {mod}}(\alpha )$ is the fundamental homomorphism [11]. (See Theorem 2.4 below for the precise statement of the classification theorem.) However, the invariant of a discrete group action $\alpha $ on a factor $\mathcal {M}$ is the triplet $(N_{\alpha }, \operatorname {\mathrm {mod}}(\alpha ), \chi (\alpha ))$ , where $N_{\alpha }=\{g\in G\mid \alpha _g\in \operatorname {\mathrm {Cnt}}_r(\mathcal {M})\}$ is a normal subgroup of G, $\operatorname {\mathrm {mod}}(\alpha )$ is the Connes–Takesaki module, and $\chi (\alpha )$ is the characteristic invariant. (See [15, 17] for details on this notation.) Thus, one can observe the similarity of both classification theorem.

It is also interesting to observe the difference between both classification theorems. Namely, the characteristic invariant $\chi (\alpha )$ does not appear in the ergodic theoretical setting. We consider the case of operator algebras first and explain how the characteristic invariant appears. Let $\alpha $ be an action of a discrete group G on a factor $\mathcal {M}$ , and assume $N_{\alpha }=\{g\in G\mid \alpha _g\in \operatorname {\mathrm {Int}}(\mathcal {M})\}$ for simplicity. By the definition of $N_{\alpha }$ , we can choose $u_n\in U(\mathcal {M})$ with $\alpha _n=\operatorname {\mathrm {Ad}} u_n$ , $n\in N_{\alpha }$ . However, there is no canonical choice of the unitary $u_n$ . Hence, we do not have $u_mu_n=u_{mn}$ and $\alpha _g(u_n)=u_{gng^{-1}}$ in general. A characteristic invariant $\chi (\alpha )$ appears as a cohomological obstruction of these relations.

Next, we consider the ergodic theoretical case. Let $\mathcal {R}_T$ be a Krieger factor associated with T. Then there are canonical homomorphisms $R\in N[T]\rightarrow \theta _R\in \operatorname {\mathrm {Aut}}(\mathcal {R}_T)$ , and $S\in [T]\rightarrow U_S\in U(\mathcal {R}_T)$ with $\operatorname {\mathrm {Ad}} U_S=\theta _S$ and $\theta _R(U_S)=U_{RSR^{-1}}$ , $S\in [T]$ , $R\in N[T]$ . If we lift an action $\alpha :G\rightarrow N[T]$ to that on $\mathcal {R}_T$ , then the invariant of the lifted action is given by $(N_{\alpha },\operatorname {\mathrm {mod}}(\alpha ),1)$ , due to the above relation between $\theta _R$ and $U_S$ . Therefore, the characteristic invariant is trivial in this case.

As we stated at the beginning of this section, our method for the proof of the classification theorem is the application of the Evans–Kishimoto type intertwining argument. To apply it, we need the characterization of full groups and their closures given in [7, 10]. In the study of group actions on operator algebras, two classes of automorphisms play important roles, that is, centrally trivial automorphisms and approximately inner automorphisms. In our case, full groups and their closures correspond to centrally trivial automorphism groups and approximately inner automorphism groups, respectively. Another important tool is the Rohlin type theorem. Combining these results, we first show the cohomology vanishing theorem by the Shapiro type argument in homology theory. Then we obtain the classification theorem by applying the Evans–Kishimoto type intertwining argument.

We expect that this work will shed new light on the relation between ergodic theory and the theory of operator algebras. For example, our result is used in [4] to classify regular subalgebras of type III injective factors.

This paper is organized as follows. In §2, we collect basic facts which will be used in this paper, and state the main results. In §3, we recall the ultraproduct construction of Connes and Krieger, and Ocneanu’s Rohlin type theorem. In §4, we show the second cohomology vanishing theorem. In §5, we apply the Evans–Kishimoto type intertwining argument [9] and classify actions of discrete amenable groups into the normalizer of a full group.

2 Preliminaries

2.1 Full groups of ergodic transformations and their normalizers

In this subsection, we collect known facts on full groups of ergodic transformations and their normalizers which will be used in this article.

Let $(X,\mathcal {B},\mu )$ be a non-atomic Lebesgue space with $\mu (X)=1$ . (Throughout this article, we treat only non-atomic Lebesgue spaces.) We denote by $\operatorname {\mathrm {Aut}}(X,\mu )$ the set of all non-singular transformations. Fix an ergodic transformation $T\in \operatorname {\mathrm {Aut}}(X,\mu )$ . Let $[T]_*$ be the set of all non-singular bijection $R:A\rightarrow B$ for some $A, B\in \mathcal {B}$ such that $Rx\in \{T^nx\}_{n\in \mathbb {Z}}$ , $x\in A$ . Define the full group of T by $[T]:=[T]_*\cap \operatorname {\mathrm {Aut}}(X,\mu )$ , that is,

$$ \begin{align*} [T]=\{R\in [T]_*\mid \mbox{the domain and the range of }R \mbox{ are both }X\}. \end{align*} $$

We say $E, F\in \mathcal {B}$ are T-equivalent if there exists $R\in [T]_*$ whose domain is E and range is F. A set $E\in \mathcal {B}$ is said to be T-infinite if there exists $F\subset E$ such that $\mu (E\backslash F)>0$ and F is T-equivalent to E. A set $E\in \mathcal {B}$ is said to be T-finite if it is not T-infinite.

When T is of type II, there exists a unique T-invariant measure m on X ( $m(X)<\infty $ when T is of type II $_1$ and $m(X)=\infty $ when T is of type II $_{\infty }$ ). In this case, the following two statements hold: (1) $E\in \mathcal {B}$ is T-finite if and only if $m(E)<\infty $ ; (2) $E,F\in \mathcal {B}$ are T-equivalent if and only if $m(E)=m(F)$ . When T is of type II $_1$ , we always assume $\mu $ is the unique T-invariant probability measure.

When T is of type III, then any $E\in \mathcal {B}$ with $\mu (E)>0$ is T-infinite, and if $E,F\in \mathcal {B}$ satisfy $\mu (E),\mu (F)>0$ , then E and F are T-equivalent. (For instance, see [12, Lemma 8].)

Let $N[T]\subset \operatorname {\mathrm {Aut}}(X,\mu )$ be the normalizer of $[T]$ . In the following, we use the notation $\hat {\alpha }(t)=\alpha t \alpha ^{-1}$ for $t\in [T]$ and $\alpha \in N[T]$ .

For $\alpha \in \operatorname {\mathrm {Aut}}(X,\mu )$ and $\xi \in L^1(X,\mu )$ , define ${\alpha }_{\mu }(\xi )\in L^1(X,\mu )$ by

$$ \begin{align*} {\alpha}_{\mu}(\xi)(x):=\xi(\alpha^{-1}x)\frac{d (\mu\circ \alpha^{-1})}{d \mu}(x), \quad \xi\in L^1(X,\mu). \end{align*} $$

Then ${\alpha }_{\mu }$ is an isometry of $L^1(X,\mu )$ , and $(\alpha \beta )_{\mu }={\alpha }_{\mu }{\beta }_{\mu }$ holds for $\alpha ,\beta \in \operatorname {\mathrm {Aut}}(X,\mu )$ .

Let $M(X,\mu )$ (respectively $M_1(X,\mu )$ ) be the set of complex-valued measures (respectively probability measures) which are absolutely continuous with respect to $\mu $ . For $\nu \in M(X,\mu )$ , let $\|\nu \|=|\nu |(X)$ , where $|\nu |$ is the total variation of $\nu $ . Then $M(X,\mu )$ is a Banach space with respect to the norm $\|\nu \|$ . For $\xi \in L^1(X,\mu )$ , let $\nu _{\xi }(f)=\int _{X}\xi (x)f(x)d\mu (x)$ . Note that $L^1(X,\mu )$ and $M(X,\mu )$ are isomorphic as Banach spaces by $\xi \mapsto \nu _{\xi }$ . Via this identification, ${\alpha }_{\mu }(\xi )$ corresponds to $\alpha (\nu _{\xi })=\nu _{\xi }\circ \alpha ^{-1}$ . In what follows, we freely use this identification, and we simply denote ${\alpha }_{\mu }(\xi )$ by $\alpha (\xi )$ for $\xi \in L^1(X,\mu )$ . Thus, $\xi (A)$ , $A\in \mathcal {B}$ , means $\nu _{\xi }(A)$ .

Recall the topology of $N[T]$ introduced in [12]. For $\alpha ,\beta \in \operatorname {\mathrm {Aut}}(X,\mu )$ , $\{\alpha \ne \beta \}$ denotes the set $\{x\in X\mid \alpha x\ne \beta x\}$ . We say a sequence $\{\alpha _n\}_n\subset N[T]$ converges to $\beta \in N[T]$ weakly if $\lim \nolimits _{n\rightarrow \infty }\|\alpha _{n}(\xi )-\beta (\xi )\|=0$ for all $\xi \in M(X,\mu )$ . Define a metric $d_{\mu }$ by $d_{\mu }(\alpha ,\beta ):=\mu (\{\alpha \ne \beta \})$ . We say $\{\alpha _n\}_n\subset N[T]$ converges to $\beta \in N[T]$ uniformly if $\lim \nolimits _{n\rightarrow \infty }d_{\mu }(\alpha _n,\beta )=0$ . This definition does not depend on the choice of equivalence classes of $\mu \in M_1(X,\mu )$ . It is shown in [11] that $[T]$ is a Polish group under $d_{\mu }$ .

Now we gift $N[T]$ with a topology as follows. We say a sequence $\{\alpha _n\}_n\subset N[T]$ converges to $\beta $ in $N[T]$ if $\{\alpha _n\}_n$ converges to $\beta $ weakly, and $\widehat {\alpha _n} (t)$ converges to $ \hat {\beta } (t) $ uniformly for all $t\in [T]$ . (In fact, we only have to require convergence for $t\in \{T^n\}_{n\in \mathbb {Z}}$ .) This is the right topology for $N[T]$ . In fact, this topology coincides with the u-topology for a Krieger factor $\mathcal {R}_{T}$ constructed from $(X,\mu , T)$ . So we also call this topology the u-topology. It is shown that $N[T]$ is a Polish group in the u-topology [12]. Indeed, let $\{\xi _k\}_{k=1}^{\infty }\subset L^1(X,\mu )$ be a countable dense subset, and define a metric d on $N[T]$ by

$$ \begin{align*} d(\alpha,\beta):= \sum_{k=1}^{\infty} \frac{1}{2^k} \frac{\|\alpha(\xi_k)-\beta(\xi_k)\|}{1+\|\alpha(\xi_k)-\beta(\xi_k)\|} +\sum_{k\in \mathbb{Z}}\frac{1}{2^{|k|}} \frac{d_{\mu}(\hat{\alpha}(T^k),\hat{\beta}(T^k))}{1+d_{\mu}(\hat{\alpha}(T^k),\hat{\beta}(T^k))}. \end{align*} $$

Then this d makes $N[T]$ a Polish group, and the topology defined by d is nothing but the u-topology on $N[T]$ .

We collect elementary results which will be frequently used in what follows. Since proof is easy, we leave it to the readers.

Lemma 2.1. The following statements hold.

1. (1) $d_{\mu }(\theta \alpha ,\theta \beta )= d_{\mu }(\alpha ,\beta )$ , $d_{\mu }(\alpha \theta ,\beta \theta )= d_{\theta (\mu )}(\alpha ,\beta )$ , $\alpha ,\beta ,\theta \in N[T]$ . In particular, we have $d_{\mu }(\alpha ,\mathrm {id})=d_{\mu }(\mathrm {id},\alpha ^{-1})=d_{\mu }(\alpha ^{-1},\mathrm {id})$ , and $d_{\mu }(\hat {\alpha }(t),\hat {\alpha }(t'))=d_{\alpha ^{-1}(\mu )}(t,t')$ , $\alpha \in N[T]$ , $t,t'\in [T]$ .

2. (2) $ d_{\nu _1}(\alpha ,\beta ) \leq \|\nu _1-\nu _2\|+d_{\nu _2}(\alpha ,\beta )$ , $\nu _1,\nu _2\in M_1(X,\mu )$ , $\alpha ,\beta \in N[T]$ .

3. (3) Let $\nu \in M_1(X,\mu )$ , $A,B,C,D\in \mathcal {B}$ . Then we have

   $$ \begin{align*} & \nu((A\cup B)\triangle (C\cup D))\leq \nu(A \triangle C)+ \nu(B\triangle D), \\ &\nu((A\cap B)\triangle (C\cap D))\leq \nu(A \triangle C)+ \nu(B\triangle D). \end{align*} $$

Recall the definition of the fundamental homomorphism [11]. Let $\tilde {X}:=X\times \mathbb {R}$ and $\mu _L$ be the Lebesgue measure on $\mathbb {R}$ . For $R\in \operatorname {\mathrm {Aut}}(X,\mu )$ and $t\in \mathbb {R}$ , define $\tilde {R},F_t\in \operatorname {\mathrm {Aut}}(\tilde {X},\mu \times \mu _L)$ by

$$ \begin{align*} \tilde{R}(x,u)=\bigg(Rx, u-\log\frac{d(\mu\circ R)}{d \mu}(x)\bigg),\quad F_t(x,u)=(x,u+t). \end{align*} $$

Let $(Y,\nu _Y)$ be the quotient space by $\tilde {T}$ . Namely, let $\zeta (\tilde {T})$ be a measurable partition of $\tilde {X}$ which generates the $\sigma $ -algebra consisting of all $\tilde {T}$ -invariant set. Then $Y=\tilde {X}/\zeta (\tilde {T})$ and $\nu _Y$ is a probability measure which is equivalent to $\mu \otimes \mu _L\circ \pi ^{-1}$ , where $\pi : \tilde {X}\rightarrow Y$ is a quotient map. The Lebesgue space $(Y,\nu _Y)$ can be also obtained by $L^{\infty }(Y,\nu _Y)=L^{\infty }(\tilde {X},\mu \times \mu _L)^{\tilde {T}}$ .

Since $\tilde {T}$ and $F_t$ commute, we get the ergodic flow $(Y,\nu _Y,F_t)$ , which is called the associated flow of $(X,T)$ . Let

$$ \begin{align*} \operatorname{\mathrm{Aut}}_F(Y,\nu_Y):=\{P\in \operatorname{\mathrm{Aut}}(Y,\nu_Y)\mid PF_t=F_tP, t\in \mathbb{R}\}. \end{align*} $$

When R is in $N[T]$ , $\tilde {R}$ induces $\operatorname {\mathrm {mod}}(R)\in \operatorname {\mathrm {Aut}}_F(Y,\nu )$ , which is called the fundamental homomorphism. If we lift R to an automorphism of a Krieger factor $\mathcal {R}_T$ , $\operatorname {\mathrm {mod}}(R)$ is nothing but a Connes–Takesaki module for R [8].

In this article, we do not use the above definition of $\operatorname {\mathrm {mod}}(R)$ explicitly, and what we need is the fact $\operatorname {\mathrm {Ker}}(\operatorname {\mathrm {mod}})=\overline {[T]}$ (closure is taken in the u-topology) and the surjectivity of $\operatorname {\mathrm {mod}}$ [10, 11].

2.2 Main results

Definition 2.2. Let G be a countable discrete group.

1. (1) A map (or 1-cochain) $v:G\rightarrow [T]$ is said to be normalized if $v(e)=\mathrm {id}$ . We denote the set of all normalized maps from G into $[T]$ by $C^1(G,[T])$ .

2. (2) A cocycle crossed action of G into $N[T]$ is a pair of maps $\alpha : G\rightarrow N[T]$ and $c:G\times G\rightarrow [T]$ such that $\alpha _g\alpha _h= c(g,h)\alpha _{gh}$ , $\alpha _e=\mathrm {id}$ , $c(e,h)=c(g,e)=\mathrm {id}$ . When $c(g,h)=\mathrm {id}$ for all $g,h\in G$ , we say $\alpha $ is an action of G into $N[T]$ .

3. (3) Let $(\alpha ,c)$ be a cocycle crossed action of G into $N[T]$ , and $v\in C^1(G,[T])$ . A perturbed crossed action $({}_v\alpha , {}_vc)$ of $(\alpha ,c)$ by v is defined by

   $$ \begin{align*} {}_v\alpha_g:=v(g)\alpha_g,\quad {}_vc(g,h)=v(g)\hat{\alpha_g}(v(h))c(g,h)v(gh)^{-1}. \end{align*} $$

4. (4) Let $\alpha $ be an action of G into $N[T]$ . We say a map $v\in C^1(G,[T])$ is a 1-cocycle for $\alpha $ if v satisfies the 1-cocycle identity $v(g)\widehat {\alpha _g}(v(h))=v(gh)$ . It is equivalent to that ${}_v\alpha $ is an action.

5. (5) Let $\alpha $ and $\beta $ be actions of G into $N[T]$ . We say they are cocycle conjugate if there exist $\theta \in N[T]$ and 1-cocycle $v(\cdot )$ such that ${}_v\alpha _g=\theta {\beta }_g\theta ^{-1}$ for all $g\in G$ . If $\theta $ is chosen in $\overline {[T]}$ , then we say they are strongly cocycle conjugate.

Remark

1. (1) Let $(\alpha ,c)$ be a cocycle crossed action of G. (Notion of a p-action is used in [3].) By $(\alpha _g\alpha _h)\alpha _k=\alpha _g(\alpha _h\alpha _k)$ , we can deduce the 2-cocycle identity $c(g,h)c(gh,k)=\widehat {\alpha _g}(c(h,k))c(g,hk)$ .

2. (2) In many works, cocycle conjugacy is said to be outer conjugacy. In fact, we must distinguish these two notions for group actions on operator algebras, However, in ergodic theory, we do not have to distinguish them. (We have the canonical homomorphism $u\in [T]$ into the normalizer of a Krieger factor arising from $(X,\mu , T)$ .)

At first, we show the following theorem.

Theorem 2.3. Let $(\alpha ,c)$ be a cocycle crossed action of a discrete amenable group into $N[T]$ with $\alpha _g\not \in [T]$ , $g\ne e$ . Then $c(g,h)$ is a coboundary, that is, there exists $v\in C^1(G,[T])$ such that ${}_vc(g,h)=\mathrm {id}$ , equivalently ${}_v\alpha $ is a genuine action of G. If $c(g,h)$ is close to $\mathrm {id}$ , then we can choose v so that it is also close to $\mathrm {id}$ .

See below for a more precise statement.

Let $N_{\alpha }:=\{g\in G\mid \alpha _g\in [T]\}$ , which is a normal subgroup of G. Our main result in this article is the following.

Theorem 2.4. Let $(X,\mu )$ be a Lebesgue space with $\mu (X)=1$ , T an ergodic transformation on $(X,\mu )$ . Let G be a countable discrete amenable group, and $\alpha $ , $\beta $ actions of G into $N[T]$ . Then $\alpha $ and $\beta $ are strongly cocycle conjugate if and only if $N_{\alpha }=N_{\beta }$ and $\operatorname {\mathrm {mod}}(\alpha )=\operatorname {\mathrm {mod}}(\beta )$ .

If $\alpha $ and $\beta $ are strongly cocycle conjugate, then it is obvious that $N_{\alpha }=N_{\beta }$ and $\operatorname {\mathrm {mod}}(\alpha _g)=\operatorname {\mathrm {mod}}({\beta }_g)$ . (Amenability of G is unnecessary for this implication.) Thus, the problem is to prove the converse implication, and a proof will be presented in subsequent sections. Here we only state the following corollary, which can be easily verified by Theorem 2.4.

Corollary 2.5. Let $\alpha $ and $\beta $ be actions of G into $N[T]$ . Then $\alpha $ and $\beta $ are cocycle conjugate if and only if $N_{\alpha }$ = $N_{\beta }$ and $\operatorname {\mathrm {mod}}(\alpha _g)=\theta \operatorname {\mathrm {mod}}({\beta }_g)\theta ^{-1}$ for some $\theta \in \operatorname {\mathrm {Aut}}_F(Y,\nu _Y)$ .

Proof. Since the ‘only if’ part is clear, we only have to prove the ‘if’ part. Suppose $N_{\alpha }$ = $N_{\beta }$ and $\operatorname {\mathrm {mod}}(\alpha _g)=\theta \operatorname {\mathrm {mod}}({\beta }_g)\theta ^{-1}$ for some $\theta \in \operatorname {\mathrm {Aut}}_F(Y,\nu _Y)$ . By the surjectivity of $\operatorname {\mathrm {mod}}$ [10], we can take $\sigma \in N[T]$ with $\operatorname {\mathrm {mod}}(\sigma )=\theta $ . Then $\operatorname {\mathrm {mod}}(\alpha _g)=\operatorname {\mathrm {mod}}(\sigma {\beta }_g\sigma ^{-1})$ holds, and hence $\alpha _g$ and $\sigma {\beta }_g\sigma ^{-1}$ are strongly cocycle conjugate by Theorem 2.4.

3 Ultraproduct of a Lebesgue space and Rohlin type theorem

We recall the ultraproduct spaces in [7].

Let $\omega \in \beta \mathbb {N}$ be a free ultrafilter on $\mathbb {N}$ . For sequences $(A_n)_n, (B_n)_n\subset \mathcal {B}$ , define an equivalence relation $(A_n)_n\sim (B_n)_n$ by $\lim \nolimits _{n\rightarrow \omega }\mu (A_n\triangle B_n)=0$ . Let . This definition depends only on the equivalence class of $\mu $ , and $\mathcal {B}^{\omega }$ is a boolean algebra.

Any $\alpha \in N[T]$ induces a transformation $\alpha ^{\omega }$ on $\mathcal {B}^{\omega }$ by $\alpha ^{\omega } ((A_n)_n):=(\alpha (A_n))_n$ . Let

$$ \begin{align*} \mathcal{B}_{\omega}:=\{\hat{A}\in \mathcal{B}^{\omega}: t^{\omega} \hat{A}=\hat{A}, t\in [T]\}. \end{align*} $$

We denote by $\alpha _{\omega }$ the restriction of $\alpha ^{\omega }$ on $\mathcal {B}_{\omega }$ .

Let $\hat {A}=(A_n)\in \mathcal {B}_{\omega }$ . Then $\lim \nolimits _{n\rightarrow \omega }\chi _{A_n}$ exists in weak- $*$ topology on $L^{\infty }(X,\nu )$ . By the ergodicity of T, this limit is in $\mathbb {C}$ , and does not depend on the choice of representative $\hat {A}=(A_n)$ . Thus, we can define $\tau : \mathcal {B}_{\omega }\rightarrow \mathbb {C}$ by $\tau (A):=\lim \nolimits _{n\rightarrow \omega }\chi _{A_n}$ . We can see $\tau \circ \alpha _{\omega } =\tau $ for $\alpha \in N[T]$ . By [7, Lemma 2.4], for $\alpha \in N[T]$ , $\alpha _{\omega }=\mathrm {id}$ if and only if $\alpha \in [T]$ . In fact, we have a stronger result. For $R\in N[T]$ , if there exists $\hat {A}\in \mathcal {B}_{\omega }$ such that $R_{\omega } \hat {B}=\hat {B}$ for any $\hat {B}\subset \hat {A}$ , $\hat {B}\in \mathcal {B}_{\omega }$ , then $R_{\omega }=\mathrm {id}$ , and hence $R\in [T]$ [7, Lemma 2.3]. This means that $R_{\omega }$ is a free transformation if $R_{\omega }\ne \mathrm {id}$ .

The main tool of this article is the following Rohlin type theorem, essentially due to Ocneanu [18]. (The following formulation is presented in [17].)

Theorem 3.1. Let $(\alpha ,c)$ be a cocycle crossed action of a discrete amenable group G into $N[T]$ such that $\alpha _{g,\omega }\ne \mathrm {id} $ for all $g\ne e$ . Let $K\Subset G$ , $\varepsilon>0$ , and S be a $(K,\varepsilon )$ -invariant set. (The notation $K\Subset G$ means that K is a finite subset of G.) Then there exists a partition of unity $\{\hat {E}_s\}_{s\in S}\subset \mathcal {B}_{\omega }$ such that:

1. (1) $\sum _{s\in S_g}\tau (\alpha _{g,\omega }\hat {E}_s\triangle \hat {E}_{gs})<5\varepsilon ^{\frac {1}{2}},\,\, g\in K;$

2. (2) $\sum _{s\in S\backslash S_{g^{-1}}}\tau (\hat {E}_s)<3\varepsilon ^{\frac {1}{2}},$

where $S_g:=S\cap g^{-1}S$ .

Note that we have $gs\in S_{g^{-1}}=S\cap gS$ for $s\in S_g$ .

The proof of [18] is based on the following two facts, that is, the freeness of actions on central sequence algebras, and the ultraproduct technique. In our case, freeness holds as we remarked before Theorem 3.1. Hence, the proof of [17, 18] can be applied in our case by a suitable modification.

Remark. The formulation of our Rohlin type theorem is different from that of the Rohlin theorem by Ornstein and Weiss [19]. The main reason is the use of the ultraproduct technique. Using the ultraproduct technique, Ocneanu showed a very strong result in [18, Lemma 6.3]. Namely, under the same assumption in Theorem 3.1, for any $ e\not \in A\Subset G$ and $\delta>0$ , he showed the existence of the partition of unity $\{E_i\}_{i=0}^N\subset \mathcal {B}_{\omega }$ such that $\tau (E_0)<\delta $ and $E_i\cap \alpha _{g,\omega }E_i=\emptyset $ for any $g\in A$ , and $i=1,2,\ldots, N$ . (Thus, $\{\alpha _{g,\omega }E_i\}_{g\in A}$ are disjoint for any $i=1,\ldots , N$ .) This lemma is an important step in constructing a Rohlin tower. (Note that $\tau \circ \alpha _{\omega }=\tau $ for $\alpha \in N[T]$ .) Combining with Zorn’s lemma, we can construct a single Rohlin tower as in Theorem 3.1.

In what follows, we say $\alpha $ is an ultrafree action of G if $\alpha _{g,\omega }\ne \mathrm {id}$ for any $g\in G$ , $g\ne e$ , to distinguish from the usual freeness of actions on Lebesgue spaces.

Lemma 3.2. Let $A,B$ be finite sets, $\{E_a\}_{a\in A}\subset \mathcal {B}_{\omega }$ a partition of X, and $\{P_{a,b}\}_{a \in A, b\in B}\subset [T]$ . Choose representatives $E_a=(E_a^n)_{n}$ such that $E_a^n \cap E_{a'}^n=\emptyset $ for $a\ne a'$ , $\bigsqcup _{a\in A}E_a^n=X$ . Then for any $\varepsilon>0$ , $\Phi \Subset M_1(X,\mu )$ , there exists $N\in \omega $ , $\{Z_a^n\}_{a\in A}\subset \mathcal {B}$ , $R^n_b \in [T]$ , $n\in N $ , $b\in B$ , such that:

1. (1) $\nu (P_{a,b}^{-1}E_a^n \triangle E_a^n)<\varepsilon $ , $n \in N$ , $\nu \in \Phi $ ;

2. (2) $Z_a^n \subset E_a^n$ , $P_{a,b}Z_a^n\subset E_a^n$ , $n \in N$ ;

3. (3) $\nu (E_a^n\backslash Z_a^n )<\varepsilon , \nu (E_a^n\backslash P_{a,b}Z_a^n )<\varepsilon , n \in N$ , $\nu \in \Phi $ ;

4. (4) $R_b^n x=P_{a,b}x, n\in N, x\in Z_a^n $ .

Proof. Since $P_{a,b}E_a=E_a$ by [7, Lemma 2.4], there exists $N\in \omega $ such that

$$ \begin{align*} P_{a,b}(\nu)\bigg(\bigg(E_a^n \cup \bigcup_{b\in B}P_{a,b}^{-1}E_a^n \bigg) \bigg\backslash \bigg(E_a^n \cap \bigcap_{b\in B}P_{a,b}^{-1}E_a^n \bigg)\bigg)< \frac{\varepsilon}{2} \end{align*} $$

for $n\in N$ , $a\in A$ , $b\in B$ , $\nu \in \Phi $ .

Let $Y_a^n:= E_a\cap \bigcap _{b\in B}P_{a,b}^{-1}E_a^n$ . Clearly we have $Y_a^n, P_{a,b}Y_a^n\subset E_a^n$ . Moreover,

$$ \begin{align*} \nu(P_{a,b}^{-1}E_a^n\triangle E_a^n)<\frac{\varepsilon}{2},\quad \nu(E_a^n\backslash Y_a^n)<\frac{\varepsilon}{2}, \quad \nu(E_a^n\backslash P_{a,b}Y_a^n) =P_{a,b}(\nu)(P_{a,b}^{-1}E_a^n\backslash Y_a^n) <\frac{\varepsilon}{2}\end{align*} $$

hold for $n \in N$ , $\nu \in \Phi $ . Let $Y^n:=\bigsqcup _{a\in A} Y_a^n$ . Thus, we can define $R^n_{0,b}\in [T]_*$ with $\mathrm {Dom}(R^n_{0,b})=Y^n$ by $R^n_{0,b}x=P_{a,b}x$ , $x\in Y_a^n$ . If $X\backslash Y^n$ and $X\backslash R_{0,b}^n Y^n$ are T-equivalent, then we can extend $R_{0,b}^n$ to an element $R_b^n\in [T]$ .

At first, let us assume that $Y^n$ is T-finite. (Thus, so is $R_{0,b}Y^n$ .) Such a case can happen if T is of type II. Then $X\backslash Y^n$ and $X\backslash R_{0,b}^n Y^n $ are T-equivalent. Hence, we can extend $R_{0,b}$ to $R_b\in [T]$ . Set $Z_a^n:=Y_a^n$ . Then all the statements in the lemma are satisfied.

Next, let us assume that $Y^n$ is T-infinite. (Hence, so is $R_{0,b}^nY^n$ .) Take $W_k\subset Y^n$ , $k\in \mathbb {N}$ , such that $W_k\subset W_{k+1}$ , $\bigcup _kW_k=Y^n$ , and $Y^n \backslash W_k$ are T-infinite for all k. Set $Z_{a,k}^n:=Y_a^n\cap ~W_k$ . Of course, we have $Z_{a,k}^n\subset Z_{a,k+1}^n$ , $\bigcup _{k}Z_{a,k}^n=Y_a^n$ , $\bigsqcup _{a\in A}Z_{a,k}^n=Y^n \cap W_k=W_k$ , and $Z_{a,k}^n, P_{a,b}Z_{a,k}^n \subset E_{a}^n$ . Thus, $\{Z_{a,k}^n \}_{a\in A}$ satisfies condition (2).

Take sufficiently large k such that

$$ \begin{align*} \nu(Y_a^n\backslash Z_{a,k}^n )<\frac{\varepsilon}{2}, \quad\nu(P_{a,b}Y_a^n \backslash P_{a,b}Z_{a,k}^n )<\frac{\varepsilon}{2} \end{align*} $$

for $a\in A$ , $b\in B$ , $\nu \in \Phi $ . Then it is clear that $\{Z_{a,k}^n \}$ satisfies condition (3). By the choice of $\{W_k\}$ , $X\backslash \bigsqcup _{a\in A}Z_{a,k}^n \supset Y^n \backslash W_k$ and $X\backslash R_{0,b}^n\bigsqcup _{a\in A}Z_{a,k}^n \supset R_{0,b}^n(Y^n \backslash W_k)$ . It follows that $X\backslash \bigsqcup _{a\in A}Z_{a,k}^n$ and $X\backslash R_{0,b}^n\bigsqcup _{a\in A}Z_{a,k}^n $ are both T-infinite and hence are equivalent. Thus, $Z_a^n:=Z_{a,k}^n$ satisfies all statements in the lemma.

Now we can combine Theorem 3.1 and Lemma 3.2 as follows.

Proposition 3.3. Let G be a discrete amenable group, and $(\alpha ,c)$ an ultrafree cocycle crossed action of G into $N[T]$ . Let $K \Subset G$ and $\varepsilon>0$ be given, and S a $(K,\varepsilon )$ -invariant set. Let $B,C$ be finite sets, $\{P_{s,b}\}_{s \in S, b\in B}\subset [T]$ , $\{\nu _s^c\}_{s\in S,c \in C}\Subset M_1(X,\mu )$ . Then for any $\delta>0$ , there exists a partition $\{E_s\}_{s\in S}\subset \mathcal {B}$ of X, $E_s\supset Z_s$ , and $R_{b}\in [T]$ , $b\in B$ , such that:

1. (1) $\sum _{s\in S_g}\nu _s^c (\alpha _gE_s\triangle E_{gs})<5\varepsilon ^{{1}/{2}}, g\in K, c\in C; $

2. (2) $ \sum _{s\in S\backslash S_{g^{-1}}}\nu _s^c(E_s)<3\varepsilon ^{{1}/{2}}, g\in K, c\in C; $

3. (3) $ \nu _s^c(P_{s,b}^{-1}E_s \triangle E_s)<\delta , s \in S, b\in B, c\in C;$

4. (4) $ P_{s,b}Z_s\subset E_s, s\in S, b\in B; $

5. (5) $\nu _s^c(E_s\backslash Z_s)<\delta , \nu _s^c(E_s\backslash P_{s,b}Z_s)<\delta , s\in S, b\in B, c\in C;$

6. (6) $ R_b x=P_{s,b}x, s\in S, b\in B, x\in Z_s,$

where $S_g:=S\cap g^{-1}S$ .

Proof. Let $\{\hat {E}_s\}_{s\in S}\subset \mathcal {B}_{\omega }$ be a Rohlin partition as in Theorem 3.1. Since $\tau (\hat {A})=\lim \nolimits _{n\rightarrow \omega }\chi _{A_n}$ for $\hat {A}=(A_n)_n\in \mathcal {B}_{\omega }$ , $\tau (\hat {A})=\lim \nolimits _{n\rightarrow \omega }\nu (A^n)$ for any $\nu \in M_1(X,\mu )$ . Choose a representative $\hat {E}_s=(E^n_s)_n$ such that $E_s^n\cap E_{s'}^n=\emptyset $ , $\bigsqcup _{s\in S}E_s^n=X$ . By Theorem 3.1:

1. (1) $\lim \nolimits _{n\rightarrow \omega }\sum _{s\in S_g}\nu _s^c(\alpha _gE_s^n\triangle E_{gs}^n)<5\varepsilon ^{{1}/{2}}, g\in K; $

2. (2) $ \lim _{n\rightarrow \omega }\sum _{s\in S\backslash S_{g^{-1}}}\nu _s^c(E_s^n)<3\varepsilon ^{{1}/{2}}, g\in K$

hold for any $\{\nu _s^c\}_{s\in S,c\in C}\subset M_1(X,\mu )$ . Thus, there exists $N_1\in \omega $ such that

$$ \begin{align*} & \sum_{s\in S_g}\nu_s^c(\alpha_gE_s^n\triangle E_{gs}^n)<5\varepsilon^{{1}/{2}}, \quad g\in K, c\in C, \\ & \sum_{s\in S\backslash S_{g^{-1}}}\nu_s^c(E_s^n)<3\varepsilon^{{1}/{2}}, \quad g\in K, c\in C \end{align*} $$

for all $n\in N_1$ . By Lemma 3.2, there exists $N_2\in \omega $ , $Z_s^n\subset E_s^n$ , and $R_b^n\in [T]$ , ( $n\in N_2$ ), such that

$$ \begin{align*} & \nu_s^c(P_{s,b}^{-1}E_s^n \triangle E_s^n)<\delta,\quad s\in S, b\in B, \\ & P_{s,b}Z_s^n\subset E_s^n,\quad s\in S, b\in B, \\ &\nu_s^c(E_s^n\backslash Z_s^n)<\delta,\quad \nu_s^c(E_s^n\backslash P_{s,b}Z_s^n)<\delta, \quad s\in S, b\in B, c\in C, \\ & R_b^n x=P_{s,b}x, \quad s\in S, b\in B, x\in Z_s^n \end{align*} $$

for any $n\in N_2$ . Fix $n\in N_1\cap N_2$ , and set $E_s:=E_s^n$ , $Z_s:=Z_s^n$ , $R_b:=R_b^n$ . Then these $E_s, Z_s, R_b$ are desired objects.

4 Cohomology vanishing

At first, we show the following second cohomology vanishing result, which is shown in [3, Theorem 1.3]. We present the proof for readers’ convenience.

Theorem 4.1. Let T be a transformation of type II $_{\infty }$ or type III, and $(\gamma ,c)$ a cocycle crossed action of a discrete group G into $N[T]$ . Then $c(g,h)$ is a coboundary, that is, there exists $u\in C^1(G,[T])$ such that ${}_uc(g,h)=\mathrm {id}$ .

Proof. Since T is of type II $_{\infty }$ or type III, there exists a partition $\{E_h\}_{h\in G}$ of X such that each $E_h$ is T-infinite. Let $\{f_{g,h}\}_{g,h\in G}\subset [T]$ be an array for $\{E_g\}_{g\in G}$ , that is, $\{E_g\}_{g\in G}$ is a partition of X and $f_{g,h}\in [T]_*$ is a bijection from $E_h$ onto $E_g$ such that $f_{g,h}f_{h,k}=f_{g,k}$ . Take $v_g^0 \in [T]_*$ with $\mathrm {Dom}(v_g^0)=\gamma _gE_e$ and $\mathrm {Ran}(v_g^0)=E_e$ . Define $v(g)\in [T]$ by $ f_{h,e}v_g^0\gamma _g(f_{e,h})$ on $\gamma _gE_h$ . Then we have ${}_{v}\gamma _g:E_h\rightarrow E_h$ and $\widehat {{}_v\gamma _g}(f_{h,k})=f_{h,k}$ for any $g,h,k\in G$ . Replacing $(\gamma ,c)$ with $({}_v\gamma ,{}_vc)$ , we may assume $\gamma _gE_k=E_k$ and $\widehat {\gamma _g}(f_{h,k})=f_{h,k}$ . Since $\gamma _g\gamma _h=c(g,h)\gamma _{gh}$ , we also have $c(g,h)E_k=E_k$ and $\widehat {c(g,h)}(f_{k,l})=f_{k,l}$ .

Next define $u(g)\in [T]$ by $u(g)=c(g,l)^{-1}f_{gl, l}$ on $E_{l}$ . Note $u(g)$ sends $E_{l}$ to $E_{gl}$ , and hence so does ${}_u\gamma _g$ . Hence, for $x\in E_l$ ,

$$ \begin{align*} {}_u\gamma_g {}_u\gamma_h x &=u(g)\gamma_g c(h,l)^{-1} f_{hl,l}\gamma_h x =u(g)\gamma_g c(h,l)^{-1} \gamma_{h} f_{hl,l} x \\ &=u(g)\widehat{\gamma_g}(c(h,l))^{-1} \gamma_g\gamma_{h} f_{hl,l} x =u(g) \widehat{\gamma_g}(c(h,l))^{-1} c(g,h)\gamma_{gh} f_{hl,l} x \\ &=c(g,hl)^{-1}f_{ghl,hl} \widehat{\gamma_g}(c(h,l))^{-1} c(g,h)\gamma_{gh} f_{hl,l} x \\ &=c(g,hl)^{-1} \widehat{\gamma_g}(c(h,l))^{-1} c(g,h)\gamma_{gh}f_{ghl,hl} f_{hl,l} x \\ &=c(gh,l)^{-1}f_{ghl,l}\gamma_{gh}x =u(gh)\gamma_{gh} x. \end{align*} $$

This implies that ${}_u\gamma $ is an action, and ${}_uc(g,h)=u(g)\widehat {\gamma _g}(u(h))c(g,h)u(gh)^{-1}=\mathrm {id}$ holds.

In Theorem 4.1, we have no estimation on the choice of $u(g)$ , even if $c(g,h)$ is close to $\mathrm {id}$ . The rest of this section is devoted to solving this problem. From now on, we always assume that G is a discrete amenable group.

For all $g\in G$ and $S\Subset G$ , fix a bijection $l(g): S\rightarrow S$ such that $l(g)s=gs$ if $gs\in S$ .

Lemma 4.2. Let $(\gamma ,c)$ be an ultrafree cocycle crossed action of G. For any $\varepsilon>0$ , $K\Subset G$ , $\mu \in \Phi \Subset M_1(X,\mu )$ , there exists $w\in C^1(G,[T])$ such that

$$ \begin{align*} d_{\nu}({}_wc(g,h),\mathrm{id})<\varepsilon, \quad g,h \in K, \nu \in \Phi. \end{align*} $$

Moreover, for given $\varepsilon>0$ , $e\in K\Subset G$ , there exist $\delta>0$ and $S\Subset G$ , which depend only on K and $\varepsilon>0$ , such that if

$$ \begin{align*} \|c(g,h)(\xi)-\xi\|<\delta, \quad d_{\nu}(\widehat{c(g,h)}(t),t)<\delta, \quad g,h\in S, t\in \Lambda, \xi,\nu\in \Phi \end{align*} $$

for some cocycle crossed action $(\gamma ,c)$ , $\Lambda \Subset [T]$ and $\Phi \Subset M_1(X,\mu )$ , then we can choose $w\in C^1(G,[T])$ so that it further satisfies

$$ \begin{align*} \|w(g)(\xi)-\xi\|<\varepsilon, \quad d_{\nu}(\widehat{w(g)}(t),t)<\varepsilon, \quad g\in K, \xi,\nu\in \Phi, t\in \Lambda. \end{align*} $$

Proof. Choose $\varepsilon '>0$ with $11\sqrt {\varepsilon '}<\varepsilon $ , and let $S'\subset G $ be a $(K\cup K^2,\varepsilon ')$ -invariant set and $S=S'\cup K$ . Choose $\delta $ such that $5\delta |S|+11\sqrt {\varepsilon '}<\varepsilon $ .

By applying Proposition 3.3, we can take a Rohlin partition $\{E_s\}_{s\in S'}\subset \mathcal {B}$ , $Z_s\subset E_s$ , $w(g)\in [T]$ , $g\in K$ , such that:

1. (1) $ E_{l(g)s}\supset c(g,s)^{-1}Z_{l(g)s}, g\in K\cup K^2, s\in S'; $

2. (2) $\nu (E_s\backslash Z_s)<\delta , \nu (E_{l(g)s}\backslash c(g,s)^{-1}Z_{l(g)s})<\delta , g\in K\cup K^2, s\in S', \nu \in \Phi; $

3. (3) $ \nu (c(gh,k)^{-1}c(g,h)^{-1} \widehat {\gamma _g}(c(h,k))(E_{ghk}\backslash Z_{ghk})) <\delta , g,h\in K, k\in S^{\prime }{gh}\cap S^{\prime }h, \nu \in ~\Phi; $

4. (4) $ \nu (c(gh,k)^{-1}c(g,h)^{-1}\gamma _g(E_{hk}\backslash Z_{hk})) <\delta , g,h\in K, k\in S^{\prime }h, \nu \in \Phi ;$

5. (5) $ \nu (E_{ghk}\triangle c(gh,k)^{-1}c(g,h)^{-1} \widehat {\gamma _g}(c(h,k))E_{ghk}) <\delta , g,h\in K, k\in S^{\prime }h, \nu \in \Phi ;$

6. (6) $ \nu (E_{ghk}\triangle c(gh,k)^{-1}c(g,h)^{-1}E_{ghk})<\delta , g,h\in K, k\in S^{\prime }{gh}\cap S^{\prime }h, \nu \in \Phi; $

7. (7) $\sum _{k\in S^{\prime }{gh}\cap S^{\prime }h} \nu (c(gh,k)^{-1}c(g,h)^{-1}(E_{ghk}\triangle \gamma _gE_{hk})) <5\sqrt {\varepsilon '}, g,h\in K, \nu \in \Phi ;$

8. (8) $ \sum _{k\in S'\backslash S^{\prime }{(gh)^{-1}}}\nu ( E_s)<3\sqrt {\varepsilon '} g\in K\cup K^2, \nu \in \Phi ;$

9. (9) $ w(g)x=c(g,s)^{-1}x, x\in Z_{l(g)s}, g\in K, s\in S'.$

Here we applied Proposition 3.3 for

$$ \begin{align*} B&=\{c(g,s)^{-1}\mid g\in K, s\in S'\} \cup \{c(gh,k)^{-1}c(g,h)^{-1}\mid g,h\in K, k\in S'\} \\ &\quad\ \ \cup \{c(gh,k)^{-1}c(g,h)^{-1}\widehat{\gamma_g}(c(h,k))\mid g,h\in K, k\in S'\} \end{align*} $$

and

$$ \begin{align*} C&= \Phi \cup \{ \nu(c(gh,k)^{-1}c(g,h)^{-1} \widehat{\gamma_g}(c(h,k)) \cdot )\mid \nu\in \Phi, g,h\in K, k\in S' \} \\ &\quad\hspace{2pt}\cup \{ \nu(c(gh,k)^{-1}c(g,h)^{-1} \gamma_g \cdot )\mid \nu\in \Phi, g,h\in K, k\in S' \} \\ &\quad\hspace{2pt}\cup \{ \nu(c(gh,k)^{-1}c(g,h)^{-1} \cdot )\mid \nu\in \Phi, g,h\in K, k\in S' \}. \end{align*} $$

We define $w(g)=\mathrm {id}$ if $g\not \in G$ .

Let

$$ \begin{align*} W^0_{g,h,k} &= c(gh,k)^{-1}Z_{ghk}\cap c(gh,k)^{-1}c(g,h)^{-1}\gamma_gZ_{hk}\\ &\quad\hspace{2pt} \cap c(gh,k)^{-1}c(g,h)^{-1} \widehat{\gamma_g}(c(h,k))Z_{ghk} \end{align*} $$

for $k\in S^{\prime }_{g,h}\cap S^{\prime }_h $ and

$$ \begin{align*} W_{g,h}= \bigcup_{k\in S^{\prime}_{gh}\cap S^{\prime}_{h}} W^0_{g,h,k}. \end{align*} $$

We can verify $w(g)\hat {\gamma _g}(w(h))c(g,h)w(gh)^{-1}{\kern-1pt}={\kern-1pt}\mathrm {id}$ on $W_{g,h}$ as follows. Take $x{\kern-1pt}\in{\kern-1pt} W^0_{g,h,k}$ . Since $x\in c(gh,k)^{-1}Z_{ghk}$ , we have $w(gh)^{-1}x=c(gh,k)x$ . Thus, we have

$$ \begin{align*} \gamma _g^{-1}c(g,h) w(gh)^{-1}x=\gamma _g^{-1}c(g,h)c(gh,k)x.\end{align*} $$

Since $x\in c(gh,k)^{-1}c(g,h)^{-1}\gamma _gZ_{hk}$ , $\gamma _g^{-1} c(g,h)c(gh,k)x\in Z_{hk}$ holds. Hence, we have

$$ \begin{align*} w(h)\gamma_g^{-1}c(g,h)c(gh,k)x= c(h,k)^{-1}\gamma_g^{-1}c(g,h)c(gh,k)x. \end{align*} $$

Since $x\in c(gh,k)^{-1}c(g,h)^{-1} \widehat {\gamma _g}(c(h,k))Z_{ghk}$ ,

$$ \begin{align*} \widehat{ \gamma_g}(w(h))c(g,h)w(gh)^{-1}x=\widehat{\gamma_g}(c(h,k))^{-1}c(g,h)c(gh,k)x\in Z_{ghk} \end{align*} $$

holds, and hence we have

$$ \begin{align*} w(g)\gamma_gc(h,k)^{-1}\gamma_g^{-1}c(g,h)c(gh,k)x= c(g,hk)^{-1}\gamma_gc(h,k)^{-1}\gamma_g^{-1}c(g,h)c(gh,k)x=x \end{align*} $$

by the 2-cocycle identity. These computations show

$$ \begin{align*} {}_wc(g,h)=w(g)\widehat {\gamma _g}(w(h))c(g,h) w(gh)^{-1}=\mathrm {id} \end{align*} $$

on $W_{g,h}$ . Thus, we have $\{{}_wc(g,h)\ne \mathrm {id}\}\subset X\backslash W_{g,h}$ .

We will show $\nu (X\backslash W_{g,h})<\varepsilon $ for $\nu \in \Phi $ . By condition (2), we have

$$ \begin{align*} \nu(E_{ghk}\backslash c(gh,k)^{-1}Z_{ghk})<\delta,\quad \nu\in \Phi, g,h\in K k\in S^{\prime}_{gh}. \end{align*} $$

For $g,h\in K$ , $k\in S_{gh}'\cap S_h'$ , $\nu \in \Phi $ , we have

$$ \begin{align*} &{ \nu(E_{ghk}\triangle c(gh,k)^{-1}c(g,h)^{-1}\gamma_gZ_{hk})} \\ &\quad\leq \nu(E_{ghk}\triangle c(gh,k)^{-1}c(g,h)^{-1}E_{ghk})+ \nu(c(gh,k)^{-1}c(g,h)^{-1}(E_{ghk}\triangle \gamma_gZ_{hk})) \\ &\quad\leq \delta+ \nu(c(gh,k)^{-1}c(g,h)^{-1}(E_{ghk}\triangle \gamma_gZ_{hk})) \quad (\mbox{by condition }(6)) \\ &\quad\leq \delta+ \nu(c(gh,k)^{-1}c(g,h)^{-1}(E_{ghk}\triangle \gamma_gE_{hk}))\\&\qquad+ \nu(c(gh,k)^{-1}c(g,h)^{-1}\gamma_g(E_{hk}\triangle Z_{hk})) \\ &\quad\leq 2\delta+ \nu(c(gh,k)^{-1}c(g,h)^{-1}(E_{ghk}\triangle \gamma_gE_{hk})) \quad (\mbox{by condition }(4)) \end{align*} $$

and

$$ \begin{align*} &{\nu (E_{ghk}\triangle c(gh,k)^{-1}c(g,h)^{-1} \widehat{\gamma_g} (c(h,k))Z_{ghk})} \\ &\quad\leq \nu (E_{ghk}\triangle c(gh,k)^{-1}c(g,h)^{-1} \widehat{\gamma_g}(c(h,k))E_{ghk}) \\ &\qquad+ \nu (c(gh,k)^{-1}c(g,h)^{-1} \widehat{\gamma_g}(c(h,k))(E_{ghk}\triangle Z_{ghk})) \\ &\quad<2\delta \quad (\mbox{by conditions (5) and (3)}).\\[-22pt] \end{align*} $$

Thus,

$$ \begin{align*} \nu(E_{ghk}\triangle W_{g,h,k}^0)&\leq \nu(E_{ghk}\backslash c(gh,k)^{-1}Z_{ghk}) +\nu(E_{ghk}\triangle c(gh,k)^{-1}c(g,h)^{-1}\gamma_gZ_{hk})\\ &\quad +\nu (E_{ghk}\triangle c(gh,k)^{-1}c(g,h)^{-1} \gamma_gc(h,k)\gamma_g^{-1}Z_{ghk}) \\ &<5\delta+ \nu(c(gh,k)^{-1}c(g,h)^{-1}(E_{ghk}\triangle \gamma_gE_{hk})) \end{align*} $$

follows. Then

$$ \begin{align*} \nu\bigg(\! \bigg(\bigcup_{k\in S^{\prime}_{gh}\cap S^{\prime}_h}E_{ghk}\!\bigg)\triangle W_{g,h}\!\bigg) &\leq \sum_{k\in S^{\prime}_{gh}\cap S^{\prime}_h} \nu( E_{ghk} \triangle W_{g,h,k}^0) \\&< 5\delta|S|+ \sum_{k\in S^{\prime}_{gh}\cap S^{\prime}_h}\! \nu(c(gh,k)^{-1}c(g,h)^{-1}(E_{ghk}\triangle \gamma_gE_{hk})) \\&<5\delta|S|+5\sqrt{\varepsilon'} \quad (\mbox{by condition (7)}) \end{align*} $$

holds.

Finally, we have

$$ \begin{align*} \nu(X\backslash W_{g,h})&\leq \nu\bigg(X\backslash \bigcup_{k\in S^{\prime}_{gh}\cap S^{\prime}_h}E_{ghk}\bigg) +\nu\bigg( \bigg(\bigcup_{k\in S^{\prime}_{gh}\cap S^{\prime}_h}E_{ghk}\bigg)\triangle W_{g,h}\bigg) \\ &=\nu\bigg(\bigcup_{k\in S\backslash (S^{\prime}_{g^{-1}}\cap S^{\prime}_{(gh)^{-1}})}E_{k} \bigg) +\nu\bigg( \bigg(\bigcup_{k\in S^{\prime}_{gh}\cap S^{\prime}_h}E_{ghk}\bigg)\triangle W_{g,h}\bigg) \\ &<\sum_{k\in S'\backslash S^{\prime}_{g^{-1}}}\nu(E_k)+\sum_{k\in S'\backslash S^{\prime}_{(gh)^{-1}}}\nu( E_s)+ 5\delta|S|+5\sqrt{\varepsilon'} \\ &<5\delta|S|+11\sqrt{\varepsilon'} \quad (\mbox{by condition (8)}) \\ &<\varepsilon. \end{align*} $$

(Note $ghk\in S^{\prime }_{g^{-1}}\cap S^{\prime }_{(gh)^{-1}}$ for $k\in S_h'\cap S_{gh}'$ .) This inequality implies

$$ \begin{align*} d_{\nu }({}_wc(g,h), \mathrm {id})<\varepsilon\quad \mathrm{for}\ g,h\in K, \nu \in \Phi. \end{align*} $$

Assume

$$ \begin{align*} \|c(g,h)(\xi)-\xi\|<\delta,\quad d_{\nu}(\widehat{c(g,h)}(t),(t))<\delta,\quad g,h\in S, t\in \Lambda, \xi,\nu\in \Phi. \end{align*} $$

We show

$$ \begin{align*} \|w(g)(\xi)-\xi\|<\varepsilon,\quad d_{\nu}(\widehat{w(g)}(t),t)<\varepsilon,\quad g\in K, \xi,\nu\in \Phi, t\in \Lambda. \end{align*} $$

Let $Z:=\bigsqcup _{s\in S'}Z_s$ . By the definition of $w(g)$ , $w(g)Z=\bigsqcup _{s\in S'}c(g,s)^{-1}Z_s$ holds. Then we have

$$ \begin{align*} &{ \|w(g)(\xi)-\xi\|} \\ &\quad=\int_{X}|w(g)(\xi)(x)-\xi(x)|\,d\mu(x) \\ &\quad=\int_{w(g) Z}|w(g)(\xi)(x)-\xi(x)|\,d\mu(x) +\int_{X\backslash w(g)Z}|w(g)(\xi)(x)-\xi(x)|\,d\mu(x) \\ &\quad= \sum_{s\in S'}\int_{c(g,s)^{-1}Z_{l(g)s}}|w(g)^{-1}(\xi)(x)-\xi(x)|\,d\mu(x)\\ &\qquad+\int_{X\backslash w(g)Z}|w(g)(\xi)(x)-\xi(x)|\,d\mu(x). \end{align*} $$

Note

$$ \begin{align*} w(g)(\xi)(x)&=\xi(w(g)^{-1}x)\frac{d(\mu \circ w(g)^{-1})}{d\mu}(x)\\ &=\xi(c(g,s)x)\frac{d(\mu\circ c(g,s))}{d\mu}(x)=c(g,s)^{-1}(\xi)(x) \end{align*} $$

for $x\in c(g,s)^{-1}Z_{l(g)s}$ , when we regard $\xi $ as an element of $L^1(X,\mu )$ . Thus, the first term is estimated as follows:

$$ \begin{align*} &{\sum_{s\in S'}\int_{c(g,s)^{-1}Z_{l(g)s}}|w(g)^{-1}(\xi)(x)-\xi(x)|\,d\mu(x)} \\ &\quad= \sum_{s\in S'}\int_{c(g,s)^{-1}Z_{l(g)s}}|c(g,s)^{-1}(\xi)(x)-\xi(x)|\,d\mu(x) \\ &\quad\leq \sum_{s\in S'}\|c(g,s)^{-1}(\xi)-\xi\| <\delta|S|. \end{align*} $$

To estimate the second term, one should note

$$ \begin{align*} \xi(X\backslash Z)=\sum_{s\in S'}\xi(E_s\backslash Z_s)<\delta|S'|,\quad \xi(X\backslash w(g)Z)=\sum_{s\in S'}\xi(E_s\backslash c(g,s)^{-1}Z_s)<\delta|S'| \end{align*} $$

by condition (2). Hence,

$$ \begin{align*} &{\int_{X\backslash w(g)Z} |w(g)(\xi)(x)-\xi(x)|\,d\mu(x) } \\ &\quad\leq \int_{X\backslash w(g)Z}w(g)(\xi)(x)\, d\mu(x)+ \int_{X\backslash w(g)Z}\xi(x)\ d\mu(x) \\ &\quad=\int_{X\backslash w(g)Z}\xi(w(g)^{-1}x)\frac{d(\mu\circ w(g)^{-1})}{d\mu} d\mu(x)+\xi(X\backslash w(g)Z) \\ &\quad= \int_{X\backslash Z}\xi(x)\,d\mu(x)+\xi(X\backslash w(g)Z) = \xi(X\backslash Z)+\xi(X\backslash w(g)Z) <2|S'|\delta, \end{align*} $$

and we obtain $\|w(g)(\xi )-\xi \|<3\delta |S'|<\varepsilon $ .

We next show

$$ \begin{align*} d_{\nu}(\widehat{w(g)}(t),t)<\varepsilon,\quad g\in G, \nu\in \Phi, t\in \Lambda. \end{align*} $$

By the assumption

$$ \begin{align*} \|c(g,s)(\nu)-\nu\|<\delta,\quad d_{\nu}(\widehat{c(g,s)}(t),t)<\delta,\quad t\in \Lambda, g,s\in S, \nu \in \Phi, \end{align*} $$

$$ \begin{align*} d_{\nu}(\widehat{c(g,s)^{-1}}(t), t)&=d_{c(g,s)(\nu)}(t, \widehat{c(g,s)}(t)) \\ &\leq \|c(g,s)(\nu)-\nu\|+d_{\nu}(\widehat{c(g,s)}(t),t) < 2\delta \end{align*} $$

holds. We can further assume

$$ \begin{align*} \nu(E_{l(g)s}\triangle c(g,s)^{-1}t^{-1}E_{l(g)s})<\delta, \quad \nu(c(g,s)^{-1}t^{-1}(E_{l(g)s}\triangle Z_{l(g)s})) < \delta\end{align*} $$

for $t\in \Lambda $ , $s\in S'$ , $g\in K$ in the choice of $Z_s$ and $E_s$ .

Let $B_{g,s,t}:=\{\widehat {c(g,s)^{-1}}(t)=t\}$ . Then we have $\nu (X\backslash B_{g,s,t})<2\delta $ , $\nu \in \Phi $ . We can see $\widehat {w(g)}(t) =\widehat {c(g,s)^{-1}}(t)$ on $c(g,s)^{-1}Z_{l(g)s}\cap c(g,s)t^{-1}Z_{l(g)s}$ as above. Thus, $\widehat {w(g)}(t)=t$ holds on

$$ \begin{align*} \bigcup_{s\in S'}c(g,s)^{-1}Z_{l(g)s}\cap c(g,s)t^{-1}Z_{l(g)s}\cap B_{g,s,t}. \end{align*} $$

We will show

$$ \begin{align*} \nu\bigg(X\backslash \bigcup_{s\in S'}c(g,s)^{-1}Z_{l(g)s}\cap c(g,s)t^{-1}Z_{l(g)s}\cap B_{g,s,t} \bigg)<\varepsilon. \end{align*} $$

At first, we have

$$ \begin{align*} &{\nu(E_{l(g)s}\triangle (c(g,s)^{-1}Z_{l(g)s}\cap c(g,s)^{-1}t^{-1}Z_{l(g)s}))} \\ &\quad\leq \nu(E_{l(g)s}\triangle c(g,s)^{-1}Z_{l(g)s})+ \nu(E_{l(g)s}\triangle c(g,s)^{-1}t^{-1}Z_{l(g)s}) \\ &\quad<\delta + \nu(E_{l(g)s}\triangle \cap c(g,s)^{-1}t^{-1}E_{l(g)s})+ \nu(c(g,s)^{-1}t^{-1}(E_{l(g)s}\triangle Z_{l(g)s}))\quad \mbox{(by (2))} \\ &\quad<3\delta. \end{align*} $$

Thus,

$$ \begin{align*} &{ \nu \bigg(X\backslash \bigcup_{s\in S'}c(g,s)^{-1}Z_{l(g)s}\cap c(g,s)^{-1}t^{-1}Z_{l(g)s}\cap B_{g,s,t}\bigg) } \\ &\quad\leq \nu \bigg(X\backslash \bigcup_{s\in S'}c(g,s)^{-1}Z_{l(g)s}\cap c(g,s)^{-1}t^{-1}Z_{l(g)s}\bigg) + \nu\bigg(X\backslash \bigcup_{s\in S'}B_{g,s,t}\bigg) \\ &\quad\leq \sum_{s\in S'} \nu(E_{l(g)s}\triangle (c(g,s)^{-1}Z_{l(g)s}\cap c(g,s)^{-1}t^{-1}Z_{l(g)s})) + \sum_{s\in S'}\nu(X\backslash B_{g,s,t}) \\ &\quad<5|S'|\delta<\varepsilon \end{align*} $$

holds, and we obtain $d_{\nu }(\widehat {w(g)}(t),t)<\varepsilon $ for $g\in K$ , $\nu \in \Phi $ , $t\in \Lambda $ .

Lemma 4.3. For any $e\in K\Subset G $ and $\varepsilon>0$ , there exist $S\Subset G$ and $\delta>0$ satisfying the following property: for any $\mu \in \Phi \Subset M_1(X,\mu )$ , an ultrafree cocycle crossed action $(\gamma , c)$ of G, and $u\in C^1(G,[T])$ with

$$ \begin{align*} d_{\nu}(\widehat{\gamma_g}(u(s))^{-1}u(g)^{-1}u(gs), \mathrm{id} )<\delta, \quad g,s\in S, \nu\in \Phi, \end{align*} $$

there exists $w\in [T]$ such that

$$ \begin{align*} d_{\nu}(w^{-1}u(g)\widehat{\gamma_g}(w),\mathrm{id})<\varepsilon, \quad \nu\in \Phi, g\in K. \end{align*} $$

Proof. Let $K\Subset G$ , $\varepsilon>0$ be given. Take $\varepsilon '>0$ such that $8\sqrt {\varepsilon '}<\varepsilon $ . Let $S'$ be a $(K,\varepsilon ')$ -invariant set and set $S=S'\cup K$ . Choose $\delta>0$ such that $4|S'|\delta +8\sqrt {\varepsilon '}<\varepsilon $ . Let a cocycle crossed action $(\gamma ,c)$ , $ \Phi \Subset M_1(X,\mu )$ , and $u\in C^1(G,[T])$ satisfying the condition

$$ \begin{align*} d_{\nu}(\widehat{\gamma_g}(u(s))^{-1}u(g)^{-1}u(gs), \mathrm{id} )<\delta, \quad g,s\in S, \nu \in \Phi \end{align*} $$

be given. By Proposition 3.3, choose a partition $\{E_s\}_{s\in S'}$ of X, $E_s\supset Z_s$ , and $w\in [T]$ such that:

1. (1) $ u(s)Z_s\subset E_s; $

2. (2) $ \nu (\gamma _g(E_s\backslash Z_s))<\delta , \nu (E_s\backslash u(s)Z_s)<\delta , g\in K, \nu \in \Phi ;$

3. (3) $ \nu (\widehat {\gamma _g}(u(s))^{-1}u(g)^{-1}(E_{gs}\backslash Z_{gs}))<\delta , g\in K, s\in S^{\prime }g, \nu \in \Phi ,$

4. (4) $ \nu (E_{gs}\triangle \widehat {\gamma _g}(u(s))^{-1}g^{-1}E_{gs})<\delta , g\in K, s\in S^{\prime }g, \nu \in \Phi ;$

5. (5) $ \sum _{s\in S^{\prime }g}\nu (E_{gs}\triangle \gamma _gE_s)<5\sqrt {\varepsilon '}, g\in K, \nu \in \Phi ;$

6. (6) $ \sum _{s\in S'\backslash S_{g^{-1}}'}\nu (E_s) <3\sqrt {\varepsilon '}, g\in K, \nu \in \Phi ;$

7. (7) $ wx=u(s)x, x\in Z_s.$

Let

$$ \begin{align*} W_g:=\bigcup_{s\in S^{\prime}_g}\{\widehat{\gamma_g}(u(s))^{-1}u(g)^{-1}u(gs)=\mathrm{id}\}\cap {\gamma_gZ_s}\cap\widehat{\gamma_g}(u(s))^{-1}u(g)^{-1}Z_{gs}. \end{align*} $$

We can verify that $w^{-1}u(g)\widehat {\gamma _g}(w)=\mathrm {id}$ on $W_g$ , $g\in K$ , as in the proof of Lemma 4.2.

Next we show $\nu (X\backslash W_g)<\varepsilon $ . We have

$$ \begin{align*} \nu(E_{gs}\triangle \gamma_gZ_s)\leq \nu(E_{gs}\triangle \gamma_gE_s)+ \nu(\gamma_gE_{s}\backslash \gamma_gZ_s)< \nu(E_{gs}\triangle \gamma_gE_s)+ \delta \end{align*} $$

by condition (2), and

$$ \begin{align*} &{ \nu(E_{gs}\triangle \widehat{\gamma_g}(u(s))^{-1}u(g)^{-1}Z_{gs}) } \\ & \quad \leq \nu(E_{gs}\triangle \widehat{\gamma_g}(u(s))^{-1}u(g)^{-1}E_{gs})+ \nu(\widehat{\gamma_g}(u(s))^{-1}u(g)^{-1}(E_{gs}\backslash Z_{gs})) <2\delta \end{align*} $$

by conditions (3) and (4). Hence, we have

$$ \begin{align*} \nu(E_{gs}\triangle(\gamma_gZ_s\cap \widehat{\gamma_g}(u(s))^{-1}u(g)^{-1}E_{gs}))<3\delta + \nu(E_{gs}\triangle \gamma_gE_s). \end{align*} $$

Then we have

$$ \begin{align*} &{ \nu\bigg(\bigcup_{s \in S^{\prime}_g}E_{gs} \triangle \bigcup_{s\in S^{\prime}_g}(\gamma_gZ_s\cap \widehat{\gamma_g}(u(s))^{-1}u(g)^{-1}E_{gs})\bigg) } \\ &\quad\leq \sum_{s\in S_g'} \nu(E_{gs} \triangle (\gamma_gZ_s\cap \widehat{\gamma_g}(u(s))^{-1}u(g)^{-1}E_{gs})) \\ &\quad<\sum_{s\in S^{\prime}_g}(3\delta+ \nu(E_{gs}\triangle \gamma_gE_s)) <3|S'|\delta+5\sqrt{\varepsilon'} \end{align*} $$

by condition (5). Hence, we get

$$ \begin{align*} &{ \nu\bigg(X\backslash \bigcup_{s\in S^{\prime}_g} (\gamma_gZ_s\cap \widehat{\gamma_g}(u(s))^{-1}u(g)^{-1}E_{gs})\bigg)} \\ &\quad\leq\sum_{s\in S'\backslash S_{g^{-1}}'}\nu(E_s) + \nu\bigg(\bigcup_{s\in S^{\prime}_g}E_{gs} \triangle \bigcup_{s\in S^{\prime}_g}(\gamma_gZ_s\cap \widehat{\gamma_g}(u(s))^{-1}u(g)^{-1}E_{gs})\bigg) \\ &\quad< 3|S'|\delta+8\sqrt{\varepsilon'} \end{align*} $$

by condition (6). By the assumption

$$ \begin{align*} d_{\nu}(\widehat{\gamma_g}(u(s))^{-1}u(g)^{-1}u(gs), \mathrm{id} )<\delta, \quad g,s\in S, \nu\in \Phi, \end{align*} $$

we have $\nu (X\backslash \bigcup _{s\in S^{\prime }g} \{\widehat {\gamma _g}(u(s))^{-1}u(g)^{-1}u(gs)=\mathrm {id}\})<|S'|\delta $ . Hence,

$$ \begin{align*} \nu(X\backslash W_g)<4|S'|\delta+8\sqrt{\varepsilon'}<\varepsilon \end{align*} $$

holds.

Theorem 4.4. Let $(\gamma ,c)$ be an ultrafree cocycle crossed action of G. Then there exists $u\in C^1(G,[T])$ such that ${}_uc(g,h)=\mathrm {id}$ , and hence ${}_u\gamma $ is an action.

Moreover, for any $e\in K\Subset G $ , $\varepsilon>0$ , there exists $S\Subset G$ , $\delta>0$ , which depends only on K and $\varepsilon $ , or on cocycle crossed action $(\gamma ,c)$ , such that if

$$ \begin{align*} d_{\nu}(c(g,h),\mathrm{id})<\delta, \quad g,h\in S, \nu\in \Phi \end{align*} $$

for some $\Phi \Subset M_1(X,\mu )$ with $\mu \in \Phi $ , then we can choose $u\in C^1(G,[T])$ so that

$$ \begin{align*} d_{\nu}(u(g),\mathrm{id})<\varepsilon, \quad g\in K, \nu\in \Phi. \end{align*} $$

Proof. At first, we treat a type II $_{\infty }$ or type III case.

Let $e{\kern-1pt}\in{\kern-1pt} K{\kern-1pt}\Subset{\kern-1pt} G$ and $\varepsilon{\kern-1pt}>{\kern-1pt}0$ be given, and take $S\Subset G$ and $\delta>0$ as in Lemma 4.3. Assume $d_{\nu }(c(g,h),\mathrm {id})<\delta $ for $g,h\in S$ , $\nu \in \Phi \Subset M_1(X,\mu )$ . There exists $v\in C^1(G,[T])$ such that ${}_vc(g,h)=\mathrm {id}$ by Theorem 4.1. Hence, $c(g,h)=\widehat {\gamma _g}(v(h))^{-1}v(g)^{-1}v(gh)$ holds and

$$ \begin{align*} d_{\nu}(\widehat{\gamma_g}(v(h))^{-1}v(g)^{-1}v(gh),\mathrm{id})<\delta,\quad g,h\in S, \nu\in \Phi. \end{align*} $$

By Lemma 4.3, there exists $w\in [T]$ such that

$$ \begin{align*} d_{\nu}(w^{-1}v(g)\widehat{\gamma_g}(w),\mathrm{id})<\varepsilon, \quad \nu\in \Phi, g\in K. \end{align*} $$

Define $u(g):=w^{-1}v(g)\widehat {\gamma _g}(w)$ . Then we obtain $d_{\nu }(u(g),\mathrm {id})<\varepsilon $ for $g\in K$ , $\nu \in \Phi $ , and

$$ \begin{align*} {}_uc(g,h)= u(g)\widehat{\gamma_g}(u(h))c(g,h)u(gh)^{-1}= w^{-1} v(g)\widehat{\gamma_g}(v(h))c(g,h)v(gh)^{-1} w=\mathrm{id}. \end{align*} $$

Hence, we have proved the theorem for the type II $_{\infty }$ and type III cases.

Next, we assume T is of type II $_1$ . In this case, we can assume that $\mu $ is the unique T-invariant probability measure and choose $\Phi $ as $\Phi =\{\mu \}$ . Let us take an increasing sequence $\{K_n\}_{n}\Subset G$ and decreasing sequence $\{\varepsilon _n\}_n$ such that $e\in K_n$ , $\bigcup _{n=1}^{\infty } K_n=G$ , and $\sum _{n}\varepsilon _n<\infty $ . Take $S_n$ and $\delta _n$ for $K_n$ and $\varepsilon _n>0$ as in Lemma 4.3. We can choose $S_n$ and $\delta _n$ so that $S_n\subset S_{n+1}$ , $\delta _n>\delta _{n+1}$ .

For given $K\Subset G$ and $\varepsilon>0$ , choose $N\in \mathbb {N}$ such that $K\subset K_N$ , $\varepsilon>\sum _{k=N}^{\infty } \varepsilon _k$ . By Lemma 4.2, take $S_N\Subset G$ and $\delta _N>0$ for $K_N$ and $\varepsilon _N>0$ . Again by Lemma 4.2, we can perturb $(\gamma ,c)$ by some $w\in C^1(G,[T])$ so that

$$ \begin{align*} &d_{\mu}({}_wc(g,h),\mathrm{id})<\varepsilon_N,\quad g,h\in K_N, \quad d_{\mu}({}_wc(g,h),\mathrm{id})<\frac{\delta_N}{2},\quad g,h\in S_N. \end{align*} $$

Set

$$ \begin{align*} (\gamma^{(N)}, c_N):=({}_w\gamma,{}_wc),\quad u_{N}(g)=1. \end{align*} $$

We will inductively construct a family of cocycle crossed actions $(\gamma ^{(n)},c_n)$ and normalized maps $\{u_{n}\}\subset C^1(G,[T])$ , $n\geq N$ , such that:

1. (1.n) $ (\gamma ^{(n)},c_n)=({}_{u_{n}}\gamma ^{(n-1)}, {}_{u_{n}}c_{n-1}); $

2. (2.n) $ d_{\mu }(c_n(g,h),\mathrm {id})<\varepsilon _n, g,h\in K_n;$

3. (3.n) $ d_{\mu }(c_n(g,h),\mathrm {id})<{\delta _n}/{2}, g,h \in S_n;$

4. (4.n) $ d_{\mu }(u_{n}(g),\mathrm {id})<\varepsilon _{n-1}, g\in K_{n-1}.$

Here we regard $\gamma ^{(N-1)}=\gamma ^{(N)}$ , $c_{N-1}(c,h)=c_N(g,h)$ . Clearly we have $(1.N)$ , $(2.N)$ , $(3.N)$ , and $(4.N)$ .

Assume we have done up to the nth step.

By Lemma 4.2, we choose $\bar {u}_{n+1}\in C^1(G,[T])$ such that:

1. (a.n + 1) $ d_{\mu }(\bar {u}_{n+1}(g)\widehat {\gamma ^{(n)}_g}(\bar {u}_{n+1}(h))c_n(g,h) \bar {u}_{n+1}(gh)^{-1},\mathrm {id}) <\varepsilon _{n+1}, g,h\in K_{n+1}; $

2. (b.n + 1) $ d_{\mu }(\bar {u}_{n+1}(g)\widehat {\gamma ^{(n)}_g}(\bar {u}_{n+1}(h))c_n(g,h) \bar {u}_{n+1}(gh)^{-1},\mathrm {id}) <{\delta _{n+1}}/{2}, g,h\in S_{n+1}.$

By condition $(b.n+1)$ , we have

$$ \begin{align*} d_{\mu}\big(\widehat{\gamma^{(n)}_g}(\bar{u}_{n+1}(h))^{-1}\bar{u}_{n+1}(g)^{-1}\bar{u}_{n+1}(gh), c_n(g,h)\big)<\frac{\delta_{n+1}}{2},\quad g,h\in S_{n+1}. \end{align*} $$

Combining with condition $(3.n)$ , we get

$$ \begin{align*} d_{\mu}\big(\widehat{\gamma_g^{(n)}}(\bar{u}_{n+1}(h))^{-1} \bar{u}_{n+1}(g)^{-1}\bar{u}_{n+1}(gh), \mathrm{id}\big)<\delta_n, \quad g,h\in S_n. \end{align*} $$

By Lemma 4.3, there exists $w{\kern-1pt}\in{\kern-1pt} [T]$ such that $d_{\mu }(w^{-1}\bar {u}_{n+1}(g)\widehat {\gamma ^{(n)}_g}(w),{\kern-1pt}\mathrm {id}){\kern-1pt}<{\kern-1pt}\varepsilon _n$ for $g{\kern-1pt}\in{\kern-1pt} K_n$ . Here set $u_{n+1}(g):=w^{-1}\bar {u}_{n+1}(g)\widehat {\gamma ^{(n)}_g}(w)$ . Then we get condition $(4.n+1)$ . Define a cocycle crossed action $(\gamma ^{(n+1)}, c_{n+1})$ as condition $(1.n+1)$ . Then we get conditions $(2.n+1)$ and $(3.n+1)$ from conditions $(a.n+1)$ and $(b.n+1)$ , respectively, and the induction is complete.

Let $v_{n}(g):=u_{n}(g)u_{n-1}(g)\cdots u_{N}(g)$ . We have $(\gamma ^n,c_n)=({}_{v_n}\gamma ^{(N)}, {}_{v_n}c_N)$ by the construction. Fix $L\in \mathbb {N}$ and take any $g\in K_L$ . By condition $(4.n)$ ,

$$ \begin{align*} d_{\mu}(v_{n}(g),v_{n-1}(g))= d_{\mu}(u_{n}(g), \mathrm{id})<\varepsilon_{n-1},\quad n\geq L+1 \end{align*} $$

holds. So $\{v_n(g)\}_n$ is a Cauchy sequence and hence $v_{n}(g)$ converges to some $v(g)\in [T]$ uniformly. Note that $v_{n}(g)^{-1}$ converges to $v(g)^{-1}$ automatically, since $\mu $ is the invariant measure for $[T]$ . Combining with condition $(2.n)$ , we obtain ${}_vc(g,h)=\mathrm {id}$ for all $g,h\in G$ .

If $g\in K_N$ , then

$$ \begin{align*} d_{\mu}(v_{n}(g),\mathrm{id} )&= d_{\mu}(v_{n}(g),v_{N}(g)) \leq \sum_{k=N}^{n-1}d_{\mu}(v_{k+1}(g),v_{k}(g)) < \sum_{k=N}^{n-1}\varepsilon_k. \end{align*} $$

Hence, we have $d_{\mu }(v(g),\mathrm {id})\leq \sum _{k=N}^{\infty } \varepsilon _k<\varepsilon $ . Set $S:=S_N\cup K_N$ , $\delta :=\min \{\delta _N/2, \varepsilon _N\}$ . If $d_{\mu }(g,h)<\delta $ for $g,h\in S$ , then we have $d_{\mu }(v(g),\mathrm {id})<\varepsilon $ for $g\in K_N$ . Note that S and $\delta $ depend only on K and $\varepsilon $ .

5 Classification

Lemma 5.1. Let $\alpha $ and $\beta $ be actions of G into $N[T]$ with $\operatorname {\mathrm {mod}}(\alpha _g)=\operatorname {\mathrm {mod}}({\beta }_g)$ . Then for any $\varepsilon>0$ , $K\Subset G$ , $\mu \in \Phi \Subset M_1(X,\mu )$ , $\Lambda \Subset [T]$ , there exists $w\in C^1(G,[T])$ such that:

1. (1) $\|{}_w\alpha _g(\xi )-{\beta }_{g}(\xi )\|<\varepsilon $ , $g\in K$ , $\xi \in \Phi $ ;

2. (2) $d_{\nu }(\widehat {{}_w\alpha _g}(t), \widehat {{\beta }_g}(t))<\varepsilon $ , $g\in K$ , $t\in \Lambda ,\nu \in \Phi $ ;

3. (3) let $c(g,h):=w(g)\widehat {\alpha _g}(w(h))w(gh)^{-1}$ . Then

   $$ \begin{align*} \|c(g,h)(\xi)-\xi\|<\varepsilon,\quad d_{\nu}(\widehat{c(g,h)}(t),t)<\varepsilon,\quad g,h\in K, \xi,\nu \in \Phi, t\in \Lambda. \end{align*} $$

Proof. By enlarging K, we may assume $e\in K=K^{-1}\Subset G$ . Let

$$ \begin{align*} \tilde{\Phi}:=\{{\beta}_{gh}(\xi)\mid g,h\in K, \xi\in \Phi\}, \quad \tilde{\Lambda}:=\{\widehat{{\beta}_{gh}}(t)\mid g,h\in K,t\in \Lambda\}. \end{align*} $$

By the assumption, ${\beta }_g\alpha _g^{-1}\in \operatorname {\mathrm {Ker}}(\operatorname {\mathrm {mod}})=\overline {[T]}$ . Hence, we can take $w\in C^1(G,[T])$ so that

$$ \begin{align*} \|{}_w\alpha_{gh}(\xi)-{\beta}_{gh}(\xi)\|<\frac{\varepsilon}{7},\quad d_{\nu}(\widehat{{}_w\alpha_{gh}} (t),\widehat{{\beta}_{gh}}(t))<\frac{\varepsilon}{7} \end{align*} $$

for $g,h\in K$ , $\nu , \xi \in \bigcup _{g\in K}{\beta }_g(\tilde {\Phi })$ , $t\in \bigcup _{g\in K}{\beta }_g(\tilde {\Lambda })$ . Obviously, we have conditions (1) and (2).

Then for $g,h\in K$ , $\eta \in \tilde {\Phi }$ , we have

$$ \begin{align*} \|{}_w{\alpha}_{g}{}_w{\alpha}_{h}(\eta)-{\beta}_{gh}(\eta)\| &\leq \|{}_w{\alpha}_{g}{}_w{\alpha}_{h}(\eta)-{}_{w}{\alpha}_{g}{\beta}_{h}(\eta)\| + \|{}_{w}{\alpha}_{g}{\beta}_{h}(\eta)-{\beta}_{g}{\beta}_{h}(\eta) \| \\ &\leq \|{}_{w}{\alpha}_{h}(\eta)-{\beta}_{h}(\eta)\| + \|{}_{w}{\alpha}_{g}{\beta}_{h}(\eta)-{\beta}_{g}{\beta}_{h}(\eta) \| < \frac{2\varepsilon}{7}. \end{align*} $$

Thus,

$$ \begin{align*} \|c(g,h){\beta}_{gh}(\eta)-{\beta}_{gh}(\eta)\| &\leq \|c(g,h){\beta}_{gh}(\eta)-c(g,h)\alpha_{gh}(\eta)\|\\&\quad+ \|c(g,h)\alpha_{gh}(\eta)-{\beta}_{gh}(\eta)\| \\ &< \frac{3\varepsilon}{7} \end{align*} $$

holds for $g,h\in K$ , $\eta \in \tilde {\Phi }$ . Hence, we get $\|c(g,h)(\xi )-\xi \|<3\varepsilon /7$ for $g,h\in K$ , $\xi \in \Phi $ .

For $g\in K$ , $t\in \tilde {\Lambda }$ , $\nu \in \tilde {\Phi }$ , we have

$$ \begin{align*} d_{\nu}(\widehat{c(g,h)}\widehat{{}_{w}{\alpha}_{gh}}(t), \widehat{{\beta}_{gh}}(t)) &=d_{\nu}(\widehat{{}_{w}{\alpha}_g}\widehat{{}_{w}{\alpha}_h} (t) ,\widehat{{\beta}_{gh}}(t)) \\ &\leq d_{\nu}(\widehat{{}_{w}{\alpha}_g}\widehat{{}_{w}{\alpha}_h} (t), \widehat{{}_{w}{\alpha}_g}\widehat{{\beta}_h}(t) )+ d_{\nu}(\widehat{{}_{w}{\alpha}_g}\widehat{{\beta}_h} (t), \widehat{{\beta}_{gh}}(t)) \\ &\leq d_{{}_{w}{\alpha}_g^{-1}(\nu)}(\widehat{{}_{w}{\alpha}_h} (t), \widehat{{\beta}_h}(t))+\frac{\varepsilon}{7} \\ &\leq d_{{\beta}_g^{-1}(\nu)}(\widehat{{}_{w}{\alpha}_h}(t), \widehat{{\beta}_h}(t) )+\|{}_{w}{\alpha}_g^{-1}(\nu)-{\beta}_g^{-1}(\nu)\|+ \frac{\varepsilon}{7} \\ &\leq d_{{\beta}_g^{-1}(\nu)}(\widehat{{}_{w}{\alpha}_h}(t), \widehat{{\beta}_h}(t) )+\frac{2\varepsilon}{7} < \frac{3\varepsilon}{7}. \end{align*} $$

By noting $\|c(g,h)(\nu )-\nu \|\leq 3\varepsilon /7$ for $g,h\in K$ , $\nu \in \Phi $ , we have

$$ \begin{align*} d_{\nu}( \widehat{c(g,h)}\widehat{{\beta}_{gh}}(t), \widehat{{\beta}_{gh}}(t)) &\leq d_{\nu} (\widehat{c(g,h)}\widehat{{\beta}_{gh}}(t), \widehat{ c(g,h)}\widehat{{}_{w}{\alpha}_{gh} }(t))\\ &\quad+ d_{\nu}(\widehat{c(g,h)}\widehat{{}_{w}{\alpha}_{gh}} (t),\widehat{{\beta}_{gh}}(t)) \\ &\leq d_{c(g,h)^{-1}(\nu)}(\widehat{{\beta}_{gh}}(t), \widehat{ {}_{w}{\alpha}_{gh}} (t)) +\frac{3\varepsilon}{7} \\ &\leq d_{\nu}(\widehat{{\beta}_{gh}}(t), \widehat{ {}_{w}{\alpha}_{gh} }(t)) +\frac{6\varepsilon}{7} <\varepsilon \end{align*} $$

for $\nu \in \Phi $ , $g,h\in K$ , $t\in \tilde {\Lambda }$ . Thus, $d_{\nu }(\widehat {c(g,h)}(t),t)<\varepsilon $ holds for $g,h\in K$ , $t\in \Lambda $ , $\nu \in \Phi $ .

Lemma 5.2. Let $\alpha $ and $\beta $ be actions of G into $N[T]$ with $\operatorname {\mathrm {mod}}(\alpha _g)=\operatorname {\mathrm {mod}}({\beta }_g)$ . For any $\varepsilon>0$ , $K\Subset G$ , $\Lambda \Subset [T]$ , $\Phi \Subset M_1(X,\mu )$ , there exists $v\in C^1(G,[T])$ such that

$$ \begin{align*} & \|{}_{v}\alpha_g(\xi)-{\beta}_g(\xi)\|<\varepsilon,\quad g\in K, \nu \in \Phi, \\ &d_{\nu}(\widehat{{}_{v}\alpha_g}(t),\widehat{{\beta}_g}(t))<\varepsilon,\quad g\in K, t\in \Lambda , \nu\in \Phi,\\ &d_{\nu}(v(g)\widehat{\alpha_g}(v(h))v(gh)^{-1},\mathrm{id})<\varepsilon,\quad g,h \in K, \nu \in \Phi. \end{align*} $$

Proof. Let $\tilde {\Phi }:=\{{\beta }_g(\xi )\mid g \in K, \xi \in \Phi \}$ , $\tilde {\Lambda }:=\{\widehat {{\beta }_g}(t)\mid g\in K, t\in \Lambda \}$ . Choose $\delta>0$ and S for $\varepsilon /3>0$ and K as in Lemma 4.2. By Lemma 5.1, there exists $u\in C^1(G,[T])$ such that

$$ \begin{align*} & c(g,h):=u(g)\widehat{\alpha_g}(u(h))u(gh)^{-1},\\ &\|{}_u\alpha_g(\xi)-{\beta}_g(\xi)\|<\frac{\varepsilon}{3},\quad g\in K, \xi \in \Phi, \\ & \|c(g,h)(\xi)-\xi\|<\delta,\quad g,h \in S, \xi\in \tilde{\Phi}, \\ & d_{\nu}(\widehat{c(g,h)}(t),t)<\delta,\quad g,h\in S, t\in \tilde{\Lambda }, \nu\in \tilde{\Phi}. \end{align*} $$

By Lemma 4.2, there exists $w\in C^1(G, [T])$ such that

$$ \begin{align*} d_{\nu}(w(g)\widehat{{}_u{\alpha}_g}(w(h))c(g,h)w(gh)^{-1},\mathrm{id})<\frac{\varepsilon}{3},\quad g,h \in K, \nu \in \Phi \end{align*} $$

and

$$ \begin{align*} \|w(g)(\xi)-\xi\|<\frac{\varepsilon}{3},\quad d_{\nu}(\widehat{w(g)}(t),t)<\frac{\varepsilon}{3},\quad g\in K, \xi,\nu\in \tilde{\Phi}, t\in \tilde{\Lambda}. \end{align*} $$

Let $v(g):=w(g)u(g)$ . Then we have

$$ \begin{align*} d_{\nu}(v(g)\widehat{\alpha_g}(v(h))v(gh)^{-1},\mathrm{id})<\varepsilon,\quad g,h \in K, \nu \in \Phi. \end{align*} $$

We can verify the first inequality as follows. For $g\in K$ , $\xi \in \Phi $ ,

$$ \begin{align*} \|{}_v\alpha_g(\xi)-{\beta}_g(\xi)\|&\leq \|w(g)u(g)\alpha_g(\xi)-w(g){\beta}_g(\xi)\|+ \|w(g){\beta}_g(\xi)-{\beta}_g(\xi)\| \\ &<\frac{2\varepsilon}{3}<\varepsilon \end{align*} $$

since ${\beta }_g(\xi )\in \tilde {\Phi }$ . Similarly, we have

$$ \begin{align*} d_{\nu}(\widehat{{}_{v}\alpha_g}(t),\widehat{{\beta}_g}(t))&\leq d_{\nu}(\widehat{w(g){}_{u}\alpha_g}(t),\widehat{w(g){\beta}_g}(t))+ d_{\nu}(\widehat{w(g){\beta}_g}(t),\widehat{{\beta}_g}(t)) \\ &\leq d_{w(g)(\nu)}(\widehat{{}_{u}\alpha_g}(t),\widehat{{\beta}_g}(t)) +\frac{\varepsilon}{3} \\ &\leq \|w(g)(\nu)-\nu\|+ d_{\nu}(\widehat{{}_{u}\alpha_g}(t),\widehat{{\beta}_g}(t)) +\frac{\varepsilon}{3} <\varepsilon \end{align*} $$

for $g\in K$ , $t\in \Lambda $ , $\nu \in \Phi $ .

Theorem 5.3. Let $\alpha $ and $\beta $ be ultrafree actions of G into $N[T]$ with $\operatorname {\mathrm {mod}}(\alpha _g)=\operatorname {\mathrm {mod}}({\beta }_g)$ . Then there exists a sequence $\{u_n(\cdot )\}$ of 1-cocycles for $\alpha _g$ such that $\lim \nolimits _{n\rightarrow \infty }{}_{{u_n}}\alpha _g={\beta }_g$ in the u-topology.

Proof. By Lemma 5.2, there exists a sequence $\{v_n\} \subset C^1(G,[T])$ of normalized maps such that $\lim \nolimits _{n\rightarrow \infty }{}_{{v_n}}\alpha _g{\kern-1pt}={\kern-1pt}{\beta }_g$ in the u-topology, and $\lim \nolimits _{n\rightarrow \infty }d_{\mu }(v_n(g)\widehat {\alpha _g} (v_n(h))v_{n}(gh)^{-1}, \mathrm {id})=0$ . Let $\alpha ^{(n)}={}_{{v_n}}\alpha $ and $c_n(g,h)=v_n(g)\widehat {\alpha _g}(v_n(h))v_{n}(gh)^{-1}$ . By Theorem 4.4, there exists a sequence $\{w_n\} \subset C^1(G,[T])$ such that

$$ \begin{align*} w_n(g)\widehat{\alpha_g^{(n)}}(w_n(h))c_n(g,h)w_{n}(gh)^{-1}=1, \quad \lim_{n\rightarrow \infty} d_{\mu}(w_n(g),\mathrm{id})=0. \end{align*} $$

Then it turns out that $u_n(g):=w_n(g) v_n(g)$ is a 1-cocycle for $\alpha _g$ , and $\lim \nolimits _{n\rightarrow \infty }{}_{{u_n}}\alpha _g={\beta }_g$ holds in the u-topology.

Lemma 5.4. Let $K\Subset G$ and $\varepsilon>0$ be given. Then there exist $S\Subset G$ and $\delta>0$ satisfying the following: for any action $\gamma $ of G, a 1-cocycle $u(\cdot )$ for $\gamma $ , $\Phi \Subset M_1(X,\mu )$ with $\mu \in \Phi $ and $\Lambda \Subset [T]$ satisfying

$$ \begin{align*}\|u(s)(\xi)-\xi\|<\delta,\quad d_{\nu}(\widehat{u(s)}(t),t)<\delta,\quad s\in S, \xi, \nu\in \Phi, t\in \Lambda, \end{align*} $$

there exists $ w\in [T]$ such that

$$ \begin{align*} d_{\nu}(u(g)\widehat{\gamma_g}(w)w^{-1},1)<\varepsilon, \|w(\xi)-\xi\|<\varepsilon, d_{\nu}(w(t),t)<\varepsilon, g\in K, \xi,\nu\in \Phi, t\in \Lambda. \end{align*} $$

Proof. Take $\varepsilon _1>0$ with $8\varepsilon _1^{1/2}<\varepsilon $ , and let S be a $(K,\varepsilon _1)$ -invariant set. Choose $\delta>0$ with $8\varepsilon _1^{1/2}+3|S|\delta <\varepsilon $ , $4|S|\delta <\varepsilon $ .

By Proposition 3.3, take a partition $\{E_s\}_{s\in S} $ of X, $Z_s\subset E_s$ , and $w\in [T]$ such that:

1. (1) $ u(s) Z_s\subset E_s, s\in S; $

2. (2) $ \nu (E_s\backslash Z_s)<\delta , \nu (E_s\backslash u(s) Z_s)<\delta , s\in S, \nu \in \Phi ;$

3. (3) $ \nu (u(gs)\gamma _g(E_{s}\backslash Z_s))<\delta , g\in K, s\in S_g, \nu \in \Phi ;$

4. (4) $ \nu (u(s)t^{-1}(E_s\backslash Z_s))<\delta , s\in S, t\in \Lambda , \nu \in \Phi ;$

5. (5) $ \nu (u(s)E_s\triangle E_s)<\delta , s\in S, \nu \in \Phi ;$

6. (6) $\nu (E_{gs}\triangle \widehat {\gamma _g} (u(s))E_{gs}) <\delta , s\in S_g, \nu \in \Phi ;$

7. (7) $ \nu (E_s\triangle u(s) t^{-1}E_s)<\delta , s\in S, t\in \Lambda , \nu \in \Phi ;$

8. (8) $ \sum _{s\in S_g}u(gs)^{-1}(\nu )(\gamma _gE_s\triangle E_{gs} )<5\varepsilon _1^{{1}/{2}}, g\in K, \nu \in \Phi ;$

9. (9) $\sum _{s\in S\backslash S_{g^{-1}}}\nu (E_s)<3\varepsilon _1^{{1}/{2}}, g\in K;$

10. (10) $ wx=u(s)x, x\in Z_s.$

In the following proof, the letters g, s, and $\nu $ denote an elements in K, S, and $\Phi $ , respectively. As in the proof of Lemma 4.2, we can see that

$$ \begin{align*} u(g)\widehat{\gamma_g}(w)w^{-1} x=u(g)\gamma_gu(s)\gamma_g^{-1}u(gs)^{-1}x=x \end{align*} $$

for $x\in u(gs)Z_{gs}\cap u(gs)\gamma _gZ_s$ .

We have

$$ \begin{align*} &{ \nu(E_{gs}\triangle (u(gs)Z_{gs}\cap u(gs)\gamma_gZ_s))} \\ &\quad\leq \nu(E_{gs}\backslash u(gs)Z_{gs})+ \nu(E_{gs}\triangle u(gs)\gamma_gZ_s) \\ &\quad< \delta+ \nu(E_{gs}\triangle u(gs)\gamma_g E_s) + \nu(u(gs)\gamma_g(E_{s})\backslash u(gs)\gamma_gZ_s)\quad (\mbox{by (2)}) \\ &\quad< 2\delta + \nu(E_{gs}\triangle u(gs) E_{gs}) + \nu(u(gs)(E_{gs}\triangle \gamma_g E_s))\quad (\mbox{by (3)}) \\ &\quad< 3\delta + u(gs)^{-1}(\nu)(E_{gs}\triangle \gamma_g E_s) \quad (\mbox{by (5)}). \end{align*} $$

Thus,

$$ \begin{align*} &{\nu\bigg(X\backslash \bigcup_{s\in S_g}u(gs)Z_{gs}\cap u(gs)\gamma_gZ_s\bigg) } \\ &\quad\leq \nu(X\backslash \bigsqcup_{s\in S_g}E_{gs})+ \sum_{s\in S_g}\nu(E_{gs}\triangle (u(gs)Z_{gs}\cap u(gs)\gamma_gZ_s)) \\ &\quad\leq \sum_{s \in S\backslash S_{g^{-1}}}\nu(E_s)+ \sum_{s\in S_g}(3\delta + u(gs)^{-1}(\nu)(E_{gs}\triangle \gamma_g E_s) )\\ &\quad< 3\varepsilon_1^{{1}/{2}}+3|S|\delta +5\varepsilon_1^{{1}/{2}} = 8\varepsilon_1^{{1}/{2}}+3|S|\delta <\varepsilon \end{align*} $$

holds. Hence, $\nu (\{u(g)\widehat {\gamma _g} (w)w^{-1}\ne \mathrm {id}\})<\varepsilon $ for $g\in K$ and $\nu \in \Phi $ , which implies

$$ \begin{align*} d_{\nu}(u(g) \widehat{\gamma_g} (w)w^{-1},\mathrm{id})<\varepsilon,\quad g\in K, \nu \in \Phi. \end{align*} $$

We next show $\|w(\xi )-\xi \|<\varepsilon $ and $d_{\nu }(\widehat {w}(t),t)<\varepsilon $ . Let $Z=\bigsqcup _{s\in S}Z_s$ . As in the proof of Lemma 4.2, we can see $w(\xi )(x)=u(s)(\xi )(x)$ on $u(s)Z_s$ , and

$$ \begin{align*} \int_{X\backslash wZ}|w(\xi)(x)-\xi(x)|\,d \mu(x) <2|S|\delta \end{align*} $$

by using conditions (2) and (10). If $u(s)$ satisfies $\|u(s)(\xi ) -\xi \|<\delta $ for $s\in S$ , then

$$ \begin{align*} \|w(\xi) -\xi \| &=\sum_{s\in S}\int_{u(s)Z_s}|w(\xi)(x)-\xi(x)|\,d \mu(x) +\int_{X\backslash wZ}|w(\xi)(x)-\xi(x)|\,d \mu(x) \\ &< \sum_{s\in S} \int_{u(s)Z_s}|u(s)(\xi)(x)-\xi(x)|\,d \mu(x) +2|S|\delta < 3|S|\delta <\varepsilon \end{align*} $$

holds for $\xi \in \Phi $ .

For $t\in \Lambda \subset [T]$ and $x\in u(s)Z_s\cap u(s) t^{-1}Z_s$ , $w^{-1}x=u(s)^{-1}x\in Z_s\cap t^{-1}Z_s$ . Hence, $tw^{-1}x=u(s)^{-1}x\in tZ_s\cap Z_s$ , and $wtw^{-1}x=u(s)tu(s)^{-1}x$ holds.

Then,

$$ \begin{align*} \nu(E_s\triangle (u(s)Z_s\cap u(s) t^{-1}Z_s))&\leq \nu(E_s\backslash u(s)Z_s)+ \nu(E_s\triangle u(s) t^{-1}Z_s) \\&< \delta + \nu(E_s\triangle u(s) t^{-1}E_s)+ \nu(u(s)t^{-1}(E_s\backslash Z_s)) \quad (\mbox{by condition (2)}) \\&< 3\delta \quad (\mbox{by conditions~(4) and (7)}). \end{align*} $$

Let us assume $d_{\nu }(\widehat {u(s)}(t), t )<\delta $ . Hence, $A_{s,t}:=\{\widehat {u(s)}(t)= t \}$ satisfies $\nu (X\backslash A_{s,t})<\delta $ . Thus,

$$ \begin{align*} &{\nu\bigg(X\backslash \bigcup_{s\in S}(u(s)Z_s\cap u(s) t^{-1}Z_s\cap A_{s,t})\bigg)} \\ &\quad\leq \sum_{s\in S} \nu(E_s\triangle (u(s)Z_s\cap u(s) t^{-1}Z_s)) +\sum_{s\in S}\nu(X\backslash A_{s,t}) \\ &\quad\leq 4\delta|S| <\varepsilon \end{align*} $$

and we have $\nu (\{\widehat {w}(t) \ne t \})<\varepsilon $ , equivalently $d_{\nu }(\widehat {w}(t),t)<\varepsilon $ .

Remark. In Lemma 5.4, we can choose $\delta $ and S so that $\delta <\delta '$ and $S'\subset S$ for any given $\delta '>0$ and $S'\Subset G$ .

Now we can classify ultrafree actions.

Theorem 5.5. Let $\alpha $ and $\beta $ be ultrafree actions of G into $N[T]$ with $\operatorname {\mathrm {mod}}(\alpha _g)=\operatorname {\mathrm {mod}}({\beta }_g)$ . Then they are strongly cocycle conjugate.

Proof. Let $\{\xi _i\}_{i=0}^{\infty }$ be a countable dense subset of $M_1(X,\mu )$ with $\xi _0=\mu $ . Take $\varepsilon _n>0$ and $K_n\Subset G$ such that $\sum _{n=0}^{\infty } \varepsilon _n<\infty $ , $\varepsilon _n>\varepsilon _{n+1}, $ $e\in K_{n}$ , $K_n\subset K_{n+1}$ , $\bigcup _{n=0}^{\infty } K_n=G$ . Then choose $S_n\Subset G$ , $\delta _n>0$ for $K_n$ , $\varepsilon _n$ as in Lemma 5.4. We can assume $S_n\subset S_{n+1}$ and $\delta _{n+1}<\delta _n$ . (See the remark after Lemma 5.4.)

Set $\gamma _g^{(0)}:=\alpha _g$ , $\gamma ^{(-1)}_g:={\beta }_g$ , and construct actions $\gamma _g^{(n)}$ of G, $v_{n}(g), \bar {v}_{n}(g), w_n,\theta _n \in [T]$ , $\Phi _n \Subset M_1(X,\mu )$ , and $\Lambda _n\Subset [T]$ as follows:

1. (1.n) $ \gamma _g^{(n)}=\bar {v}_{n}(g)w_n\gamma _g^{(n-2)}w_n^{-1};$

2. (2.n) $ \theta _n=w_n\theta _{n-2};$

3. (3.n) $ v_{n}(g)= \bar {v}_{n}(g)\widehat {w_n}(v_{n-2}(g));$

4. (4.n) $ \|\gamma _g^{(n)}(\xi )-\gamma _g^{(n-1)}(\xi )\|<\varepsilon _n, g\in K_n, \xi \in \Phi _{n-1};$

5. (5.n) $ d_{\mu }(\widehat {\gamma _g^{(n)}}(t),\widehat {\gamma _{g}^{(n-1)}}(t))<\varepsilon _n, g\in K_n, t\in \Lambda _{n-1};$

6. (6.n) $ \|\gamma _g^{(n)}(\xi )-\gamma _g^{(n-1)}(\xi )\|<{\delta _{n-1}}/{2}, g\in S_{n-1}, \xi \in \bigcup _{g\in S_{n-1}}\gamma ^{(n-1)}_{g^{-1}}(\Phi _{n-1});$

7. (7.n) $ d_{\nu }(\widehat {\gamma _g^{(n)}}(t),\widehat {\gamma _g^{(n-1)}}(t)){\kern-1.2pt}<{\kern-1.2pt}{\delta _{n-1}}/{2}, g{\kern-1.2pt}\in{\kern-1.2pt} S_{n-1}, t {\kern-1.2pt}\in{\kern-1.2pt} \bigcup _{s\in S_{n-1}}\gamma ^{(n-1)}_{g^{-1}}(\Lambda _{n-1}), \nu {\kern-1.2pt}\in{\kern-1.2pt} \Phi _{n-1};$

8. (8.n) $ d_{\nu }(\bar {v}_{n}(g),\mathrm {id})<\varepsilon _{n-2}, g\in K_{n-2}, \nu \in \Phi _{n-2}, (n\geq 2);$

9. (9.n) $ \|w_n(\xi )-\xi \|<\varepsilon _{n-2}, \xi \in \Phi _{n-2}, (n\geq 2);$

10. (10.n) $d_{\nu }(\widehat {w_n}(t),t)<\varepsilon _{n-2}, \nu \in \Phi _{n-2}, t\in \Lambda _{n-2}, (n\geq 2);$

11. (11.n) $ \Phi _n=\{\xi _i\}_{i=0}^n \cup \{\theta _{n}(\xi _i)\}_{i=0}^n\cup \{v_{n}(g)(\mu )\}_{g\in K_n};$

12. (12.n) $ \Lambda _n=\{T^i\}_{i=-n}^n \cup \{\theta _n(T^i)\}_{i=-n}^n \cup \{v_{n}(g), v_{n}(g)^{-1}\}_{g\in K_n}.$

1st step. Let $\theta _{-1}=\theta _0=\mathrm {id}$ , $v_{-1}(g)=v_{0}(g)=\mathrm {id}$ . By Theorem 5.3, take a 1-cocycle $u_{1}(\cdot )$ for $\gamma ^{(-1)}$ such that:

1. (a.1) $\|{}_{{u_1}}\gamma _g^{(-1)}(\xi )-\gamma _g^{(0)}(\xi )\|<\varepsilon _{1}, g\in K_{1}, \xi \in \Phi _{0};$

2. (b.2) $ d_{\mu } (\widehat {{}_{{u_1}}\gamma _g^{(-1)}}(t), \widehat {\gamma _g^{(0)}}(t))<\varepsilon _{1}, g\in K_{1}, t\in \Lambda _{0};$

3. (c.2) $\|{}_{{u_1}}\gamma _g^{(-1)}(\xi )-\gamma _g^{(0)}(\xi )\|<{\delta _{0}}/{2}, g\in S_{0}, \xi \in \bigcup _{g\in S_{0}}\gamma _{g^{-1}}^{(0)}(\Phi _{0});$

4. (d.1) $d_{\nu }(\widehat {{}_{{u_1}}\gamma _g^{(-1)}}(t), \widehat {\gamma _g^{(0)}}(t))<{\delta _{0}}/{2}, g\in S_{0}, t \in \bigcup _{g\in S_{0}}\gamma ^{(0)}_{g^{-1}}(\Lambda _0), \nu \in \Phi _{0}.$

Set $w_1=\mathrm {id}$ , $\bar {v}_{1}(g)=u_{1}(g)$ , and define

$$ \begin{align*} \gamma_g^{(1)}& :=\bar{v}_{1}(g)w_1\gamma_g^{(-1)}w_1^{-1} ={}_{{u_1}}\gamma_g^{(-1)}, \\ \theta_1&:=w_1\theta_{-1}=\mathrm{id}, \\ v_{1}(g)&:=\bar{v}_{1}(g)\widehat{w_1}(v_{-1}(g))=u_{1}(g) \end{align*} $$

as in conditions $(1.1)$ , $(1.2)$ , and $(1.3)$ , respectively. By conditions $(a.1)$ , $(b.1)$ , $(c.1)$ , and $(d.1)$ , we get conditions $(4.1)$ , $(5.1)$ , $(6.1)$ , and $(7.1)$ , respectively. Define $\Phi _1$ and $\Lambda _1$ as in conditions $(11.1)$ and $(12.1)$ , respectively. Then we have finished the 1st step of the induction.

Assume that we have done up to the nth step. By Theorem 5.3, let us take a $\gamma ^{(n-1)}$ -cocycle $u_{n+1}(\cdot )$ such that:

1. (a.n + 1) $\|{}_{{u_{n+1}}}\gamma _g^{(n-1)}(\xi )-\gamma _g^{(n)}(\xi )\|<\varepsilon _{n+1}, g\in K_{n+1}, \xi \in \Phi _{n}; $

2. (a.n + 2) $d_{\mu } (\widehat {{}_{{u_{n+1}}}\gamma _g^{(n-1)}}(t), \widehat {\gamma _g^{(n)}}(t)) <\varepsilon _{n+1}, \, g\in K_{n+1}, t\in \Lambda _{n}; $

3. (c.n + 1) $ \|{}_{{u_{n+1}}}\gamma _g^{(n-1)}(\xi )-\gamma _g^{(n)}(\xi )\|<{\delta _{n}}/{2}, g\in S_{n}, \xi \in \bigcup _{g\in S_{n}}\gamma _{g^{-1}}^{(n)}(\Phi _{n}); $

4. (d.n + 1) $d_{\nu }( \widehat {{}_{{u_{n+1}}}\gamma _g^{(n-1)}}(t), \widehat {\gamma _g^{(n)}}(t))<{\delta _{n}}/{2}, g\in S_{n}, t \in \bigcup _{g\in S_{n}}\gamma ^{(n)}_{g^{-1}}(\Lambda _n), \nu \in \Phi _{n}; $

5. (e.n + 1) $ \|{}_{{u_{n+1}}}\gamma _g^{(n-1)}(\xi )-\gamma _g^{(n)}(\xi )\|<{\delta _{n-1}}/{2}, g\in S_{n-1}, \xi \in \bigcup _{g\in S_{n-1}}\gamma _{g^{-1}}^{(n-1)}(\Phi _{n-1}); $

6. (f.n + 1) $d_{\nu }(\widehat {{}_{u_{n+1}}\gamma _g^{(n-1)}}(t), \widehat {\gamma _g^{(n)}}(t))<{\delta _{n-1}}/{2}, g\in S_{n-1}, t \in \bigcup _{g\in S_{n-1}}\widehat {\gamma ^{(n-1)}_{g^{-1}}}(\Lambda _{n-1}), \nu \in \Phi _{n-1}.$

By conditions $(6.n)$ and $(e.n+1)$ , we have

$$ \begin{align*} \|u_{n+1}(g)\gamma_g^{(n-1)}(\xi)-\gamma_g^{(n-1)}(\xi)\|<\delta_{n-1},\quad g\in S_{n-1}, \xi\in \bigcup_{g\in S_{n-1}}\gamma^{(n-1)}_{g^{-1}}(\Phi_{n-1}) \end{align*} $$

and hence

$$ \begin{align*} \|u_{n+1}(g)(\xi)-\xi\|<\delta_{n-1},\quad g\in S_{n-1}, \xi\in \Phi_{n-1}. \end{align*} $$

By conditions $(7.n)$ and $(f.n+1)$ ,

$$ \begin{align*} d_{\nu}(\widehat{u_{n+1}(g)}\widehat{\gamma_g^{(n-1)}}(t),\widehat{\gamma_g^{(n-1)}}(t))<\delta_{n-1}, \!\!\quad g\in S_{n-1}, t \in \bigcup_{g\in S_{n-1}}\widehat{\gamma^{(n-1)}_{g^{-1}}}(\Lambda_{n-1}), \nu \in \Phi_{n-1}, \end{align*} $$

and hence

$$ \begin{align*} d_{\nu}(\widehat{u_{n+1}(g)}(t),t)<\delta_{n-1}, \quad g\in S_{n-1}, t \in \Lambda_{n-1}, \nu \in \Phi_{n-1}. \end{align*} $$

By Lemma 5.4, there exists $w_{n+1}\in [T]$ such that

$$ \begin{align*} d_{\nu}(u_{n+1}(g)\widehat{\gamma^{(n-1)}_g}(w_{n+1})w_{n+1}^{-1},\mathrm{id})<\varepsilon_{n-1}, \quad g\in K_{n-1}, \nu\in \Phi_{n-1}, \end{align*} $$

$$ \begin{align*} \|w_{n+1}(\xi)-\xi\|<\varepsilon_{n-1}, \quad d_{\nu}(\widehat{w_{n+1}}(t),t)<\varepsilon_{n-1}, \quad \xi,\nu\in \Phi_{n-1}, t\in K_{n-1}. \end{align*} $$

Set

$$ \begin{align*} \bar{v}_{n+1}(g)&:=u_{n+1}(g)\widehat{\gamma^{(n-1)}_g}(w_{n+1})w_{n+1}^{-1}, \\ \gamma^{(n+1)}_g&:={}_{{u_{n+1}}}\gamma_g^{(n-1)} =\bar{v}_{n+1}(g)w_{n+1}\gamma^{(n-1)}_g w_{n+1}^{-1}, \\ \theta_{n+1}&:=w_{n+1}\theta_{n-1}. \end{align*} $$

We clearly have conditions $(1.n+1)$ , $(2.n+1)$ , $(3.n+1)$ , $(8.n+1)$ , $(9.n+1)$ , and $(10.n+1)$ . From conditions $(a.n+1)$ , $(b.n+1)$ , $(c.n+1)$ , and $(d.n+1)$ , we obtain conditions $(4.n+1)$ , $(5.n+1)$ , $(6.n+1)$ , and $(7.n+1)$ , respectively. We define $\Phi _{n+1}$ and $\Lambda _{n+1}$ as in conditions $(11.n+1)$ and $(12.n+1)$ , respectively. Then we have finished the $(n+1)$ st step, and the induction is complete.

By the construction, we have

$$ \begin{align*} \gamma_{g}^{(2n)}=v_{2n}(g)\theta_{2n}\alpha_{g}\theta_{2n}^{-1},\quad \gamma_{g}^{(2n+1)}= v_{2n+1}(g)\theta_{2n+1}{\beta}_{g}\theta_{2n+1}^{-1}. \end{align*} $$

We will show that sequences $\{\theta _{2n}\}_n$ , $\{\theta _{2n+1}\}_n$ , $\{v_{2n}(g)\}_n$ , and $\{v_{2n+1}(g)\}_n$ will converge. Fix $k\in \mathbb {N}$ and take $\xi \in \{\xi _i\}_{i=1}^k$ , $t\in \{T^l\}_{|l|\leq k}$ . For $n>k+2$ , we have $\xi , \theta _{n-2}(\xi )\in \Phi _{n-2}$ , $ \widehat {\theta _{n-2}}(t)\in \Lambda _{n-2}$ . Then

$$ \begin{align*} \|\theta_{n}(\xi)-\theta_{n-2}(\xi)\|=\|w_n(\theta_{n-2}(\xi))-\theta_{n-2}(\xi)\|<\varepsilon_{n-2}, \end{align*} $$

$$ \begin{align*} \|\theta_{n}^{-1}(\xi)-\theta_{n-2}^{-1}(\xi)\|=\|w_n^{-1}(\xi)-\xi\|<\varepsilon_{n-2}, \end{align*} $$

and

$$ \begin{align*} d_{\mu}(\widehat{\theta_n}(t),\widehat{\theta_{n-2}}(t))= d_{\mu}(\widehat{w_n}(\widehat{\theta_{n-2}}(t)),\widehat{\theta_{n-2}}(t))<\varepsilon_{n-2} \end{align*} $$

hold by conditions $(9.n)$ and $(10.n)$ . It follows that $\{\theta _{2n}\}_n$ and $\{\theta _{2n+1}\}_n$ are both Cauchy sequences with respect to the metric d on $N[T]$ . (See §2.1 for the definition of d.) Hence, both $\{\theta _{2n}\}_n$ and $\{\theta _{2n+1}\}_n$ converge to some $\sigma _0,\sigma _1\in \overline {[T]}$ , respectively, in the u-topology.

Fix $l\in \mathbb {N}$ and take any $g\in K_l$ . Then for $n>l+2$ , we have $v_{n-2}(g), v_{n-2}(g)^{-1} \in \Lambda _{n-2}$ , $v_{n-2}(g)(\mu )\in \Phi _{n-2}$ . Thus,

$$ \begin{align*} &{ d_{\mu}(v_{n}(g),v_{n-2}(g))} \\ &\quad\leq d_{\mu}(\bar{v}_{n}(g)\widehat{w_{n}}(v_{n-2}(g)), \bar{v}_{n}(g)v_{n-2}(g))+ d_{\mu}(\bar{v}_{n}(g)v_{n-2}(g),v_{n-2}(g))\\ &\quad= d_{\mu}(\widehat{w_{n}}(v_{n-2}(g)), v_{n-2}(g))+ d_{v_{n-2}(g)(\mu)}(\bar{v}_{n}(g),\mathrm{id})\\ &\quad< 2\varepsilon_{n-2} \end{align*} $$

and

$$ \begin{align*} &{d_{\mu}(v_{n}(g)^{-1},v_{n-2}(g)^{-1})} \\ &\quad\leq d_{\mu}(\widehat{w_n}(v_{n-2}(g)^{-1})\bar{v}_{n}(g)^{-1}, \widehat{w_n}(v_{n-2}(g)^{-1})) + d_{\mu}(\widehat{w_n}(v_{n-2}(g)^{-1}),v_{n-2}(g)^{-1})\\ &\quad= d_{\mu}(\bar{v}_{n}(g)^{-1}, \mathrm{id}) +d_{\mu}(\widehat{w_n}(v_{n-2}(g)^{-1}),v_{n-2}(g)^{-1})\\ &\quad<2\varepsilon_{n-2} \end{align*} $$

by conditions $(8.n)$ and $(10.n)$ . Thus, both $\{v_{2n}(g)\}_n$ and $\{v_{2n+1}(g)\}_n$ are Cauchy sequences with respect to $d_{\mu }$ , and hence converge to some $z_0(g), z_1(g)\in [T]$ uniformly, respectively.

Summarizing these results, we have

$$ \begin{align*} \lim_{n\rightarrow \infty}\gamma_{g}^{(2n)}&= \lim_{n\rightarrow \infty}v_{2n}(g)\theta_{2n}\alpha_{g}\theta_{2n}^{-1}= z_0(g)\sigma_0\alpha_g\sigma_0^{-1}, \\ \lim_{n\rightarrow \infty}\gamma_{g}^{(2n+1)}&= \lim_{n\rightarrow \infty}v_{2n+1}(g)\theta_{2n+1}{\beta}_{g}\theta_{2n+1}^{-1}= z_1(g)\sigma_1{\beta}_g\sigma_1^{-1}. \end{align*} $$

By conditions $(4.n)$ and $(5.n)$ , we have $z_0(g)\sigma _0\alpha _g\sigma _0^{-1}=z_1(g)\sigma _1{\beta }_g\sigma _1^{-1}$ . Hence, $\alpha $ and $\beta $ are cocycle conjugate.

Proof of Theorem 2.4

Let $N:=N_{\alpha }=N_{\beta }$ , $Q:=G/N$ , and $\pi :G\rightarrow Q$ be the quotient map. Fix a section $s:Q\rightarrow G$ such that $s(e)=e$ . Then $\alpha _{s(p)}$ is an ultrafree cocycle crossed action of Q. By Theorem 4.4, there exists $v\in C^1(Q,[T])$ such that $\bar {\alpha }_p:=v(p)\alpha _{s(p)}$ is a genuine action of Q. Here define $v(g):=v(p)\alpha _n^{-1}\in [T]$ , where $g=ns(p)$ with $p=\pi (g)$ and $n\in N$ . Then $v(g) \alpha _g=v(p)\alpha _{s(p)}=\bar {\alpha }_{\pi (g)}$ , and $\bar {\alpha }_{\pi (g)}$ is an action of G. Thus, $\alpha _g$ is strongly cocycle conjugate to $\bar {\alpha }_{\pi (g)}$ for some ultrafree action $\bar {\alpha }$ of Q. In the same way, ${\beta }_g$ is strongly cocycle conjugate to $\bar {\beta }_{\pi (g)}$ for some ultrafree action $\bar {\beta }$ of Q. Since $\operatorname {\mathrm {mod}}(\bar {\alpha }_p)=\operatorname {\mathrm {mod}}(\bar {\beta }_p)$ , $\bar {\alpha }$ and $\bar {\beta }$ are strongly cocycle conjugate as actions of Q by Theorem 5.5, and hence also are as actions of G. Therefore, the two actions $\alpha $ and $\beta $ of G are strongly cocycle conjugate.