CONCRETE CLASSIFICATION AND CENTRALIZERS OF
CERTAIN Z2 ⋊ SL(2,Z)-ACTIONS
HIROKI SAKO
Abstract. We introduce a new class of actions of the group Z2 ⋊ SL(2,Z) on
finite von Neumann algebras and call them twisted Bernoulli shift actions. We
classify these actions up to conjugacy and give an explicit description of their
centralizers. We also distinguish many of those actions on the AFD II1 factor in
view of outer conjugacy.
1. Introduction
We consider the classification of Z2 ⋊ SL(2,Z)-actions on finite von Neumann
algebras in this paper. Mainly, we concentrate on the case that the finite von
Neumann algebra is the AFD factor of type II1 or non-atomic abelian.
There are two difficulties for analyzing discrete group actions on operator algebras.
The first is that we do not have various ways to construct actions. The second is
that we can not analyze them by concrete calculation in most cases. To give many
examples of actions which admit concrete analysis, we introduce a class of trace
preserving Z2 ⋊ SL(2,Z)-actions on finite von Neumann algebras and call them
twisted Bernoulli shift actions. We classify those actions up to conjugacy and study
them up to outer conjugacy.
An action β(H, µ, χ) in the class is defined for a triplet (H, µ, χ), where H is
an abelian countable discrete group, µ is a normalized scalar 2-cocycle of H and χ
is a character of H . We obtain the action by restricting the so-called generalized
Bernoulli shift action to a subalgebra N(H, µ) and “twisting” it by the character χ.
The process of restriction has a vital role in concrete analysis of these actions.
A ∗-isomorphism which gives conjugacy between two twisted Bernoulli shift ac-
tions β(Ha, µa, χa) and β(Hb, µb, χb) must be induced from an isomorphism between
the two abelian groups Ha and Hb. We prove this by concrete calculation (Section
4). It turns out that there exist continuously many, non-conjugate Z2 ⋊ SL(2,Z)-
actions on the AFD factor of type II1 (Section 5). By using the same technique, we
describe the centralizers of all twisted Bernoulli shift actions. Here we should men-
tion that the present work was motivated by the previous ones [Ch], [NPS], where
similar studies were carried out in the case of SL(n,Z).
In Section 6, we distinguish many twisted Bernoulli shift actions in view of outer
conjugacy. The classification for actions of discrete amenable groups on the AFD
factor of type II1 was given by Ocneanu [Oc]. Outer actions of countable amenable
groups are outer conjugate. In the contrast to this, V. F. R. Jones [Jon] proved that
any discrete non-amenable group has at least two non outer conjugate actions on
2000 Mathematics Subject Classification. Primary 46L40; Secondary 46L10.
Key words and phrases. von Neumann algebras; automorphisms.
http://arxiv.org/abs/0704.1533v3
2 HIROKI SAKO
the AFD factor of type II1. S. Popa ([Po3], [Po4], [PoSa], etc.) used the malleabil-
ity/deformation arguments for the Bernoulli shift actions to study (weak) 1-cocycles
for the actions. For some of twisted Bernoulli shift actions, which we introduce in
this paper, it is shown that (weak) 1-cocycles are represented in simple forms under
some assumption on the (weak) 1-cocycles. We prove that there exist continuously
many twisted Bernoulli shift actions which are mutually non outer conjugate. This
strengthens the above mentioned result due to Jones in the Z2 ⋊ SL(2,Z) cases.
2. Preparations
2.1. Functions det and gcd. For the definition of twisted Bernoulli shift actions in
Section 3, we define two Z-valued functions det and gcd. The function det is given
by the following equation:
= qr0 − rq0,
∈ Z2.
The value of the function gcd at k ∈ Z2 is the greatest common divisor of the two
entries. For 0 ∈ Z2, let the value of gcd be 0.
Lemma 2.1.
(1) The action of SL(2,Z) on Z2 preserves the functions det and gcd, that is,
det(k, k0) = det(γ · k, γ · k0),
gcd(k) = gcd(γ · k), k, k0 ∈ Z2, γ ∈ SL(2,Z).
(2) The following equation holds true:
det(k, k0) = gcd(k) + gcd(k0)− gcd(k + k0) mod 2, k, k0 ∈ Z2.
Proof. The claim (1) is a well-known fact, so we prove the claim (2). For the function
gcd, we get
1 mod 2, (either q or r is odd),
0 mod 2, (both q and r are even).
Since the action of SL(2,Z) on Z2 preserves the functions det and gcd, it suffices to
show the desired equation against the following four pairs:
(k, k0) =
mod 2.
2.2. Scalar 2-cocycles for abelian groups. We fix some notations for countable
abelian groups and their scalar 2-cocycles. For the rest of this paper, let H be an
abelian countable discrete group and suppose that any scalar 2-cocycle µ : H×H →
T = {z ∈ C | |z| = 1} is normalized, that is, µ(g, 0) = 1 = µ(0, g) for g ∈ H . We
denote by µ∗ the 2-cocycle for H given by µ∗(g, h) = µ(h, g), g, h ∈ H . Let µ∗µ be
the function on H ×H defined by
µ∗µ(g, h) = µ(h, g)µ(g, h), g, h ∈ H.
CLASSIFICATION AND CENTRALIZERS 3
This is a bi-character, that is, µ∗µ(g, ·) and µ∗µ(·, h) are characters of H . By using
this function, we can describe the cohomology class of µ. See [OPT] for the proof
of the following Proposition:
Proposition 2.2. Two scalar 2-cocycles µ1 and µ2 of H are cohomologous if and
only if µ∗1µ1 = µ
Let Cµ(H) be the twisted group algebra of H with respect to the 2-cocycle µ. We
denote by {uh | h ∈ H} the standard basis for Cµ(H) as C-linear space. We recall
that the C-algebra Cµ(H) has a structure of ∗-algebra defined by
ug uh = µ(g, h) ug+h, u
g = µ(g,−g)u−g, g, h ∈ H.
Let µ̃ be the T-valued function on ⊕Z2H ×⊕Z2H defined by
µ̃(λ1, λ2) =
µ(λ1(k), λ2(k)), λ1, λ2 ∈ ⊕Z2H.(Eq1)
The function µ̃ is a normalized scalar 2-cocycle for ⊕Z2H . Let Λ(H) be the abelian
group defined by
Λ(H) =
λ : Z2 → H
∣∣∣∣∣ finitely supported and
λ(k) = 0
Its additive rule is defined by pointwise addition.
2.3. Definition of a Z2 ⋊ SL(2,Z)-action on Λ(H). The group SL(2,Z) acts on
Z2 as matrix-multiplication and the group Z2 also does on Z2 by addition. These
two actions define the action of Z2 ⋊ SL(2,Z) on Z2 which is explicitly described as
q + xq0 + yr0
r + zq0 + wr0
for all ((
∈ Z2 ⋊ SL(2,Z),
∈ Z2.
We define an action of Z2 ⋊ SL(2,Z) on ⊕Z2H as
(γ · λ)(k) = λ(γ−1 · k), k ∈ Z2,
for γ ∈ Z2 ⋊ SL(2,Z) and λ ∈ ⊕Z2H .
2.4. On the relative property (T) of Kazhdan. We give the definition of the
relative property (T) of Kazhdan for a pair of discrete groups.
Definition 2.3. Let G ⊂ Γ be an inclusion of discrete groups. We say that the pair
(Γ, G) has the relative property (T) if the following condition holds:
There exist a finite subset F of Γ and δ > 0 such that if π : Γ → U(H) is a
unitary representation of Γ on a Hilbert space H with a unit vector ξ ∈ H satisfying
‖π(g)ξ − ξ‖ < δ for g ∈ F , then there exists a non-zero vector η ∈ H such that
π(h)η = η for h ∈ G.
Instead of this original definition, we use the following condition.
4 HIROKI SAKO
Proposition 2.4. ([Jol]) Let G ⊂ Γ be an inclusion of discrete groups. The pair
(Γ, G) has the relative property (T) if and only if the following condition holds:
For any ǫ > 0, there exist a finite subset F of Γ and δ > 0 such that if π : Γ → U(H)
is a unitary representation of Γ on a Hilbert space H with a unit vector ξ ∈ H
satisfying ‖π(g)ξ − ξ‖ < δ for g ∈ F , then ‖π(h)ξ − ξ‖ < ǫ for h ∈ G.
The pair (Z2⋊SL(2,Z),Z2) is a typical example of group with the relative property
(T). See [Bu] or [Sh] for the proof.
2.5. Weakly mixing actions. An action of a countable discrete group G on a von
Neumann algebra N is said to be ergodic if any G-invariant element of N is a scalar
multiple of 1. The weak mixing property is a stronger notion of ergodicity.
Definition 2.5. Let N be a von Neumann algebra with a faithful normal state φ.
A state preserving action (ρg)g∈G of a countable discrete group G on N is said to be
weakly mixing if for every finite subset {a1, a2, . . . , an} ⊂ N and ǫ > 0, there exists
g ∈ G such that |φ(aiρg(aj))− φ(ai)φ(aj)| < ǫ, i, j = 1, . . . , n.
The following is a basic characterization of the weak mixing property. Between
two von Neumann algebra N and M , N ⊗ M stands for the tensor product von
Neumann algebra.
Proposition 2.6 (Proposition D.2 in [Vaes]). Let a countable discrete group G act
on a finite von Neumann algebra (N, tr) by trace preserving automorphisms (ρg)g∈G.
The following statements are equivalent:
(1) The action (ρg) is weakly mixing.
(2) The only finite-dimensional invariant subspace of N is C1.
(3) For any action (αg) of G on a finite von Neumann algebra (M, τ), we have
(N ⊗ M)ρ⊗α = 1 ⊗ Mα, where (N ⊗ M)ρ⊗α and Mα are the fixed point
subalgebras.
2.6. A remark on group von Neumann algebras. Let Γ be a discrete group
and let µ be a scalar 2-cocycle of a countable group Γ. A group Γ acts on the Hilbert
space ℓ2Γ by the following two ways;
uγ(δg) = µ(γ, g)δγg, ργ(δg) = µ(g, γ
−1)δgγ−1 , γ, g ∈ Γ.
These two representations commute with each other. The von Neumann algebra
Lµ(Γ) generated by the image of u is called the group von Neumann algebra of
Γ twisted by µ. The normal state 〈·δe, δe〉 is a trace on Lµ(Γ). The vector δe is
separating for Lµ(Γ). For any element a ∈ Lµ(Γ), we define the square summable
function a(·) on Γ by aδe =
a(g)δg. The function a(·) is called the Fourier
coefficient of a. We write a =
g∈Γ a(g)ug and call this the Fourier expansion of
a. The Fourier expansion of a∗ is given by a∗ =
g∈Γ µ(g, g
−1)a(g−1)ug, since the
Fourier coefficient a∗(g) = 〈a∗δe, δg〉 is described as
〈δe, aρg−1δe〉 =
ρ∗g−1δe,
a(g)δg
= µ(g, g−1)a(g−1).
CLASSIFICATION AND CENTRALIZERS 5
Here we used the equation ρ∗
= µ(g, g−1)ρg, which is verified by direct computa-
tion. For two elements a, b, the Fourier coefficient of ab is given by
ab(γ) = 〈bδe, ργ−1a∗δe〉 =
a∗(g)µ(g, γ)b(gγ) =
µ(g−1, gγ)a(g−1)b(gγ).
This equation allows us to calculate the Fourier coefficient algebraically, that is,
µ(g−1, gγ)a(g−1)b(gγ)
a(g)b(h)
For a subgroup Λ ⊂ Γ, the subalgebra {uλ | λ ∈ Λ}′′ ⊂ Lµ(Γ) is isomorphic to
Lµ(Λ). We sometimes identify them. An element a ∈ Lµ(Γ) is in the subalgebra
Lµ(Λ) if and only if the Fourier expansion a(·) : Γ → C is supported on Λ, since
the trace 〈·δe, δe〉 preserving conditional expectation E from Lµ(Γ) onto Lµ(Λ) is
described as E(a) =
λ∈Λ a(λ)uλ.
3. Definition of twisted Bernoulli shift actions
In this section, we introduce twisted Bernoulli shift actions of Z2 ⋊ SL(2,Z) on
finite von Neumann algebras. The action is defined for a triplet i = (H, µ, χ), where
H 6= {0} is an abelian countable discrete group, µ is a normalized scalar 2-cocycle
of H and χ is a character of H . The finite von Neumann algebra, on which the
group Z2 ⋊ SL(2,Z) acts, is defined by the pair (H, µ).
We introduce a group structure on the set Γ0 = Ĥ × Z2 × SL(2,Z) as
(c1, k, γ1)(c2, l, γ2) =
c1c2χ
det(k,γ1·l), k + γ1 · l, γ1γ2
for any c1, c2 ∈ Ĥ, k, l ∈ Z2, γ1, γ2 ∈ SL(2,Z). The associativity is verified by Lemma
2.1. It turns out that the subsets Ĥ = Ĥ × {0} × {e} and G0 = Ĥ × Z2 × {e} are
subgroups in Γ0. It is easy to see that G0 is a normal subgroup of Γ0 and that Ĥ is a
normal subgroup of G0 and Γ0. We get a normal inclusion of groups G0/Ĥ ⊂ Γ0/Ĥ
and this is isomorphic to Z2 ⊂ Z2 ⋊ SL(2,Z).
Before stating the definition of the twisted Bernoulli shift action, we define a Γ0-
action ρ on the von Neumann algebra Lµ̃(⊕Z2H). We denote by u(λ) ∈ Lµ̃(⊕Z2H)
the unitary corresponding to λ ∈ ⊕Z2H . We define a faithful normal trace tr of
Lµ̃(⊕Z2H) in the usual way. For c ∈ Ĥ, k ∈ Z2, γ ∈ SL(2,Z), let ρ(c), ρ(k), ρ(γ) be
the linear transformations on Cµ̃(⊕Z2H) given by,
ρ(c)(u(λ)) =
c(λ(l))
u(λ),
ρ(k)(u(λ)) =
χ(λ(m))det(k,m)
u(k · λ),
ρ(γ)(u(λ)) = u(γ · λ), λ ∈ Λ(H),
These maps are compatible with the multiplication rule and the ∗-operation of
Cµ̃(⊕Z2H) . Since these maps preserve the trace, they extend to ∗-automorphisms
6 HIROKI SAKO
on Lµ̃(⊕Z2H). It is immediate to see that ρ(c) commutes with ρ(k) and ρ(γ). For
k, l ∈ Z2, we have the following relation:
ρ(k) ◦ ρ(l)(u(λ)) =
χ(λ(m))det(l,m)ρ(k)(u(l · λ))
χ(λ(m))det(l,m)
χ((l · λ)(m))det(k,m)u(k · (l · λ))
χ(λ(m))det(l,m)χ(λ(m))det(k,m+l)u((k + l) · λ).
By det(l, m) + det(k,m+ l) = det(k, l) + det(k + l, m), this equals to
χ(λ(m))
)det(k,l) ∏
χ(λ(m))det(k+l,m)u((k + l) · λ)
= ρ(χdet(k,l)) ◦ ρ(k + l)(u(λ)).
Since det is SL(2,Z)-invariant (Lemma 2.1), for k ∈ Z2, γ ∈ SL(2,Z), we get
ρ(γ · k) ◦ ρ(γ)(u(λ)) = ρ(γ · k)(u(γ · λ))
χ((γ · λ)(l))det(γ·k,l)u((γ · k) · (γ · λ))
χ(λ(l))det(γ·k,γ·l)u(γ · (k · λ))
χ(λ(l))det(k,l)ρ(γ)(u(k · λ))
= ρ(γ) ◦ ρ(k)(u(λ)), λ ∈ Λ(H).
By using the above two equations, ρ satisfies the following formula:
(ρ(c1) ◦ ρ(k) ◦ ρ(γ1)) ◦ (ρ(c2) ◦ ρ(l) ◦ ρ(γ2))
= ρ(c1) ◦ ρ(c2) ◦ ρ(k) ◦ ρ(γ1) ◦ ρ(l) ◦ ρ(γ2)
= ρ(c1) ◦ ρ(c2) ◦ ρ(k) ◦ ρ(γ1 · l) ◦ ρ(γ1) ◦ ρ(γ2)
= ρ(c1) ◦ ρ(c2) ◦ ρ(χdet(k,γ1·l)) ◦ ρ(k + (γ1 · l)) ◦ ρ(γ1γ2)
= ρ(c1c2χ
det(k,γ1·l)) ◦ ρ(k + (γ1 · l)) ◦ ρ(γ1γ2).
With ρ(c, k, γ) = ρ(c) ◦ ρ(k) ◦ ρ(γ), ρ gives a Γ0-action on Lµ̃(⊕Z2H).
We define the finite von Neumann algebra N(H, µ) as the group von Neumann
algebra Lµ̃(Λ(H)). By using Fourier coefficients, we can prove that N(H, µ) is the
fixed point algebra under the Ĥ-action ρ(Ĥ, 0, e) on Lµ̃(⊕Z2H). We get a Z2 ⋊
SL(2,Z)-action on N(H, µ) by
β(k, γ)(x) = ρ(1, k, γ)(x), k ∈ Z2, γ ∈ SL(2,Z), x ∈ N(H, µ).
This is the definition of the twisted Bernoulli shift action β = β(H, µ, χ) on N(H, µ).
We obtained the actions β(H, µ, χ) not only by twisting generalized Bernoulli shift
actions but also restricting to subalgebras N(H, µ) ⊂ Lµ̂(⊕Z2H) =
Lµ(H).
This restriction allows us to classify the actions up to conjugacy in the next section.
CLASSIFICATION AND CENTRALIZERS 7
In order to give a variety of the actions, we twisted the shift actions by the character
χ of the abelian group H .
Remark 3.1. The action β|Z2 = β(H, µ, χ)|Z2 has the weak mixing property. In
definition 2.5, we may assume that the Fourier coefficients of ai (i = 1, 2, · · · , n) are
finitely supported, by approximating in the L2-norm. Then for appropriate k ∈ Z2,
we get tr(aiβ(k)(aj)) = tr(ai)tr(aj), i, j = 1, 2, · · · , n.
4. Classification up to conjugacy
In this section, we classify the twisted Bernoulli shift actions {β(H, µ, χ)} up to
conjugacy (Theorem 4.1). We prove that an isomorphism which gives conjugacy
between two twisted Bernoulli shift actions is of a very special form. In fact it
comes from an isomorphism in the level of base groups H . We also determine the
centralizer of the Z2 ⋊ SL(2,Z)-action β(H, µ, χ) on N(H, µ) (Theorem 4.4).
We fix some notations for the proofs. We define 0, e1, e2 ∈ Z2 as
, e1 =
, e2 =
Let ξ be the element of Z2 ⋊ SL(2,Z) satisfying
ξ · 0 = e1, ξ · e1 = e2, ξ · e2 = 0.
The elements ξ and ξ2 are explicitly described as
−1 −1
−1 −1
The order of ξ is 3. Let η, δ ∈ SL(2,Z) be given by η =
, δ =
Let D be the subset of all elements of Z2 fixed under the action of δ, that is,
n ∈ Z
Then we get
ξ ·D =
n ∈ Z
, ξ2 ·D =
n ∈ Z
We define the subgroup ΛD(H) of Λ(H) by
ΛD(H) = {λ ∈ Λ(H) | λ : Z2 → H is supported on D}.
Let (Ha, µa, χa) and (Hb, µb, χb) be triplets of countable abelian groups, their
normalized 2-cocycles and characters. For h ∈ Ha, we define λh ∈ ΛD(Ha) as
λh(k) =
h (k = e1),
−h (k = 0),
0 (k 6= e1, 0).
8 HIROKI SAKO
For g ∈ Hb, we define σg ∈ ΛD(Hb) as
σg(k) =
g (k = e1),
−g (k = 0),
0 (k 6= e1, 0).
We denote by v(σ) ∈ N(Hb, µb) the unitary corresponding to σ ∈ Λ(Hb).
Theorem 4.1. If π : N(Ha, µa) → N(Hb, µb) is a ∗-isomorphism giving conjugacy
between βa = β(Ha, µa, χa) and βb = β(Hb, µb, χb), then there exists a group isomor-
phism φ = φπ : Ha → Hb satisfying
(1) π(u(λ)) = v(φ ◦ λ) mod T for λ ∈ Λ(Ha),
(2) the 2-cocycles µa(·, ·) and µb(φ(·), φ(·)) of Ha are cohomologous,
(3) χ2a = (χb ◦ φ)2.
Conversely, given a group isomorphism φ : Ha → Hb satisfying (2) and (3), there
exists a ∗-isomorphism π = πφ : N(Ha, µa) → N(Hb, µb) which satisfies condition
(1) and gives conjugacy between βa, βb.
We note that by Proposition 2.2 condition (2) for φ is equivalent to
(2)′ µ∗aµa(g, h) = µ
bµb(φ(g), φ(h)), g, h ∈ Ha.
Proof for the first half of Theorem 4.1.
Suppose that there exists a (not necessarily trace preserving) ∗-isomorphism π from
N(Ha, µa) onto N(Hb, µb) such that π ◦ βa(γ) = βb(γ) ◦ π, γ ∈ Z2 ⋊ SL(2,Z).
We prove that for every h ∈ Ha there exists φ(h) ∈ Hb satisfying π(u(λh)) =
v(σφ(h)) mod T. Let Uh denote the unitary in N(Hb, µb)
Uh = π
µa(h,−h) u(λh)
, h ∈ Ha.
We identify N(H, µ) with the subalgebra of the infinite tensor product
Lµ(H),
which is canonically isomorphic to Lµ̃ (⊕Z2H). The preimage π−1(Uh) can be written
as u∗h ⊗ uh. Here uh is the unitary corresponding to h ∈ Ha and placed on 1 ∈ Z2
and the unitary u∗h is placed on 0 ∈ Z2. We describe Uh as the Fourier expansion
σ∈Λ(Hb)
c(σ)v(σ). Since e1 and 0 are fixed under the action of δ, one has
βb(δ)
n(Uh) = π ◦ βa(δ)n(π−1(Uh)) = Uh.
It follows that the Fourier expansion Uh =
σ∈Λ(Hb)
c(σ)v(σ) must satisfy that
c(σ) = c(δ−n · σ) for every σ ∈ Λ(Hb) and n ∈ Z. For σ ∈ Λ(Hb) \ ΛD(Hb), the
orbit of σ under the action of δ−1 is an infinite set, since the support supp(σ) ⊂ Z2
is not included in D. It turns out that c(σ) = 0 for all σ ∈ Λ(Hb) \ ΛD(Hb) due to
Σ|c(σ)|2 = 1 < +∞, so that Uh =
σ∈ΛD(Hb)
c(σ)v(σ).
The unitary χa(h)U
h is also fixed under the action of δ and can be written as
χa(h)U
h = π
χa(h)µa(h,−h)u(−λh)
χa(h)µa(h,−h)u(ξ · λh)
µa(h,−h)u(ξ2 · λh)
= βb(ξ)(Uh) βb(ξ
2)(Uh).
CLASSIFICATION AND CENTRALIZERS 9
Letting ne1 = (n, 0)
T ∈ Z2, we get
βb(ξ)(Uh) = βb(ξ)
c(σ) v(σ)
σ∈ΛD(Hb)
c(σ) v(ξ · σ)
χb(σ(ne1))
2)(Uh) = βb(ξ
c(σ) v(σ)
σ∈ΛD(Hb)
c(σ) v(ξ2 · σ).
Since Fourier expansion admits algebraical calculation as in subsection 2.6, the ex-
pansion of χa(h)U
χa(h)U
h = βb(ξ)(Uh) βb(ξ
2)(Uh)
σ1,σ2∈ΛD(Hb)
c(σ1) c(σ2) v(ξ · σ1) v(ξ2 · σ2)
χb(σ1(ne1))
σ1,σ2∈ΛD(Hb)
c(σ1) c(σ2) µ̃b(ξ · σ1, ξ2 · σ2)
χb(σ1(ne1))
n v(ξ · σ1 + ξ2 · σ2).
The map ΛD(Hb)×ΛD(Hb) ∋ (σ1, σ2) 7→ ξ ·σ1+ξ2·σ2 ∈ Λ(Hb) is injective. Indeed, σ1
is uniquely determined by ξ ·σ1+ξ2·σ2, since σ1(k) = (ξ ·σ1+ξ2·σ2)(ξ ·k), k ∈ D\{e1}
and σ1(e1) = −
k∈D\{e1}
σ1(k). Here we used the condition
σ1(k) = 0. The
element σ2 is also determined by ξ · σ1 + ξ2 · σ2. Thus the index (σ1, σ2) uniquely
determines ξ · σ1 + ξ2 · σ2.
We take arbitrary elements σ1, σ2 ∈ ΛD(Hb) and suppose that c(σ1) 6= 0, c(σ2) 6= 0.
Since the unitary χa(h)U
h is invariant under the action of δ and the coefficient of
ξ ·σ1+ξ2 ·σ2 is not zero, ξ ·σ1+ξ2 ·σ2 is supported on D. It follows that the elements
σ1 and σ2 can be written as σ1 = σφ(h) = σ2, by some φ(h) ∈ Hb. Indeed, since the
subsets D \ {0, e1}, ξD \ {e1, e2} and ξ2D \ {e2, 0} are mutually disjoint, the element
ξ · σ1 must be supported on {e1, e2} and the element ξ2 · σ2 must be supported on
{e2, 0}. By the assumption
k∈Z2 σi(k) = 0 (i = 1, 2), σi can be written as σφ(hi).
Then using the fact that (ξ · σ1 + ξ2 · σ2)(e2) = σ1(e1) + σ2(0) = 0, we get that
σ1 = σφ(h) = σ2 for some h ∈ Hb. This means that there exists only one σ ∈ Λ(Hb)
such that c(σ) 6= 0 and that it is of the form σ = σφ(h). Then the unitary Uh satisfies
Uh = π(u(λh)) = v(σφ(h)) mod T.
We claim that the map φ = φπ : Ha → Hb is a group isomorphism. For all
h1, h2 ∈ Ha, we get
π(u(λh1+h2)) = π(u(λh1)) π(u(λh2)) = v(σφ(h1)) v(σφ(h2))
= v(σφ(h1) + σφ(h2)) = v(σφ(h1)+φ(h2)) mod T.
On the other hand, we get π(u(λh1+h2)) = v(σφ(h1+h2)) mod T. Since {v(σ)} are
linearly independent, we get σφ(h1+h2) = σφ(h1)+φ(h2), and hence
φ(h1 + h2) = φ(h1) + φ(h2).
This means that the map φ is a group homomorphism. The bijectivity of the ∗-
isomorphism π leads to that of the group homomorphism φ = φπ. Since {γ ·λh | γ ∈
Z2⋊SL(2,Z), h ∈ Ha} ⊂ Λ(Ha) generates Λ(Ha), we get π(u(λ)) = v(φ◦λ) mod T
for λ ∈ Λ(Ha).
10 HIROKI SAKO
We prove that the group isomorphism φ = φπ satisfies conditions (2) and (3) in
the theorem. For all h ∈ Ha, there exists c(h) ∈ T satisfying
Uh = π
µa(h,−h) u(λh)
= c(h)µb(φ(h),−φ(h)) v(σφ(h)).
Since (e1, η) ∈ Z2 ⋊ SL(2,Z) acts on Z2 as (e1, η) · e1 = 0, (e1, η) · 0 = e1, we get
Uh βb(e1, η)(Uh) = π
µa(h,−h) u(λh)µa(h,−h) u(−λh)
= µa(h,−h)
µ̃a(λh,−λh) = 1.
The following equation also holds:
Uh βb(e1, η)(Uh) = c(h)µb(φ(h),−φ(h)) v(σφ(h)) c(h)µb(φ(h),−φ(h)) v(σ−φ(h))
= c(h)2 µb(φ(h),−φ(h))
µ̃b(σφ(h),−σφ(h)) = c(h)2.
Thus we have c(h) ∈ {1,−1} for h ∈ Ha.
Since ξ · e1 = e2, ξ · e2 = 0 and ξ · 0 = e1, we have
Uh βb(ξ)(Uh) βb(ξ
2)(Uh)
µa(h,−h) u(λh)
χa(h)µa(h,−h) u(ξ · λh)
µa(h,−h) u(ξ2 · λh)
= χa(h).
On the other hand, we have the following:
Uh βb(ξ)(Uh) βb(ξ
2)(Uh) = c(h)µb(φ(h),−φ(h)) v(σφ(h))
c(h)χb(φ(h))µb(φ(h),−φ(h)) v(ξ · σφ(h))
c(h)µb(φ(h),−φ(h)) v(ξ2 · σφ(h))
= c(h)3 χb(φ(h)) = c(h)χb(φ(h)).
It follows that
c(h) = χb(φ(h))χa(h)(Eq2)
and χb(φ(h))
2 = χa(h)
2, for all h ∈ Ha.
We recall that the algebra Lµ̃a(⊕Z2Ha) is canonically identified wit the infinite
tensor product
Lµa(Ha). The unitary π
−1(Uh) ∈ N(Ha, µa) ⊂
Lµa(Ha)
can be written as 1⊗ u∗h ⊗ uh, where 1 is placed on −e1 ∈ Z2, u∗h is placed on 0 and
uh is placed on e1. Since η ∈ Z2 ⋊ SL(2,Z) acts on Z2 as η · e1 = −e1, η · 0 = 0,
the unitary π−1(βb(η)(Ug)) can be written as ug ⊗ u∗g ⊗ 1. We have the following
equation:
Ug βb(η)(Uh)U
g βb(η)(Uh)
= π((1⊗ u∗g ⊗ ug)(uh ⊗ u∗h ⊗ 1)(1⊗ u∗g ⊗ ug)∗(uh ⊗ u∗h ⊗ 1)∗) = µ∗aµa(g, h).
The unitary Uh can be written as c(h)(1⊗ v∗φ(h) ⊗ vφ(h)) ∈ N(Hb, µb) ⊂
Lµb(Hb).
Here we write vφ(h) for the unitary in Lµb(Hb) corresponding to φ(h). The unitary
βb(η)(Ug) can be written as c(g)(vφ(g) ⊗ v∗φ(g) ⊗ 1). Then we get
Ug βb(η)(Uh)U
g βb(η)(U
= (1⊗ v∗φ(g) ⊗ vφ(g))(vφ(h) ⊗ v∗φ(h) ⊗ 1)(1⊗ v∗φ(g) ⊗ vφ(g))∗(vφ(h) ⊗ v∗φ(h) ⊗ 1)∗
= µ∗bµb(φ(g), φ(h)).
CLASSIFICATION AND CENTRALIZERS 11
Thus we get µ∗aµa(g, h) = µ
bµb(φ(g), φ(h)), for all g, h ∈ Ha. We proved that the
group isomorphism φ = φπ satisfies conditions (1), (2) and (3). �
From a group homomorphism which satisfies conditions (2) and (3), we construct
a ∗-homomorphism from N(Ha, µa) to N(Hb, µb) with condition (1). In the con-
struction, the function µ̂ on Λ(H) given below is useful. We fix an index for Z2 as
Z2 = {k0, k1, k2, · · · } throughout the rest of this section. For a scalar 2-cocycle µ of
H , we define the function µ̂ by
µ̂(λ) =
λ(ki), λ(kj)
, λ ∈ Λ(H),
where λ is supported on {k0, k1, k2, · · · , kn}. This definition depends on the choice
of an order on Z2. Since
i λ(ki) = 0, the function µ̂ is also given by the following
relation in Cµ(H):
µ̂(λ)1 = uλ(k0)uλ(k1)uλ(k2) · · ·uλ(kn), λ ∈ Λ(H).
If µ is a coboundary, then the definition of µ̂ does not depend on the order on Z2,
since Cµ(H) is commutative.
Lemma 4.2. Let µ0 be another normalized scalar 2-cocycle for H. Let µ̃0 be the
scalar 2-cocycle on Λ(H)×Λ(H) given in the same way as equation (Eq1) in subsec-
tion 2.2 and let µ̂0 be the function on Λ(H) constructed from µ0 in the above manner.
If the scalar 2-cocycles µ and µ0 are cohomologous, then for all λ1, λ2 ∈ Λ(H), we
have the equation
µ̃(λ1, λ2) µ̂(λ1) µ̂(λ2) µ̂(λ1 + λ2) = µ̃0(λ1, λ2) µ̂0(λ1) µ̂0(λ2) µ̂0(λ1 + λ2).
Proof. We denote by {ν(g, h)} the scalar 2-cocycle {µ0(g, h)µ(g, h)} of H . Since ν
is a 2-coboundary, there exists {c(g)}g∈H ⊂ T satisfying ν(g, h) = b(g)b(h)b(g + h).
Then the map ν̂ becomes ν̂(λ) =
b(λ(ki)). Since
ν̂(λ1) ν̂(λ2) =
b(λ1(ki)) b(λ2(ki)),
ν̃(λ1, λ2) =
b(λ1(ki)) b(λ2(ki)) b(λ1(ki) + λ2(ki)),
ν̂(λ1 + λ2) =
b(λ1(ki) + λ2(ki)),
we get ν̂(λ1) ν̂(λ2) = ν̃(λ1, λ2) ν̂(λ1 + λ2). By the definitions of µ̃, µ̃0, µ̂ and µ̂0, the
maps ν̂ and ν̃ are given by
ν̂(λ) = µ̂(λ) µ̂0(λ), ν̃(λ1, λ2) = µ̃(λ1, λ2) µ̃0(λ1, λ2),
Thus the desired equality immediately follows. �
Proof for the second half of Theorem 4.1.
Suppose that there exists a group isomorphism φ satisfying conditions (2) and (3)
in the theorem. We prove that there exists a ∗-isomorphism π = πφ from N(Ha, µa)
onto N(Hb, µb) preserving the Z
2 ⋊ SL(2,Z)-actions with condition (1).
We define a group homomorphism cφ from Ha to {1,−1} ⊂ T by
cφ(h) = χb(φ(h))χa(h), h ∈ Ha.
12 HIROKI SAKO
Let c̃φ be the group homomorphism from Λ(Ha) to {1,−1} ⊂ T given by
c̃φ(λ) =
cφ(λ(k))
gcd(k) =
χa(λ(k))
gcd(k)χb(φ(λ(k)))
gcd(k)
, λ ∈ Λ(Ha).
We define a linear map π from the group algebra Cµ̃a(Λ(Ha)) onto Cµ̃b(Λ(Hb)) by
µ̂a(λ) u(λ)
= c̃φ(λ) µ̂b(φ ◦ λ) v(φ ◦ λ), λ ∈ Λ(Ha).
By direct computations, for all λ1, λ2 ∈ Λ(Ha), we get
µ̂a(λ1) u(λ1)
µ̂a(λ2)u(λ2)
= c̃φ(λ1) c̃φ(λ2) µ̂b(φ ◦ λ1) µ̂b(φ ◦ λ2) v(φ ◦ λ1) v(φ ◦ λ2)
= c̃φ(λ1 + λ2) µ̂b(φ ◦ λ1) µ̂b(φ ◦ λ2) µ̃b(φ ◦ λ1, φ ◦ λ2) v(φ ◦ (λ1 + λ2)).
On the other hand, we have the following equation:
µ̂a(λ1) µ̂a(λ2)u(λ1) u(λ2)
µ̂a(λ1) µ̂a(λ2) µ̃a(λ1, λ2) u(λ1 + λ2)
= µ̂a(λ1) µ̂a(λ2) µ̃a(λ1, λ2) µ̂a(λ1 + λ2) π
µ̂a(λ1 + λ2) u(λ1 + λ2)
= c̃φ(λ1 + λ2) µ̂a(λ1) µ̂a(λ2) µ̃a(λ1, λ2) µ̂a(λ1 + λ2)
µ̂b(φ ◦ (λ1 + λ2)) v(φ ◦ (λ1 + λ2)).
By Lemma 4.2 and condition (2) for the group isomorphism φ in the theorem, we
have that
µ̂b(φ ◦ λ1) µ̂b(φ ◦ λ2) µ̃b(φ ◦ λ1, φ ◦ λ2)
= µ̂a(λ1) µ̂a(λ2) µ̃a(λ1, λ2) µ̂a(λ1 + λ2) µ̂b(φ ◦ (λ1) + φ ◦ (λ2)).
Therefore we get π(u(λ1)) π(u(λ2)) = π(u(λ1) u(λ2)). The linear map π also pre-
serves the ∗-operation. As a consequence, π is a ∗-isomorphism from Cµ̃a(Λ(Ha))
onto Cµ̃b(Λ(Hb)) and this preserves the trace. The map π = πφ is extended to a
normal ∗-isomorphism from N(Ha, µa) onto N(Hb, µb).
We next prove that this π preserves the Z2 ⋊ SL(2,Z)-actions. The group ho-
momorphism c̃φ from Λ(Ha) to {1,−1} is invariant under the action of SL(2,Z),
by Lemma 2.1 (1). The scalar 2-cocycle ν(g, h) = µa(g, h)µb(φ(h), φ(g)) satisfies
ν(g, h) = ν(h, g) by condition (2), so the function ν̂(·) = µ̂a(·)µ̂b(φ ◦ ·) on Λ(Ha)
does not depend on the order on Z2 chosen before. Since
π ◦ βa(γ)(u(λ)) = π(u(γ · λ))
= c̃φ(γ · λ)µ̂a(γ · λ)µ̂b(φ ◦ (γ · λ))v(φ ◦ (γ · λ))
= c̃φ(λ)µ̂a(λ)µ̂b(φ ◦ λ)v(γ · (φ ◦ λ))
= βb(γ)
c̃φ(λ)µ̂a(λ)µ̂b(φ ◦ λ)v(φ ◦ λ)
= βb(γ) ◦ π(u(λ)), γ ∈ SL(2,Z), λ ∈ Λ(Ha),
it turns out that the ∗-isomorphism π preserves the SL(2,Z)-action.
CLASSIFICATION AND CENTRALIZERS 13
For all λ ∈ Λ(Ha) and k ∈ Z2, we have
π ◦ βa(k)(u(λ)) = π
χa(λ(l))
det(k,l)u(k · λ)
cφ((k · λ)(l))gcd(l)
χa(λ(l))
det(k,l)µ̂a(k · λ) µ̂b(φ ◦ (k · λ)) v(φ ◦ (k · λ)).
Since cφ(h)
det(k,l)cφ(h)
gcd(k+l) = cφ(h)
gcd(k)cφ(h)
gcd(l), by Lemma 2.1 (2), the unitary
π ◦ βa(k)(u(λ)) equals to
cφ(λ(l))
gcd(k+l)
cφ(λ(l))
det(k,l)χb(φ ◦ λ(l))det(k,l)
µ̂a(k · λ) µ̂b(φ ◦ (k · λ)) v(φ ◦ (k · λ))
cφ(λ(l))
gcd(k)
cφ(λ(l))
gcd(l)
χb(φ ◦ λ(l))det(k,l)
µ̂a(k · λ) µ̂b(φ ◦ (k · λ)) v(k · (φ ◦ λ))
= c̃φ(λ) µ̂a(k · λ) µ̂b(k · (φ ◦ λ))βb(k)(v(φ · λ))
= βb(k) ◦ π(u(λ)).
This means that the ∗-isomorphism π preserves the Z2-actions.
We get the ∗-isomorphism π = πφ fromN(Ha, µa) ontoN(Hb, µb) giving conjugacy
between βa and βb. �
Remark 4.3. The proof of the first half of Theorem 4.1 shows that any isomorphism
π giving conjugacy between βa and βb is of the form πφ. This means that an
isomorphism which gives conjugacy between two twisted Bernoulli shift actions must
be trace preserving.
This proof shows that an isomorphism giving conjugacy between the two actions
β(Ha, µa, χa), β(Hb, µb, χb) is of a very special form derived from a group isomor-
phism between Ha and Hb. Taking notice of this fact, we can describe the centralizer
of a twisted Bernoulli shift action. We define two topological groups before we state
Theorem 4.4.
Let β be a trace preserving action of some group Γ on a separable finite von
Neumann algebra (N, tr). We denote by Aut(N, β) the group of all automorphisms
which commute with the action β, that is,
{α ∈ Aut(N) | β(γ) ◦ α = α ◦ β(γ), γ ∈ Γ}.
We regard the group Aut(N, β) as a topological group equipped with the pointwise-
strong topology. When β is a twisted Bernoulli shift action on N , an automorphism
α commuting with β is necessarily trace preserving by Remark 4.3. We consider
that Aut(N, β) is equipped with the pointwise-2-norm topology.
Let Aut(H, µ, χ) be the group of all automorphisms of an abelian group H which
preserve its 2-cocycle µ and character χ, that is,
{φ ∈ Aut(H) | µ(g, h) = µ(φ(g), φ(h)), χ(g) = χ(φ(g)), g, h ∈ H}.
We define the topology of Aut(H, µ, χ) by pointwise convergence.
14 HIROKI SAKO
Theorem 4.4. For π ∈ Aut(N(H, µ), β(H, µ, χ)), there exists a unique element
φ = φπ ∈ Aut(H, µ∗µ, χ2) satisfying π(u(λ)) = u(φ ◦ λ) mod T for λ ∈ Λ(H). The
map π 7→ φπ gives an isomorphism between two topological groups
Aut(N(H, µ), β(H, µ, χ)) ∼= Aut(H, µ∗µ, χ2).
Proof. We use the notations in the proof of the previous theorem letting Ha = Hb =
H , µa = µb = µ and χa = χb = χ. Denote N = N(H, µ) and β = β(H, µ, χ). We
have already have shown the first claim. Let
Aut(H, µ∗µ, χ2) ∋ φ 7→ πφ ∈ Aut(N, β),
be the map given as in the proof of Theorem 4.1, that is,
µ̂(λ)u(λ)
= c̃φ(λ) µ̂(φ ◦ λ)u(φ ◦ λ), λ ∈ Λ(H),
where
c̃φ(λ) =
χ(λ(k))gcd(k) χ(φ ◦ λ(k))
gcd(k)
It is easy to prove that φπφ = φ by the definition. Thus the map π 7→ φπ is
surjective. This map is also injective. Let φ be an element of Aut(H, µ∗µ, χ2).
Suppose that π is an arbitrary element of Aut(N, β) satisfying φ = φπ. The set
{β(γ)(u(λh)) | h ∈ H, γ ∈ Z2 ⋊ SL(2,Z)} generates N , so we have only to prove
the uniqueness of c(h) ∈ T satisfying
µ(h,−h) u(λh)
= c(h)µ(φ(h),−φ(h))u(λφ(h)),
for all h ∈ H . In the proof of the first half of the previous theorem (equation
(Eq2)), we have already shown that c(h) = χ(h)χ(φ(h)). Thus the ∗-isomorphism
π is uniquely determined and the map π 7→ φπ is injective.
We prove the two maps φ 7→ πφ and π 7→ φπ are continuous. Let (φi) be a net in
Aut(H, µ∗µ, χ2) converging to φ. For all h ∈ H , we have
χ(h)µ(h,−h) u(λh)
= χ(φi(h))µ(φi(h),−φi(h))u(λφi(h)).
The right side of the equation converges to
χ(φ(h))µ(φ(h),−φ(h))u(λφ(h)) = πφ
χ(h)µ(h,−h) u(λh)
This proves that πφi converges to πφ in pointwise 2-norm topology on the generating
set {β(γ)(u(λh)) | h ∈ H, γ ∈ Z2 ⋊SL(2,Z)} of N . Thus πφi converges to πφ on N .
Conversely, let (πi) be a net in Aut(N, β) converging to π. For all h ∈ H ,
we get πi(u(λh)) = u(λφπi(h)) mod T. The left side of the equation converges to
π(u(λh)) = u(λφπ(h)). If φπi(h) 6= φπ(h), then the distance between Tu(λφπ(h)) and
Tu(λφπi (h)) is
2 in the 2-norm. Thus φπi(h) = φπ(h) for large enough i. This
means that (φπi) converges to φπ.
As a consequence, the two maps φ 7→ πφ and π 7→ φπ are continuous group
homomorphisms and inverse maps of each other. �
CLASSIFICATION AND CENTRALIZERS 15
5. Examples
5.1. Twisted Bernoulli shift actions on L∞(X). In this subsection, we consider
the case of µ = 1 and H 6= {0}. Then the algebra N(H, 1) is abelian and has a
faithful normal state, so it is isomorphic to L∞(X), whereX is a standard probability
space. The measure of X is determined by the trace on N(H, 1). Furthermore, X is
non-atomic, since N(H, 1) is infinite dimensional and the action β(H, 1, χ) is ergodic.
As corollaries of Theorems 4.1 and 4.4, we get trace preserving Z2⋊SL(2,Z)-actions
on L∞(X) whose centralizers are isomorphic to some prescribed groups.
Remark 5.1. The Z2 ⋊ SL(2,Z)-action on X defined by β = β(H, 1, χ) is free.
An automorphism α ∈ Aut(L∞(X), β) is free or the identity map for any twisted
Bernoulli shift action β on L∞(X). This is proved as follows. We identify β(γ)
(γ ∈ Z2 ⋊ SL(2,Z)) and α with measure preserving Borel isomorphisms on X here.
Suppose that there exists a non-null Borel subset Y ⊂ X whose elements are fixed
under α. All elements in Ỹ = ∪{β(γ)(Y ) | γ ∈ Z2⋊SL(2,Z)} are fixed under α. By
the ergodicity of β, the measure of Ỹ is 1. Then α is the identity map of L∞(X).
Corollary 5.2. For any abelian countable discrete group H 6= {0}, there exists a
trace preserving essentially free ergodic action β of Z2⋊SL(2,Z) on L∞(X) satisfying
Aut(L∞(X), β) ∼= Aut(H).
Proof. When we define β = β(H, 1, 1), we have the above relation by Theorem
4.4. �
In the next corollary we use the effect of twisting by a character χ.
Corollary 5.3. For every abelian countable discrete group H 6= {0}, there exist
continuously many trace preserving essentially free ergodic actions {βc} of Z2 ⋊
SL(2,Z) on L∞(X) which are mutually non-conjugate and satisfy
Aut(L∞(X), βc) ∼= H ⋊Aut(H).
Here the topology of H ⋊ Aut(H) is the product of the discrete topology on H and
the pointwise convergence topology on Aut(H).
Proof. Let c ∈ {eiπt | t ∈ (0, 1/2) \Q}. We put βc = β(H ⊕ Z, 1, 1× χc), where the
character χc of Z is defined as χc(n) = c
n. By Theorem 4.4, we get
Aut(L∞(X), βc) ∼= Aut(H ⊕ Z, 1, 1× χ2c).
Since the character χ2c is injective, a group automorphism α ∈ Aut(H⊕Z, 1, 1×χ2c)
preserves the second entry. For all α ∈ Aut(H⊕Z, 1, 1×χ2c), there exist φα ∈ Aut(H)
and hα ∈ H satisfying
α(h, n) = (φα(h) + nhα, n), (h, n) ∈ H ⊕ Z.
The map Aut(H ⊕ Z, 1, 1× χ2c) ∋ α 7→ (hα, φα) ∈ H ⋊ Aut(H) is a homeomorphic
group isomorphism.
If c1, c2 ∈ {eiπt | t ∈ (0, 1/2) \ Q} and c1 6= c2, then there exists no isomorphism
from H ⊕ Z to H ⊕ Z whose pull back of the character 1 × χ2c2 is equal to 1 × χ
The two actions βc1 and βc2 are not conjugate by Theorem 4.1. �
16 HIROKI SAKO
Corollary 5.4. There exist continuously many trace preserving essentially free er-
godic actions {βc} of Z2 ⋊ SL(2,Z) on L∞(X) which are mutually non-conjugate
and have the trivial centralizer Aut(L∞(X), βc) = {idL∞X}.
Proof. Let {χc | c = eiπt, t ∈ (0, 1/2)} be characters of Z such that χc(m) = cm.
Since χ2c(1) is in the upper half plane, the identity map is the only automorphism
of Z preserving χ2c . By Theorem 4.4, we get Aut(β(Z, 1, χc)) = {id}.
If β(Z, 1, χc1), β(Z, 1, χc2) are conjugate, then there exists a group isomorphism
on Z whose pull back of χ2c2 is χ
by Theorem 4.1. This means c1 = c2. Thus the
actions {β(Z, 1, χc)} are mutually non-conjugate. �
5.2. Twisted Bernoulli shift actions on the AFD factor of type II1. Firstly,
we find a condition that the finite von Neumann algebra N(H, µ) is the AFD factor
of type II1.
Lemma 5.5. For an abelian countable discrete group H 6= {0} and its normalized
scalar 2-cocycle µ, the following statements are equivalent:
(1) The algebra N(H, µ) is the AFD factor of type II1.
(2) The group von Neumann algebra Lµ(H) twisted by the scalar 2-cocycle µ is
a factor (of type II1 or In).
(3) For all g ∈ H \ {0}, there exists h ∈ H such that µ(g, h) 6= µ(h, g).
Proof. The amenability of the group Λ(H) leads the injectivity for N(H, µ). The
injectivity for N(H, µ) implies that N(H, µ) is approximately finite dimensional
([Co]). We have only to show the equivalence of conditions (2), (3) and
(1)′ The algebra N(H, µ) is a factor.
By using Fourier expansion it is easy to see that condition (2) holds true if and only
if for any g ∈ H \ {0} there exists h ∈ H satisfying uguh 6= uhug. This is equivalent
to condition (3). Similarly, condition (1)′ is equivalent to
(1)′′ For any λ1 ∈ Λ(H) \ {0}, there exists λ2 ∈ Λ(H) satisfying
µ̃(λ2, λ1)µ̃(λ1, λ2) 6= 1.
Suppose condition (3). For any λ1, choose element k, l ∈ Z2 so that k ∈ supp(λ1)
and l /∈ supp(λ1). By condition (3), there exists h ∈ H satisfying µ∗µ(λ1(k), h) 6= 1.
Let λ2 be the element in Λ(H) which takes h at k, −h at l and 0 for the other
places. The element λ2 satisfies µ̃(λ2, λ1)µ̃(λ1, λ2) = µ
∗µ(λ1(k), h) 6= 1. Here we get
condition (1)′′. The implication from (1)′′ to (3) is easily shown. �
Remark 5.6. The twisted Bernoulli shift action β = β(H, µ, χ) is an outer action
of Z2 ⋊ SL(2,Z). Any non-trivial automorphism in Aut(R, β(H, µ, χ)) is also outer.
This is proved by the weak mixing property of the action β(H, µ, χ) as follows. If
α ∈ Aut(R, β(H, µ, χ)) is an inner automorphism Ad(u), then we have
Ad(β(γ)(u))(x) = β(γ)(uβ(γ)−1(x)u∗) = β(γ) ◦ α ◦ β(γ)−1(x)
= α(x) = Ad(u)(x),
for all x ∈ R and γ ∈ Z2 ⋊ SL(2,Z). Since Ad(β(γ)(u)u∗) = id, Cu ⊂ R is an
invariant subspace of the action β. The only subspace invariant under the weakly
mixing action β is C1 (Proposition 2.6), thus we get α = id.
CLASSIFICATION AND CENTRALIZERS 17
Using Theorems 4.1 and 4.4, we give continuously many actions of Z2 ⋊ SL(2,Z)
on R such that there exists no commuting automorphism except for trivial one.
Corollary 5.7. There exist continuously many ergodic outer actions {βc} of Z2 ⋊
SL(2,Z) on the AFD factor R of type II1 which are mutually non-conjugate and
have the trivial centralizer Aut(R, βc) = {idR}.
Proof. We can choose and fix a character χ on Z2 such that χ2 is injective. Let
{µc | c = eiπt, t ∈ (0, 1/2) \Q} be scalar 2-cocycles for Z2 defined by
= cs1t2−t1s2, s1, t1, s2, t2 ∈ Z.
We put βc = β(Z
2, µc, χ). The 2-cocycle µc satisfies condition (3) in Lemma 5.5.
Thus βc defines a Z
2 ⋊ SL(2,Z)-action on R. By Theorem 4.4, we get the following
isomorphism between topological groups:
Aut(R, βc) ∼= Aut(Z2, µ∗cµc, χ2) = Aut(Z2, µc2, χ2).
Since the character χ2 of Z2 is injective, so the group of the right side is {id|Z2}.
This means that the action βc has trivial centralizers.
Finally, we prove that the actions {βc | c = eiπt, t ∈ (0, 1/2) \ Q} are mutually
non-conjugate. Suppose that actions βc1 and βc2 are conjugate. By Theorem 4.1,
there exists a group isomorphism φ of Z2 satisfying
(g, h) = µc2
(φ(g), φ(h)), g, h ∈ Z2.
A group isomorphism of Z2 is given by an element of GL(2,Z). If the automorphism
φ is given by an element of SL(2,Z), we get c22 = c
1. If φ is given by an element
of GL(2,Z) \ SL(2,Z), then we get c22 = −c21. Since both c1 and c2 have the form
eiπt, t ∈ (0, 1/2), we get c1 = c2. �
Any cyclic group of an odd order can be realized as the centralizer of a twisted
Bernoulli shift actions on R.
Corollary 5.8. Let q be an odd natural number ≥ 3 and denote by Hq the abelian
group (Z/qZ)2. We define the 2-cocycle µq and the character χq on Hq as
= exp (2πis1t2/q), χq
= exp (2πis1/q).
Then the algebra N(Hq, µq) is the AFD factor R of type II1 and the centralizer of
the twisted Bernoulli shift action βa = β(Hq, µq, χq) is isomorphic to Z/qZ.
Proof. By Lemma 5.5, it is shown that the algebra N(HQ, µQ) is the AFD factor of
type II1. Using Theorem 4.4, we have only to prove that Aut(Hq, µ
qµq, χ
∼= Z/qZ.
Let φ be in Aut(Hq, µ
qµq, χ
q). The automorphism φ of Hq is given by a 2 × 2
matrix A of Z/qZ. Since φ preserves µ∗qµq, the determinant of A must be 1. Since
q is odd, the value of χ2q determines the first entry of (Z/qZ)
2 and φ preserves χ2q .
The matrix A is of the form(
, tφ ∈ Z/qZ.
The map φ 7→ tφ is an isomorphism. In turn, if the matrix A is of this form, it
defines an element in Aut(Hq, µ
qµq, χ
q). �
18 HIROKI SAKO
Corollary 5.9. For a set Q consisting of odd prime numbers, let βQ be the tensor
product
q∈Q βq of the actions βq on the AFD factor of type II1. The centralizer
of βQ is isomorphic to
q∈Q Z/qZ.
Proof. The action βQ is the twisted Bernoulli shift action β(HQ, µQ, χQ), where HQ
is the abelian group ⊕q∈QHq and the scalar 2-cocycle µq and a character χQ on HQ
are given by
µQ((sq), (tq)) =
µq(sq, tq),
χQ((sq)) =
χq(sq), (sq), (tq) ∈ HQ, sq, tq ∈ Hq.
Using Theorem 4.4, we have only to prove
Aut(HQ, µ
QµQ, χ
Z/qZ.
A group automorphism φ of HQ = ⊕q∈QHq has a form φ((kq)) = (φq(kq)), for some
{φq ∈ Aut(Hq)}. Thus we get
Aut(HQ, µ
QµQ, χ
Aut(Hq, µ
qµq, χ
Together with the previous corollary, we get the conclusion. �
Remark 5.10. If Q1 6= Q2, then the two groups
Z/qZ and
Z/qZ are not
isomorphic. The continuously many outer actions {βQ} are distinguished in view of
conjugacy only by using the centralizers {Aut(R, βQ)}.
6. Malleability and rigidity arguments
In this section, we give malleability and rigidity type arguments invented by S.
Popa, in order to examine weak 1-cocycles for actions. See Popa [Po2], [Po3], [Po4]
and Popa–Sasyk [PoSa] for the references. S. Popa in [Po3] showed that every
1-cocycle for a Connes-Størmer Bernoulli shift by property (T) group (or w-rigid
group like Z2 ⋊ SL(2,Z)) vanishes modulo scalars. As a consequence, two such
actions are cocycle conjugate if and only if they are conjugate. In our case, 1-
cocycles do not vanish modulo scalars but they are still in the situation that cocycle
(outer) conjugacy implies conjugacy. We need the following notion to examine outer
conjugacy of two group actions.
Definition 6.1. Let α be an action of discrete group Γ on a von Neumann algebra
M. A weak 1-cocycle for α is a map w : Γ → U(M) satisfying
wgh = wgαg(wh) mod T, g, h ∈ Γ.
The weak 1-cocycle w is called a weak 1-coboundary if there exists a unitary v ∈
U(M) satisfying wg = vαg(v)∗ mod T. Two weak 1-cocycles w and w′ are said to be
equivalent when w′g = vwgαg(v)
∗ mod T for some v ∈ U(M).
Let N be a finite von Neumann algebra with a faithful normal trace. The following
is directly obtained by combining Lemmas 2.4 and 2.5 in [PoSa], although these
Lemmas were proved for Bernoulli shift actions on standard probability space. The
CLASSIFICATION AND CENTRALIZERS 19
following can be also regarded as a weak 1-cocycle version of Proposition 3.2 in
[Po4].
Proposition 6.2. Let G be a countable discrete group. Let β be a trace preserving
weakly mixing action of G on N . A weak 1-cocycle {wg}g∈G ⊂ N for β is a weak
1-coboundary if only if there exists a non-zero element x̃0 ∈ N ⊗N satisfying
(wg ⊗ 1)(βg ⊗ βg)(x̃0)(1⊗ w∗g) = x̃0, g ∈ G.
The following is a weak 1-cocycle version of Proposition 3.6.3◦ in [Po4].
Proposition 6.3. Let Γ be a countable discrete group and G be a normal subgroup
of Γ. The group Γ acts on a finite von Neumann algebra N in a trace-preserving
way by β. Suppose that the restriction of β to G is weakly mixing. Let {wγ}γ∈Γ be a
weak 1-cocycle for β. If w|G is a weak 1-coboundary, then w is a weak 1-coboundary
for the Γ-action.
Proof. Suppose that w|G is a weak 1-coboundary, that is, there exists a unitary
element v in N such that wg = vβg(v
∗) mod T for g ∈ G. It suffices to show
that {w′γ} = {v∗wγβγ(v)} is in T for all γ ∈ Γ. Take arbitrary γ ∈ Γ, g ∈ G. Write
h = γ−1gγ ∈ G. Let πγ be the unitary on L2(N) induced from βγ . Since w′h, w′g ∈ T,
we get
w′γπgw
= (w′γπγ)(w
hπh)(w
∗ = w′γhγ−1πγhγ−1 = πg mod T,
By applying these operators to 1̂ ∈ N̂ ⊂ L2(N), it follows that w′γβg(w′γ
) ∈ T.
Since the G-action is weakly mixing, we have w′γ ∈ T. �
By using the above propositions, we will “untwist” some weak 1-cocycles later.
We require some ergodicity assumption on the weak 1-cocycles.
Definition 6.4. Let Γ be a discrete group and G be a subgroup of Γ. Suppose that
its restriction to G is ergodic. Let β be a trace preserving action of Γ on N . A weak
1-cocycle w = {wg}g∈Γ for β is said to be ergodic on G, if the action βw of G is still
ergodic, where βw is defined by βwg = Adwg ◦ βg, g ∈ G.
Let β be a Γ-action onN . Suppose that the diagonal action β⊗β on (N⊗N, tr⊗tr)
has an extension β̃ on a finite von Neumann algebra (Ñ , τ). The algebra Ñ is not
necessarily identical with N⊗N . When the action β̃ is ergodic on a normal subgroup
G ⊂ Γ, we get the following:
Proposition 6.5. Let {wγ}γ∈Γ ⊂ N be a weak 1-cocycle for β. Let α be a trace
preserving continuous action of R on Ñ satisfying the following properties:
• α1(x⊗ 1) = 1⊗ x, for all x ∈ N .
• αt ◦ β̃(γ)(x̃) = β̃(γ) ◦ αt(x̃), for all t ∈ R, γ ∈ Γ and x̃ ∈ Ñ .
Suppose that the weak 1-cocycle {wγ⊗1} ⊂ Ñ is ergodic for the G-action β̃|G. If the
group inclusion G ⊂ Γ has the relative property (T) of Kazhdan, then there exists a
non-zero element x̃0 ∈ Ñ so that (wg ⊗ 1)β̃g(x̃0)(1⊗ w∗g) = x̃0, g ∈ G.
This is proved in the same way for Bernoulli shift actions on the infinite tensor
product of abelian von Neumann algebras ([PoSa], Lemma 3.5). Since we are in-
terested in actions on the AFD II1 factor, we require the ergodicity assumption on
20 HIROKI SAKO
weak 1-cocycle {wγ ⊗ 1}. For the self-containedness and in order to make it clear
where the ergodicity assumption works, we write down a complete proof.
Proof. For t ∈ (0, 1], let Kt be the convex weak closure of
{(wg ⊗ 1)αt(w∗g ⊗ 1) | g ∈ G} ⊂ Ñ
and x̃t ∈ Kt be the unique element whose 2-norm is minimum in Kt. Since
(wg ⊗ 1)β̃g((wg1 ⊗ 1)αt(w∗g1 ⊗ 1))αt(w
g ⊗ 1)
= (wgβg(wg1)⊗ 1)αt(βg(w∗g1)w
g ⊗ 1)
= (wgg1 ⊗ 1)αt(w∗gg1 ⊗ 1), g, g1 ∈ G,
we have (wg ⊗ 1)β̃g(Kt)αt(w∗g ⊗ 1) = Kt, for g ∈ G. By the uniqueness of x̃t, we get
(wg ⊗ 1)β̃g(x̃t)αt(w∗g ⊗ 1) = x̃t, g ∈ G.(Eq3)
By the assumption, the action (Ad(wg ⊗ 1) ◦ β̃g)g∈G is ergodic on Ñ . By the calcu-
lation
(wg ⊗ 1)β̃g(x̃tx̃t∗)(w∗g ⊗ 1)
= (wg ⊗ 1)β̃g(x̃t)αt(w∗g ⊗ 1)αt(wg ⊗ 1)β̃g(x̃t
)(w∗g ⊗ 1)
= x̃tx̃t
, g ∈ G,
we get x̃tx̃t
∗ ∈ C1. The element x̃t is a scalar multiple of a unitary in Ñ .
We shall next prove that x̃1/n is not zero for some positive integer n. The pair
(Γ, G) has the relative property (T) of Kazhdan. By proposition 2.4, we can find a
positive number δ and a finite subset F ⊂ Γ satisfying the following condition: If a
unitary representation (π,H) of Γ and a unit vector ξ of H satisfy ‖π(γ)ξ − ξ‖ ≤
δ (γ ∈ F ), then ‖π(g)ξ− ξ‖ ≤ 1/2 (g ∈ G). By the continuity of the action α, there
exists n such that
‖(wγ ⊗ 1)α1/n(w∗γ ⊗ 1)− 1‖tr,2 ≤ δ, γ ∈ F,
The actions β and (αl1/n)l∈Z on Ñ give a Γ× Z action on Ñ . Let P be the crossed
product von Neumann algebra P = Ñ ⋊ (Γ × Z). Let (Uγ)γ∈Γ and W be the
implementing unitaries in P for Γ and 1 ∈ Z respectively. We put Vγ = (wγ ⊗
1)Uγ , γ ∈ Γ. We regard AdV· as a unitary representation of Γ on L2(P ). Since
‖AdVγ(W )−W‖L2(P ) = ‖(wγ ⊗ 1)W (w∗γ ⊗ 1)W ∗ − 1‖L2(P )
= ‖(wγ ⊗ 1)α1/n(w∗γ ⊗ 1)− 1‖L2(Ñ) ≤ δ, γ ∈ F,
we have the following inequality:
1/2 ≥ ‖AdVg(W )−W‖L2(P ) = ‖(wg ⊗ 1)α1/n(w∗g ⊗ 1)− 1‖L2(Ñ), g ∈ G.
We get 1/2 ≥ ‖x̃1/n − 1‖L2(Ñ) and x̃1/n 6= 0.
Let ũ1/n be the unitary of Ñ given by a scalar multiple of x̃1/n. By equation
(Eq3), the unitary satisfies
(wg ⊗ 1)β̃g(ũ1/n)α1/n(w∗g ⊗ 1) = ũ1/n, g ∈ G.
CLASSIFICATION AND CENTRALIZERS 21
Let x̃0 be the unitary defined by
x̃0 = ũ1/nα1/n(ũ1/n)α2/n(ũ1/n) . . . α(n−1)/n(ũ1/n).
By direct computations, we have the following desired equality:
(wg ⊗ 1)β̃g(x̃0)(1⊗ w∗g) = (wg ⊗ 1)β̃g(x̃0)α1(w∗g ⊗ 1) = x̃0, g ∈ G.
Theorem 6.6. Let β = β(H, µ, χ) be a twisted Bernoulli shift action on N(H, µ).
Suppose that N(H, µ) is the AFD factor of type II1 and that there exists a continuous
R-action (α
t )t∈R on Lµ(H)⊗ Lµ(H) satisfying the following properties:
• For any x ∈ Lµ(H), α(0)1 (x⊗ 1) = 1⊗ x,
• The automorphism α(0)t commutes with the diagonal action of Ĥ.
Let β(1) be another action of Z2 ⋊ SL(2,Z) on the AFD factor N (1) of type II1 and
suppose that its restriction to Z2 is ergodic. The action β(1) is outer conjugate to β,
if and only if β(1) is conjugate to β.
Proof. We deduce from outer conjugacy to conjugacy in the above situation. Let θ
be a ∗-isomorphism from N (1) onto N(H, µ) which gives the outer conjugacy of the
action β(1) and β = β(H, µ, χ). There exists a weak 1-cocycle {wγ}γ∈Z2⋊SL(2,Z) for β
satisfying
θ ◦ β(1)(γ) = Adwγ ◦ β(γ) ◦ θ, γ ∈ Z2 ⋊ SL(2,Z).
Since the action β(1) is ergodic on Z2, the weak 1-cocycle w is ergodic on Z2.
We use the notations Γ0, G0 given in Section 3. Let ρ̃ be the diagonal action ρ⊗ρ
of Γ0 on the tensor product algebra M̃ = Lµ̃(⊕Z2H)⊗ Lµ̃(⊕Z2H):
ρ̃(γ0)(a⊗ b) = ρ(γ0)(a)⊗ ρ(γ0)(b).
The fixed point algebra Ñ ⊂ M̃ of the diagonal Ĥ-action contains N(H, µ) ⊗
N(H, µ). Since Z2⋊SL(2,Z) = Γ0/Ĥ, the action ρ̃ gives a Z
2⋊SL(2,Z)-action β̃ on
Ñ . The action β̃ is the extension of the diagonal action β⊗β on N(H, µ)⊗N(H, µ).
We denote by αt the action on M̃ ∼=
(Lµ(H)⊗Lµ(H)) given by the infinite tensor
product of the R-action α
t . By the assumption on α
t , the R-action αt commutes
with the action ρ̃. It follows that the subalgebra Ñ is globally invariant under αt.
The set of unitary {Wγ = wγ ⊗ 1}γ∈Z2⋊SL(2,Z) ⊂ Ñ is a weak 1-cocycle for β̃. We
shall prove that this weak 1-cocycle is ergodic on Z2. Let a be an element in Ñ fixed
under β̃|Z2 . The element a can be written as a =
H aλ⊗u(λ) in L2M̃ , where
aλ ⊗ 1 = EM⊗C(a(1⊗ u(λ))∗). Since a is fixed under the action of Z2, we have
a = β̃W (k)(a) =
Adwk ◦ ρ(1, k)(aλ)⊗ ρ(1, k)(u(λ))
Adwk ◦ ρ(1, k)(aλ)⊗
χ(λ(l))det(k,l)u(k · λ).
Since Adwk ◦ ρ(1, k) preserves the 2-norm, we get ‖aλ‖2 = ‖ak−1·λ‖2. Since ‖a‖22 =∑
‖aλ‖22 < ∞ and the set {ak−1·λ | k ∈ Z2} is infinite for λ 6= 0, it turns out that
aλ = 0 for λ 6= 0 and thus a ∈ Ñ ∩ (M ⊗ C) = N(H, µ) ⊗ C. By the ergodicity of
22 HIROKI SAKO
the Z2-action {Adwk ◦ β(k)}, we get a ∈ C. We conclude that the weak 1-cocycle
{Wγ} ⊂ Ñ is ergodic on Z2.
By the relative property (T) for the inclusion Z2 ⊂ Z2⋊SL(2,Z) and Proposition
6.5, there exists a non-zero element x̃0 ∈ Ñ satisfying
(wk ⊗ 1)β̃(k)(x̃0)(1⊗ w∗k) = x̃0, k ∈ Z2.
The element x̃0 can be written as the following Fourier expansion:
x̃0 =
c(λ1, λ2)u(λ1)⊗ u(λ2) ∈ Ñ ⊂ Lµ̃(⊕Z2H)⊗ Lµ̃(⊕Z2H).
Here c(λ1, λ2) is a complex number and (λ1, λ2) ∈ (⊕Z2H)2 runs through all pairs
satisfying
k∈Z2(λ1(k) + λ2(k)) = 0. Choose and fix a pair (λ1, λ2) satisfying
λ1(k) = h =
λ2(k), c(λ1, λ2) 6= 0.
Let v′h ∈ M be the unitary written as v′h = uh ⊗ 1 ⊗ 1 ⊗ · · · , where uh ∈ Lµ(H) is
placed on 0 ∈ Z2. The following unitaries {w′γ} ⊂ N(H, µ) give a weak 1-cocycle
for β:
w′(k,γ0) = v
hw(k,γ0)ρ(1, k, γ0)(v
), (k, γ0) ∈ Z2 ⋊ SL(2,Z).
Letting ỹ = (v′h ⊗ 1)x̃0(1⊗ v′h)∗ ∈ M̃ , we get
ỹ = (w′k ⊗ 1)β̃(k)(ỹ)(1⊗ w′k
), k ∈ Z2.
Applying the trace preserving conditional expectation E = EN(H,µ)⊗N(H,µ), we get
E(ỹ) = (w′k ⊗ 1)E(β̃(k)(ỹ))(1⊗ w′k
= (w′k ⊗ 1)β̃(k)(E(ỹ))(1⊗ w′k
), k ∈ Z2.
Since the Fourier coefficient of x̃0 at (λ1, λ2) ∈ (⊕Z2H)2 is not zero, that of E(ỹ)
at (λ1 + δh,0, λ2 − δh,0) ∈ Λ(H)2 is also non-zero, where δh,0 ∈ ⊕Z2H is zero on
Z2 \ {0} and is h on 0 ∈ Z2. By Proposition 6.2, it follows that the weak 1-cocycle
{w′(k,e)}k∈Z2 ⊂ N(H, µ) is a weak 1-coboundary of β|Z2. Since the Z2-action β|Z2 is
weakly mixing, w′ is a weak 1-coboundary on Z2 ⋊ SL(2,Z), by Proposition 6.3. In
other words, there exists v ∈ N(H, µ) satisfying
w′γ = vβ(γ)(v
∗) mod T,
wγ = v
vρ(1, γ)(v∗v′h) mod T, γ ∈ Z2 ⋊ SL(2,Z).
Noting that u = v∗v′h ∈ M is a normalizer of N(H, µ), we get
(Ad(u) ◦ θ) ◦ β(1)(γ) = Ad(u) ◦ Ad(wγ) ◦ β(γ) ◦ θ
= Ad(ρ(1, γ)(u)) ◦ β(γ) ◦ θ
= ρ(1, γ) ◦ Ad(u) ◦ θ
= β(γ) ◦ (Ad(u) ◦ θ), γ ∈ Z2 ⋊ SL(2,Z).
Thus we get the conjugacy of two Z2 ⋊ SL(2,Z)-actions β(0) and β. �
We can always apply Theorem 6.6 if H is finite.
CLASSIFICATION AND CENTRALIZERS 23
Corollary 6.7. Let H be a finite abelian group and let β = β(H, µ, χ) be a twisted
Bernoulli shift action on N(H, µ). Suppose that N(H, µ) is the AFD factor of type
II1. Let β
(1) be an action of Z2 ⋊ SL(2,Z) on the AFD factor N (1) of type II1 and
suppose that its restriction to Z2 is ergodic. The action β(1) is outer conjugate to β,
if and only if β(1) is conjugate to β.
Proof. We have only to construct an R-action on Lµ(H) ⊗ Lµ(H) satisfying the
properties in Theorem 6.6. Let U be an element of Lµ(H)⊗ Lµ(H) defined by
|H|1/2
uh ⊗ u∗h.
We note that µ∗µ(g, ·) is a character of H and that it is not identically 1 provided
g 6= 0 by Lemma 5.5. The element U is self-adjoint and unitary, since
|H|1/2
u∗h ⊗ uh =
|H|1/2
µ(h,−h)u−h ⊗ µ(h,−h)u∗−h = U,
g,h∈H
uguh ⊗ u∗gu∗h =
g,h∈H
µ∗µ(g, h)ug+h ⊗ u∗g+h
µ∗µ(g, h− g)
ug ⊗ u∗g = 1.
The operator U is a fixed point under the action of Ĥ , so the projections P1 =
(1 + U)/2 and P−1 = (1 − U)/2 are also fixed points. Thus the R-action α(0)t =
Ad(P1+exp (iπt)P−1) commutes with the Ĥ-action. The automorphism α
1 satisfies
1 (ug ⊗ 1) = U(ug ⊗ 1)U∗ = (1⊗ ug)UU∗ = 1⊗ ug, g ∈ H.
This verifies the first condition for α(0). �
Corollary 6.8. Let Q be a set consisting of odd prime numbers and βQ be the twisted
Bernoulli shift action defined in Corollary 5.9. Let β be a Z2 ⋊ SL(2,Z)-action on
the AFD factor of type II1 whose restriction to Z
2 is ergodic. The actions βQ and
β are outer conjugate if and only if they are conjugate. In particular, {βQ} is an
uncountable family of Z2⋊SL(2,Z)-actions which are mutually non outer conjugate.
Proof. We will use the notation given in Corollary 5.8 and 5.9. Let α
t be the R-
action on Lµq (Hq)⊗Lµq(Hq) constructed as in the previous corollary. We define the
R-action α(Q) on LµQ(HQ)⊗LµQ(HQ) by α
t (⊗q∈Qxq) = ⊗q∈Qα
t (xq), where xq ∈
Lµq (Hq) ⊗ Lµq (Hq) and xq 6= 1 only for finitely many q. The R-action satisfies the
conditions in Theorem 6.6. By Corollary 5.9, {βQ} are mutually non conjugate and
their restriction to Z2 is ergodic. Thus they are mutually non outer conjugate. �
Acknowledgment . The author would like to thank Professor Yasuyuki Kawahigashi
for helpful conversations. He thanks the referee for careful reading and numerous
detailed comments. He is supported by JSPS Research Fellowships for Young Sci-
entists.
24 HIROKI SAKO
References
[Bu] M. Burger, Kazhdan constants for SL3(Z), J. Reine Angew. Math., 413 (1991), 36–67.
[Ch] M. Choda, A continuum of non-conjugate property T actions of SL(n,Z) on the hyperfinite
II1-factor, Math. Japon., 30 (1985), 133–150.
[Co] A. Connes, Classification of injective factors. Cases II1, II∞, IIIλ, λ 6= 1. Ann. of Math. (2)
104 (1976), no. 1, 73–115.
[Jol] P. Jolissaint, On property (T) for pairs of topological groups. Enseign. Math., (2) 51 (2005),
no. 1-2, 31E5.
[Jon] V. F. R. Jones, A converse to Ocneanu’s theorem, J. Operator Theory, 10 (1983), no. 1,
61–63.
[Ka] Y. Kawahigashi, Cohomology of actions of discrete groups on factors of type II1, Pac. J.
Math., 149 (1991), no. 1, 303–317.
[NPS] R. Nicoara, S. Popa, R. Sasyk, On II1 factors arising from 2-cocycles of w-rigid groups. J.
Funct. Anal. 242 (2007),no. 1, 230–246.
[Oc] A. Ocneanu, Actions of discrete amenable groups on von Neumann algebras, Lecture Notes
in Mathematics, 1138. Springer-Verlag, Berlin, 1985.
[OPT] D. Olesen, G. K. Pedersen, M. Takesaki, Ergodic actions of compact abelian groups, J.
Operator Theory, 3 (1980), no. 2, 237–269.
[Po1] S. Popa, On a class of type II1 factors with Betti numbers invariants, Ann. of Math., (2)
163 (2006), no. 3, 809–899.
[Po2] S. Popa, Some computations of 1-cohomology groups and construction of non orbit equivalent
actions, J. Inst. Math. Jussieu, 5 (2006), 309–332.
[Po3] S. Popa, Some rigidity results for non-commutative Bernoulli shifts, J. Funct. Anal., 230
(2006), no. 2, 273–328.
[Po4] S. Popa, Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups,
Invent. Math. 170 (2007), no. 2, 243–295.
[PoSa] S. Popa, R. Sasyk, On the cohomology of Bernoulli actions, Erg. Theory Dyn. Sys., 26
(2007), 1–11.
[Sh] Y. Shalom, Bounded generation and Kazhdan’s Property (T). Publ. Math. IHES, 90 (1999),
145–168.
[Vaes] S. Vaes, Rigidity results for Bernoulli actions and their von Neumann algebras (after Sorin
Popa), Sminaire Bourbaki, exp. no. 961, Astrisque 311 (2007) 237–294.
Department of Mathematical Sciences, University of Tokyo, Komaba, Tokyo,
153-8914, Japan
E-mail address : hiroki@ms.u-tokyo.ac.jp
	1. Introduction
	2. Preparations
	2.1. Functions det and gcd
	2.2.  Scalar 2-cocycles for abelian groups 
	2.3. Definition of a Z2 SL(2,Z)-action on (H)
	2.4. On the relative property (T) of Kazhdan
	2.5. Weakly mixing actions
	2.6. A remark on group von Neumann algebras
	3.  Definition of twisted Bernoulli shift actions
	4. Classification up to conjugacy
	5. Examples
	5.1. Twisted Bernoulli shift actions on L(X)
	5.2. Twisted Bernoulli shift actions on the AFD factor of type II1
	6. Malleability and rigidity arguments
	References

  We introduce a new class of actions of the group $\G$ on finite von Neumann
algebras and call them twisted Bernoulli shift actions. We classify these
actions up to conjugacy and give an explicit description of their centralizers.
We also distinguish many of those actions on the AFD $\mathrm{II}_1$ factor in
view of outer conjugacy.

Introduction
We consider the classification of Z2 ⋊ SL(2,Z)-actions on finite von Neumann
algebras in this paper. Mainly, we concentrate on the case that the finite von
Neumann algebra is the AFD factor of type II1 or non-atomic abelian.
There are two difficulties for analyzing discrete group actions on operator algebras.
The first is that we do not have various ways to construct actions. The second is
that we can not analyze them by concrete calculation in most cases. To give many
examples of actions which admit concrete analysis, we introduce a class of trace
preserving Z2 ⋊ SL(2,Z)-actions on finite von Neumann algebras and call them
twisted Bernoulli shift actions. We classify those actions up to conjugacy and study
them up to outer conjugacy.
An action β(H, µ, χ) in the class is defined for a triplet (H, µ, χ), where H is
an abelian countable discrete group, µ is a normalized scalar 2-cocycle of H and χ
is a character of H . We obtain the action by restricting the so-called generalized
Bernoulli shift action to a subalgebra N(H, µ) and “twisting” it by the character χ.
The process of restriction has a vital role in concrete analysis of these actions.
A ∗-isomorphism which gives conjugacy between two twisted Bernoulli shift ac-
tions β(Ha, µa, χa) and β(Hb, µb, χb) must be induced from an isomorphism between
the two abelian groups Ha and Hb. We prove this by concrete calculation (Section
4). It turns out that there exist continuously many, non-conjugate Z2 ⋊ SL(2,Z)-
actions on the AFD factor of type II1 (Section 5). By using the same technique, we
describe the centralizers of all twisted Bernoulli shift actions. Here we should men-
tion that the present work was motivated by the previous ones [Ch], [NPS], where
similar studies were carried out in the case of SL(n,Z).
In Section 6, we distinguish many twisted Bernoulli shift actions in view of outer
conjugacy. The classification for actions of discrete amenable groups on the AFD
factor of type II1 was given by Ocneanu [Oc]. Outer actions of countable amenable
groups are outer conjugate. In the contrast to this, V. F. R. Jones [Jon] proved that
any discrete non-amenable group has at least two non outer conjugate actions on
2000 Mathematics Subject Classification. Primary 46L40; Secondary 46L10.
Key words and phrases. von Neumann algebras; automorphisms.
http://arxiv.org/abs/0704.1533v3
2 HIROKI SAKO
the AFD factor of type II1. S. Popa ([Po3], [Po4], [PoSa], etc.) used the malleabil-
ity/deformation arguments for the Bernoulli shift actions to study (weak) 1-cocycles
for the actions. For some of twisted Bernoulli shift actions, which we introduce in
this paper, it is shown that (weak) 1-cocycles are represented in simple forms under
some assumption on the (weak) 1-cocycles. We prove that there exist continuously
many twisted Bernoulli shift actions which are mutually non outer conjugate. This
strengthens the above mentioned result due to Jones in the Z2 ⋊ SL(2,Z) cases.
2. Preparations
2.1. Functions det and gcd. For the definition of twisted Bernoulli shift actions in
Section 3, we define two Z-valued functions det and gcd. The function det is given
by the following equation:
= qr0 − rq0,
∈ Z2.
The value of the function gcd at k ∈ Z2 is the greatest common divisor of the two
entries. For 0 ∈ Z2, let the value of gcd be 0.
Lemma 2.1.
(1) The action of SL(2,Z) on Z2 preserves the functions det and gcd, that is,
det(k, k0) = det(γ · k, γ · k0),
gcd(k) = gcd(γ · k), k, k0 ∈ Z2, γ ∈ SL(2,Z).
(2) The following equation holds true:
det(k, k0) = gcd(k) + gcd(k0)− gcd(k + k0) mod 2, k, k0 ∈ Z2.
Proof. The claim (1) is a well-known fact, so we prove the claim (2). For the function
gcd, we get
1 mod 2, (either q or r is odd),
0 mod 2, (both q and r are even).
Since the action of SL(2,Z) on Z2 preserves the functions det and gcd, it suffices to
show the desired equation against the following four pairs:
(k, k0) =
mod 2.
2.2. Scalar 2-cocycles for abelian groups. We fix some notations for countable
abelian groups and their scalar 2-cocycles. For the rest of this paper, let H be an
abelian countable discrete group and suppose that any scalar 2-cocycle µ : H×H →
T = {z ∈ C | |z| = 1} is normalized, that is, µ(g, 0) = 1 = µ(0, g) for g ∈ H . We
denote by µ∗ the 2-cocycle for H given by µ∗(g, h) = µ(h, g), g, h ∈ H . Let µ∗µ be
the function on H ×H defined by
µ∗µ(g, h) = µ(h, g)µ(g, h), g, h ∈ H.
CLASSIFICATION AND CENTRALIZERS 3
This is a bi-character, that is, µ∗µ(g, ·) and µ∗µ(·, h) are characters of H . By using
this function, we can describe the cohomology class of µ. See [OPT] for the proof
of the following Proposition:
Proposition 2.2. Two scalar 2-cocycles µ1 and µ2 of H are cohomologous if and
only if µ∗1µ1 = µ
Let Cµ(H) be the twisted group algebra of H with respect to the 2-cocycle µ. We
denote by {uh | h ∈ H} the standard basis for Cµ(H) as C-linear space. We recall
that the C-algebra Cµ(H) has a structure of ∗-algebra defined by
ug uh = µ(g, h) ug+h, u
g = µ(g,−g)u−g, g, h ∈ H.
Let µ̃ be the T-valued function on ⊕Z2H ×⊕Z2H defined by
µ̃(λ1, λ2) =
µ(λ1(k), λ2(k)), λ1, λ2 ∈ ⊕Z2H.(Eq1)
The function µ̃ is a normalized scalar 2-cocycle for ⊕Z2H . Let Λ(H) be the abelian
group defined by
Λ(H) =
λ : Z2 → H
∣∣∣∣∣ finitely supported and
λ(k) = 0
Its additive rule is defined by pointwise addition.
2.3. Definition of a Z2 ⋊ SL(2,Z)-action on Λ(H). The group SL(2,Z) acts on
Z2 as matrix-multiplication and the group Z2 also does on Z2 by addition. These
two actions define the action of Z2 ⋊ SL(2,Z) on Z2 which is explicitly described as
q + xq0 + yr0
r + zq0 + wr0
for all ((
∈ Z2 ⋊ SL(2,Z),
∈ Z2.
We define an action of Z2 ⋊ SL(2,Z) on ⊕Z2H as
(γ · λ)(k) = λ(γ−1 · k), k ∈ Z2,
for γ ∈ Z2 ⋊ SL(2,Z) and λ ∈ ⊕Z2H .
2.4. On the relative property (T) of Kazhdan. We give the definition of the
relative property (T) of Kazhdan for a pair of discrete groups.
Definition 2.3. Let G ⊂ Γ be an inclusion of discrete groups. We say that the pair
(Γ, G) has the relative property (T) if the following condition holds:
There exist a finite subset F of Γ and δ > 0 such that if π : Γ → U(H) is a
unitary representation of Γ on a Hilbert space H with a unit vector ξ ∈ H satisfying
‖π(g)ξ − ξ‖ < δ for g ∈ F , then there exists a non-zero vector η ∈ H such that
π(h)η = η for h ∈ G.
Instead of this original definition, we use the following condition.
4 HIROKI SAKO
Proposition 2.4. ([Jol]) Let G ⊂ Γ be an inclusion of discrete groups. The pair
(Γ, G) has the relative property (T) if and only if the following condition holds:
For any ǫ > 0, there exist a finite subset F of Γ and δ > 0 such that if π : Γ → U(H)
is a unitary representation of Γ on a Hilbert space H with a unit vector ξ ∈ H
satisfying ‖π(g)ξ − ξ‖ < δ for g ∈ F , then ‖π(h)ξ − ξ‖ < ǫ for h ∈ G.
The pair (Z2⋊SL(2,Z),Z2) is a typical example of group with the relative property
(T). See [Bu] or [Sh] for the proof.
2.5. Weakly mixing actions. An action of a countable discrete group G on a von
Neumann algebra N is said to be ergodic if any G-invariant element of N is a scalar
multiple of 1. The weak mixing property is a stronger notion of ergodicity.
Definition 2.5. Let N be a von Neumann algebra with a faithful normal state φ.
A state preserving action (ρg)g∈G of a countable discrete group G on N is said to be
weakly mixing if for every finite subset {a1, a2, . . . , an} ⊂ N and ǫ > 0, there exists
g ∈ G such that |φ(aiρg(aj))− φ(ai)φ(aj)| < ǫ, i, j = 1, . . . , n.
The following is a basic characterization of the weak mixing property. Between
two von Neumann algebra N and M , N ⊗ M stands for the tensor product von
Neumann algebra.
Proposition 2.6 (Proposition D.2 in [Vaes]). Let a countable discrete group G act
on a finite von Neumann algebra (N, tr) by trace preserving automorphisms (ρg)g∈G.
The following statements are equivalent:
(1) The action (ρg) is weakly mixing.
(2) The only finite-dimensional invariant subspace of N is C1.
(3) For any action (αg) of G on a finite von Neumann algebra (M, τ), we have
(N ⊗ M)ρ⊗α = 1 ⊗ Mα, where (N ⊗ M)ρ⊗α and Mα are the fixed point
subalgebras.
2.6. A remark on group von Neumann algebras. Let Γ be a discrete group
and let µ be a scalar 2-cocycle of a countable group Γ. A group Γ acts on the Hilbert
space ℓ2Γ by the following two ways;
uγ(δg) = µ(γ, g)δγg, ργ(δg) = µ(g, γ
−1)δgγ−1 , γ, g ∈ Γ.
These two representations commute with each other. The von Neumann algebra
Lµ(Γ) generated by the image of u is called the group von Neumann algebra of
Γ twisted by µ. The normal state 〈·δe, δe〉 is a trace on Lµ(Γ). The vector δe is
separating for Lµ(Γ). For any element a ∈ Lµ(Γ), we define the square summable
function a(·) on Γ by aδe =
a(g)δg. The function a(·) is called the Fourier
coefficient of a. We write a =
g∈Γ a(g)ug and call this the Fourier expansion of
a. The Fourier expansion of a∗ is given by a∗ =
g∈Γ µ(g, g
−1)a(g−1)ug, since the
Fourier coefficient a∗(g) = 〈a∗δe, δg〉 is described as
〈δe, aρg−1δe〉 =
ρ∗g−1δe,
a(g)δg
= µ(g, g−1)a(g−1).
CLASSIFICATION AND CENTRALIZERS 5
Here we used the equation ρ∗
= µ(g, g−1)ρg, which is verified by direct computa-
tion. For two elements a, b, the Fourier coefficient of ab is given by
ab(γ) = 〈bδe, ργ−1a∗δe〉 =
a∗(g)µ(g, γ)b(gγ) =
µ(g−1, gγ)a(g−1)b(gγ).
This equation allows us to calculate the Fourier coefficient algebraically, that is,
µ(g−1, gγ)a(g−1)b(gγ)
a(g)b(h)
For a subgroup Λ ⊂ Γ, the subalgebra {uλ | λ ∈ Λ}′′ ⊂ Lµ(Γ) is isomorphic to
Lµ(Λ). We sometimes identify them. An element a ∈ Lµ(Γ) is in the subalgebra
Lµ(Λ) if and only if the Fourier expansion a(·) : Γ → C is supported on Λ, since
the trace 〈·δe, δe〉 preserving conditional expectation E from Lµ(Γ) onto Lµ(Λ) is
described as E(a) =
λ∈Λ a(λ)uλ.
3. Definition of twisted Bernoulli shift actions
In this section, we introduce twisted Bernoulli shift actions of Z2 ⋊ SL(2,Z) on
finite von Neumann algebras. The action is defined for a triplet i = (H, µ, χ), where
H 6= {0} is an abelian countable discrete group, µ is a normalized scalar 2-cocycle
of H and χ is a character of H . The finite von Neumann algebra, on which the
group Z2 ⋊ SL(2,Z) acts, is defined by the pair (H, µ).
We introduce a group structure on the set Γ0 = Ĥ × Z2 × SL(2,Z) as
(c1, k, γ1)(c2, l, γ2) =
c1c2χ
det(k,γ1·l), k + γ1 · l, γ1γ2
for any c1, c2 ∈ Ĥ, k, l ∈ Z2, γ1, γ2 ∈ SL(2,Z). The associativity is verified by Lemma
2.1. It turns out that the subsets Ĥ = Ĥ × {0} × {e} and G0 = Ĥ × Z2 × {e} are
subgroups in Γ0. It is easy to see that G0 is a normal subgroup of Γ0 and that Ĥ is a
normal subgroup of G0 and Γ0. We get a normal inclusion of groups G0/Ĥ ⊂ Γ0/Ĥ
and this is isomorphic to Z2 ⊂ Z2 ⋊ SL(2,Z).
Before stating the definition of the twisted Bernoulli shift action, we define a Γ0-
action ρ on the von Neumann algebra Lµ̃(⊕Z2H). We denote by u(λ) ∈ Lµ̃(⊕Z2H)
the unitary corresponding to λ ∈ ⊕Z2H . We define a faithful normal trace tr of
Lµ̃(⊕Z2H) in the usual way. For c ∈ Ĥ, k ∈ Z2, γ ∈ SL(2,Z), let ρ(c), ρ(k), ρ(γ) be
the linear transformations on Cµ̃(⊕Z2H) given by,
ρ(c)(u(λ)) =
c(λ(l))
u(λ),
ρ(k)(u(λ)) =
χ(λ(m))det(k,m)
u(k · λ),
ρ(γ)(u(λ)) = u(γ · λ), λ ∈ Λ(H),
These maps are compatible with the multiplication rule and the ∗-operation of
Cµ̃(⊕Z2H) . Since these maps preserve the trace, they extend to ∗-automorphisms
6 HIROKI SAKO
on Lµ̃(⊕Z2H). It is immediate to see that ρ(c) commutes with ρ(k) and ρ(γ). For
k, l ∈ Z2, we have the following relation:
ρ(k) ◦ ρ(l)(u(λ)) =
χ(λ(m))det(l,m)ρ(k)(u(l · λ))
χ(λ(m))det(l,m)
χ((l · λ)(m))det(k,m)u(k · (l · λ))
χ(λ(m))det(l,m)χ(λ(m))det(k,m+l)u((k + l) · λ).
By det(l, m) + det(k,m+ l) = det(k, l) + det(k + l, m), this equals to
χ(λ(m))
)det(k,l) ∏
χ(λ(m))det(k+l,m)u((k + l) · λ)
= ρ(χdet(k,l)) ◦ ρ(k + l)(u(λ)).
Since det is SL(2,Z)-invariant (Lemma 2.1), for k ∈ Z2, γ ∈ SL(2,Z), we get
ρ(γ · k) ◦ ρ(γ)(u(λ)) = ρ(γ · k)(u(γ · λ))
χ((γ · λ)(l))det(γ·k,l)u((γ · k) · (γ · λ))
χ(λ(l))det(γ·k,γ·l)u(γ · (k · λ))
χ(λ(l))det(k,l)ρ(γ)(u(k · λ))
= ρ(γ) ◦ ρ(k)(u(λ)), λ ∈ Λ(H).
By using the above two equations, ρ satisfies the following formula:
(ρ(c1) ◦ ρ(k) ◦ ρ(γ1)) ◦ (ρ(c2) ◦ ρ(l) ◦ ρ(γ2))
= ρ(c1) ◦ ρ(c2) ◦ ρ(k) ◦ ρ(γ1) ◦ ρ(l) ◦ ρ(γ2)
= ρ(c1) ◦ ρ(c2) ◦ ρ(k) ◦ ρ(γ1 · l) ◦ ρ(γ1) ◦ ρ(γ2)
= ρ(c1) ◦ ρ(c2) ◦ ρ(χdet(k,γ1·l)) ◦ ρ(k + (γ1 · l)) ◦ ρ(γ1γ2)
= ρ(c1c2χ
det(k,γ1·l)) ◦ ρ(k + (γ1 · l)) ◦ ρ(γ1γ2).
With ρ(c, k, γ) = ρ(c) ◦ ρ(k) ◦ ρ(γ), ρ gives a Γ0-action on Lµ̃(⊕Z2H).
We define the finite von Neumann algebra N(H, µ) as the group von Neumann
algebra Lµ̃(Λ(H)). By using Fourier coefficients, we can prove that N(H, µ) is the
fixed point algebra under the Ĥ-action ρ(Ĥ, 0, e) on Lµ̃(⊕Z2H). We get a Z2 ⋊
SL(2,Z)-action on N(H, µ) by
β(k, γ)(x) = ρ(1, k, γ)(x), k ∈ Z2, γ ∈ SL(2,Z), x ∈ N(H, µ).
This is the definition of the twisted Bernoulli shift action β = β(H, µ, χ) on N(H, µ).
We obtained the actions β(H, µ, χ) not only by twisting generalized Bernoulli shift
actions but also restricting to subalgebras N(H, µ) ⊂ Lµ̂(⊕Z2H) =
Lµ(H).
This restriction allows us to classify the actions up to conjugacy in the next section.
CLASSIFICATION AND CENTRALIZERS 7
In order to give a variety of the actions, we twisted the shift actions by the character
χ of the abelian group H .
Remark 3.1. The action β|Z2 = β(H, µ, χ)|Z2 has the weak mixing property. In
definition 2.5, we may assume that the Fourier coefficients of ai (i = 1, 2, · · · , n) are
finitely supported, by approximating in the L2-norm. Then for appropriate k ∈ Z2,
we get tr(aiβ(k)(aj)) = tr(ai)tr(aj), i, j = 1, 2, · · · , n.
4. Classification up to conjugacy
In this section, we classify the twisted Bernoulli shift actions {β(H, µ, χ)} up to
conjugacy (Theorem 4.1). We prove that an isomorphism which gives conjugacy
between two twisted Bernoulli shift actions is of a very special form. In fact it
comes from an isomorphism in the level of base groups H . We also determine the
centralizer of the Z2 ⋊ SL(2,Z)-action β(H, µ, χ) on N(H, µ) (Theorem 4.4).
We fix some notations for the proofs. We define 0, e1, e2 ∈ Z2 as
, e1 =
, e2 =
Let ξ be the element of Z2 ⋊ SL(2,Z) satisfying
ξ · 0 = e1, ξ · e1 = e2, ξ · e2 = 0.
The elements ξ and ξ2 are explicitly described as
−1 −1
−1 −1
The order of ξ is 3. Let η, δ ∈ SL(2,Z) be given by η =
, δ =
Let D be the subset of all elements of Z2 fixed under the action of δ, that is,
n ∈ Z
Then we get
ξ ·D =
n ∈ Z
, ξ2 ·D =
n ∈ Z
We define the subgroup ΛD(H) of Λ(H) by
ΛD(H) = {λ ∈ Λ(H) | λ : Z2 → H is supported on D}.
Let (Ha, µa, χa) and (Hb, µb, χb) be triplets of countable abelian groups, their
normalized 2-cocycles and characters. For h ∈ Ha, we define λh ∈ ΛD(Ha) as
λh(k) =
h (k = e1),
−h (k = 0),
0 (k 6= e1, 0).
8 HIROKI SAKO
For g ∈ Hb, we define σg ∈ ΛD(Hb) as
σg(k) =
g (k = e1),
−g (k = 0),
0 (k 6= e1, 0).
We denote by v(σ) ∈ N(Hb, µb) the unitary corresponding to σ ∈ Λ(Hb).
Theorem 4.1. If π : N(Ha, µa) → N(Hb, µb) is a ∗-isomorphism giving conjugacy
between βa = β(Ha, µa, χa) and βb = β(Hb, µb, χb), then there exists a group isomor-
phism φ = φπ : Ha → Hb satisfying
(1) π(u(λ)) = v(φ ◦ λ) mod T for λ ∈ Λ(Ha),
(2) the 2-cocycles µa(·, ·) and µb(φ(·), φ(·)) of Ha are cohomologous,
(3) χ2a = (χb ◦ φ)2.
Conversely, given a group isomorphism φ : Ha → Hb satisfying (2) and (3), there
exists a ∗-isomorphism π = πφ : N(Ha, µa) → N(Hb, µb) which satisfies condition
(1) and gives conjugacy between βa, βb.
We note that by Proposition 2.2 condition (2) for φ is equivalent to
(2)′ µ∗aµa(g, h) = µ
bµb(φ(g), φ(h)), g, h ∈ Ha.
Proof for the first half of Theorem 4.1.
Suppose that there exists a (not necessarily trace preserving) ∗-isomorphism π from
N(Ha, µa) onto N(Hb, µb) such that π ◦ βa(γ) = βb(γ) ◦ π, γ ∈ Z2 ⋊ SL(2,Z).
We prove that for every h ∈ Ha there exists φ(h) ∈ Hb satisfying π(u(λh)) =
v(σφ(h)) mod T. Let Uh denote the unitary in N(Hb, µb)
Uh = π
µa(h,−h) u(λh)
, h ∈ Ha.
We identify N(H, µ) with the subalgebra of the infinite tensor product
Lµ(H),
which is canonically isomorphic to Lµ̃ (⊕Z2H). The preimage π−1(Uh) can be written
as u∗h ⊗ uh. Here uh is the unitary corresponding to h ∈ Ha and placed on 1 ∈ Z2
and the unitary u∗h is placed on 0 ∈ Z2. We describe Uh as the Fourier expansion
σ∈Λ(Hb)
c(σ)v(σ). Since e1 and 0 are fixed under the action of δ, one has
βb(δ)
n(Uh) = π ◦ βa(δ)n(π−1(Uh)) = Uh.
It follows that the Fourier expansion Uh =
σ∈Λ(Hb)
c(σ)v(σ) must satisfy that
c(σ) = c(δ−n · σ) for every σ ∈ Λ(Hb) and n ∈ Z. For σ ∈ Λ(Hb) \ ΛD(Hb), the
orbit of σ under the action of δ−1 is an infinite set, since the support supp(σ) ⊂ Z2
is not included in D. It turns out that c(σ) = 0 for all σ ∈ Λ(Hb) \ ΛD(Hb) due to
Σ|c(σ)|2 = 1 < +∞, so that Uh =
σ∈ΛD(Hb)
c(σ)v(σ).
The unitary χa(h)U
h is also fixed under the action of δ and can be written as
χa(h)U
h = π
χa(h)µa(h,−h)u(−λh)
χa(h)µa(h,−h)u(ξ · λh)
µa(h,−h)u(ξ2 · λh)
= βb(ξ)(Uh) βb(ξ
2)(Uh).
CLASSIFICATION AND CENTRALIZERS 9
Letting ne1 = (n, 0)
T ∈ Z2, we get
βb(ξ)(Uh) = βb(ξ)
c(σ) v(σ)
σ∈ΛD(Hb)
c(σ) v(ξ · σ)
χb(σ(ne1))
2)(Uh) = βb(ξ
c(σ) v(σ)
σ∈ΛD(Hb)
c(σ) v(ξ2 · σ).
Since Fourier expansion admits algebraical calculation as in subsection 2.6, the ex-
pansion of χa(h)U
χa(h)U
h = βb(ξ)(Uh) βb(ξ
2)(Uh)
σ1,σ2∈ΛD(Hb)
c(σ1) c(σ2) v(ξ · σ1) v(ξ2 · σ2)
χb(σ1(ne1))
σ1,σ2∈ΛD(Hb)
c(σ1) c(σ2) µ̃b(ξ · σ1, ξ2 · σ2)
χb(σ1(ne1))
n v(ξ · σ1 + ξ2 · σ2).
The map ΛD(Hb)×ΛD(Hb) ∋ (σ1, σ2) 7→ ξ ·σ1+ξ2·σ2 ∈ Λ(Hb) is injective. Indeed, σ1
is uniquely determined by ξ ·σ1+ξ2·σ2, since σ1(k) = (ξ ·σ1+ξ2·σ2)(ξ ·k), k ∈ D\{e1}
and σ1(e1) = −
k∈D\{e1}
σ1(k). Here we used the condition
σ1(k) = 0. The
element σ2 is also determined by ξ · σ1 + ξ2 · σ2. Thus the index (σ1, σ2) uniquely
determines ξ · σ1 + ξ2 · σ2.
We take arbitrary elements σ1, σ2 ∈ ΛD(Hb) and suppose that c(σ1) 6= 0, c(σ2) 6= 0.
Since the unitary χa(h)U
h is invariant under the action of δ and the coefficient of
ξ ·σ1+ξ2 ·σ2 is not zero, ξ ·σ1+ξ2 ·σ2 is supported on D. It follows that the elements
σ1 and σ2 can be written as σ1 = σφ(h) = σ2, by some φ(h) ∈ Hb. Indeed, since the
subsets D \ {0, e1}, ξD \ {e1, e2} and ξ2D \ {e2, 0} are mutually disjoint, the element
ξ · σ1 must be supported on {e1, e2} and the element ξ2 · σ2 must be supported on
{e2, 0}. By the assumption
k∈Z2 σi(k) = 0 (i = 1, 2), σi can be written as σφ(hi).
Then using the fact that (ξ · σ1 + ξ2 · σ2)(e2) = σ1(e1) + σ2(0) = 0, we get that
σ1 = σφ(h) = σ2 for some h ∈ Hb. This means that there exists only one σ ∈ Λ(Hb)
such that c(σ) 6= 0 and that it is of the form σ = σφ(h). Then the unitary Uh satisfies
Uh = π(u(λh)) = v(σφ(h)) mod T.
We claim that the map φ = φπ : Ha → Hb is a group isomorphism. For all
h1, h2 ∈ Ha, we get
π(u(λh1+h2)) = π(u(λh1)) π(u(λh2)) = v(σφ(h1)) v(σφ(h2))
= v(σφ(h1) + σφ(h2)) = v(σφ(h1)+φ(h2)) mod T.
On the other hand, we get π(u(λh1+h2)) = v(σφ(h1+h2)) mod T. Since {v(σ)} are
linearly independent, we get σφ(h1+h2) = σφ(h1)+φ(h2), and hence
φ(h1 + h2) = φ(h1) + φ(h2).
This means that the map φ is a group homomorphism. The bijectivity of the ∗-
isomorphism π leads to that of the group homomorphism φ = φπ. Since {γ ·λh | γ ∈
Z2⋊SL(2,Z), h ∈ Ha} ⊂ Λ(Ha) generates Λ(Ha), we get π(u(λ)) = v(φ◦λ) mod T
for λ ∈ Λ(Ha).
10 HIROKI SAKO
We prove that the group isomorphism φ = φπ satisfies conditions (2) and (3) in
the theorem. For all h ∈ Ha, there exists c(h) ∈ T satisfying
Uh = π
µa(h,−h) u(λh)
= c(h)µb(φ(h),−φ(h)) v(σφ(h)).
Since (e1, η) ∈ Z2 ⋊ SL(2,Z) acts on Z2 as (e1, η) · e1 = 0, (e1, η) · 0 = e1, we get
Uh βb(e1, η)(Uh) = π
µa(h,−h) u(λh)µa(h,−h) u(−λh)
= µa(h,−h)
µ̃a(λh,−λh) = 1.
The following equation also holds:
Uh βb(e1, η)(Uh) = c(h)µb(φ(h),−φ(h)) v(σφ(h)) c(h)µb(φ(h),−φ(h)) v(σ−φ(h))
= c(h)2 µb(φ(h),−φ(h))
µ̃b(σφ(h),−σφ(h)) = c(h)2.
Thus we have c(h) ∈ {1,−1} for h ∈ Ha.
Since ξ · e1 = e2, ξ · e2 = 0 and ξ · 0 = e1, we have
Uh βb(ξ)(Uh) βb(ξ
2)(Uh)
µa(h,−h) u(λh)
χa(h)µa(h,−h) u(ξ · λh)
µa(h,−h) u(ξ2 · λh)
= χa(h).
On the other hand, we have the following:
Uh βb(ξ)(Uh) βb(ξ
2)(Uh) = c(h)µb(φ(h),−φ(h)) v(σφ(h))
c(h)χb(φ(h))µb(φ(h),−φ(h)) v(ξ · σφ(h))
c(h)µb(φ(h),−φ(h)) v(ξ2 · σφ(h))
= c(h)3 χb(φ(h)) = c(h)χb(φ(h)).
It follows that
c(h) = χb(φ(h))χa(h)(Eq2)
and χb(φ(h))
2 = χa(h)
2, for all h ∈ Ha.
We recall that the algebra Lµ̃a(⊕Z2Ha) is canonically identified wit the infinite
tensor product
Lµa(Ha). The unitary π
−1(Uh) ∈ N(Ha, µa) ⊂
Lµa(Ha)
can be written as 1⊗ u∗h ⊗ uh, where 1 is placed on −e1 ∈ Z2, u∗h is placed on 0 and
uh is placed on e1. Since η ∈ Z2 ⋊ SL(2,Z) acts on Z2 as η · e1 = −e1, η · 0 = 0,
the unitary π−1(βb(η)(Ug)) can be written as ug ⊗ u∗g ⊗ 1. We have the following
equation:
Ug βb(η)(Uh)U
g βb(η)(Uh)
= π((1⊗ u∗g ⊗ ug)(uh ⊗ u∗h ⊗ 1)(1⊗ u∗g ⊗ ug)∗(uh ⊗ u∗h ⊗ 1)∗) = µ∗aµa(g, h).
The unitary Uh can be written as c(h)(1⊗ v∗φ(h) ⊗ vφ(h)) ∈ N(Hb, µb) ⊂
Lµb(Hb).
Here we write vφ(h) for the unitary in Lµb(Hb) corresponding to φ(h). The unitary
βb(η)(Ug) can be written as c(g)(vφ(g) ⊗ v∗φ(g) ⊗ 1). Then we get
Ug βb(η)(Uh)U
g βb(η)(U
= (1⊗ v∗φ(g) ⊗ vφ(g))(vφ(h) ⊗ v∗φ(h) ⊗ 1)(1⊗ v∗φ(g) ⊗ vφ(g))∗(vφ(h) ⊗ v∗φ(h) ⊗ 1)∗
= µ∗bµb(φ(g), φ(h)).
CLASSIFICATION AND CENTRALIZERS 11
Thus we get µ∗aµa(g, h) = µ
bµb(φ(g), φ(h)), for all g, h ∈ Ha. We proved that the
group isomorphism φ = φπ satisfies conditions (1), (2) and (3). �
From a group homomorphism which satisfies conditions (2) and (3), we construct
a ∗-homomorphism from N(Ha, µa) to N(Hb, µb) with condition (1). In the con-
struction, the function µ̂ on Λ(H) given below is useful. We fix an index for Z2 as
Z2 = {k0, k1, k2, · · · } throughout the rest of this section. For a scalar 2-cocycle µ of
H , we define the function µ̂ by
µ̂(λ) =
λ(ki), λ(kj)
, λ ∈ Λ(H),
where λ is supported on {k0, k1, k2, · · · , kn}. This definition depends on the choice
of an order on Z2. Since
i λ(ki) = 0, the function µ̂ is also given by the following
relation in Cµ(H):
µ̂(λ)1 = uλ(k0)uλ(k1)uλ(k2) · · ·uλ(kn), λ ∈ Λ(H).
If µ is a coboundary, then the definition of µ̂ does not depend on the order on Z2,
since Cµ(H) is commutative.
Lemma 4.2. Let µ0 be another normalized scalar 2-cocycle for H. Let µ̃0 be the
scalar 2-cocycle on Λ(H)×Λ(H) given in the same way as equation (Eq1) in subsec-
tion 2.2 and let µ̂0 be the function on Λ(H) constructed from µ0 in the above manner.
If the scalar 2-cocycles µ and µ0 are cohomologous, then for all λ1, λ2 ∈ Λ(H), we
have the equation
µ̃(λ1, λ2) µ̂(λ1) µ̂(λ2) µ̂(λ1 + λ2) = µ̃0(λ1, λ2) µ̂0(λ1) µ̂0(λ2) µ̂0(λ1 + λ2).
Proof. We denote by {ν(g, h)} the scalar 2-cocycle {µ0(g, h)µ(g, h)} of H . Since ν
is a 2-coboundary, there exists {c(g)}g∈H ⊂ T satisfying ν(g, h) = b(g)b(h)b(g + h).
Then the map ν̂ becomes ν̂(λ) =
b(λ(ki)). Since
ν̂(λ1) ν̂(λ2) =
b(λ1(ki)) b(λ2(ki)),
ν̃(λ1, λ2) =
b(λ1(ki)) b(λ2(ki)) b(λ1(ki) + λ2(ki)),
ν̂(λ1 + λ2) =
b(λ1(ki) + λ2(ki)),
we get ν̂(λ1) ν̂(λ2) = ν̃(λ1, λ2) ν̂(λ1 + λ2). By the definitions of µ̃, µ̃0, µ̂ and µ̂0, the
maps ν̂ and ν̃ are given by
ν̂(λ) = µ̂(λ) µ̂0(λ), ν̃(λ1, λ2) = µ̃(λ1, λ2) µ̃0(λ1, λ2),
Thus the desired equality immediately follows. �
Proof for the second half of Theorem 4.1.
Suppose that there exists a group isomorphism φ satisfying conditions (2) and (3)
in the theorem. We prove that there exists a ∗-isomorphism π = πφ from N(Ha, µa)
onto N(Hb, µb) preserving the Z
2 ⋊ SL(2,Z)-actions with condition (1).
We define a group homomorphism cφ from Ha to {1,−1} ⊂ T by
cφ(h) = χb(φ(h))χa(h), h ∈ Ha.
12 HIROKI SAKO
Let c̃φ be the group homomorphism from Λ(Ha) to {1,−1} ⊂ T given by
c̃φ(λ) =
cφ(λ(k))
gcd(k) =
χa(λ(k))
gcd(k)χb(φ(λ(k)))
gcd(k)
, λ ∈ Λ(Ha).
We define a linear map π from the group algebra Cµ̃a(Λ(Ha)) onto Cµ̃b(Λ(Hb)) by
µ̂a(λ) u(λ)
= c̃φ(λ) µ̂b(φ ◦ λ) v(φ ◦ λ), λ ∈ Λ(Ha).
By direct computations, for all λ1, λ2 ∈ Λ(Ha), we get
µ̂a(λ1) u(λ1)
µ̂a(λ2)u(λ2)
= c̃φ(λ1) c̃φ(λ2) µ̂b(φ ◦ λ1) µ̂b(φ ◦ λ2) v(φ ◦ λ1) v(φ ◦ λ2)
= c̃φ(λ1 + λ2) µ̂b(φ ◦ λ1) µ̂b(φ ◦ λ2) µ̃b(φ ◦ λ1, φ ◦ λ2) v(φ ◦ (λ1 + λ2)).
On the other hand, we have the following equation:
µ̂a(λ1) µ̂a(λ2)u(λ1) u(λ2)
µ̂a(λ1) µ̂a(λ2) µ̃a(λ1, λ2) u(λ1 + λ2)
= µ̂a(λ1) µ̂a(λ2) µ̃a(λ1, λ2) µ̂a(λ1 + λ2) π
µ̂a(λ1 + λ2) u(λ1 + λ2)
= c̃φ(λ1 + λ2) µ̂a(λ1) µ̂a(λ2) µ̃a(λ1, λ2) µ̂a(λ1 + λ2)
µ̂b(φ ◦ (λ1 + λ2)) v(φ ◦ (λ1 + λ2)).
By Lemma 4.2 and condition (2) for the group isomorphism φ in the theorem, we
have that
µ̂b(φ ◦ λ1) µ̂b(φ ◦ λ2) µ̃b(φ ◦ λ1, φ ◦ λ2)
= µ̂a(λ1) µ̂a(λ2) µ̃a(λ1, λ2) µ̂a(λ1 + λ2) µ̂b(φ ◦ (λ1) + φ ◦ (λ2)).
Therefore we get π(u(λ1)) π(u(λ2)) = π(u(λ1) u(λ2)). The linear map π also pre-
serves the ∗-operation. As a consequence, π is a ∗-isomorphism from Cµ̃a(Λ(Ha))
onto Cµ̃b(Λ(Hb)) and this preserves the trace. The map π = πφ is extended to a
normal ∗-isomorphism from N(Ha, µa) onto N(Hb, µb).
We next prove that this π preserves the Z2 ⋊ SL(2,Z)-actions. The group ho-
momorphism c̃φ from Λ(Ha) to {1,−1} is invariant under the action of SL(2,Z),
by Lemma 2.1 (1). The scalar 2-cocycle ν(g, h) = µa(g, h)µb(φ(h), φ(g)) satisfies
ν(g, h) = ν(h, g) by condition (2), so the function ν̂(·) = µ̂a(·)µ̂b(φ ◦ ·) on Λ(Ha)
does not depend on the order on Z2 chosen before. Since
π ◦ βa(γ)(u(λ)) = π(u(γ · λ))
= c̃φ(γ · λ)µ̂a(γ · λ)µ̂b(φ ◦ (γ · λ))v(φ ◦ (γ · λ))
= c̃φ(λ)µ̂a(λ)µ̂b(φ ◦ λ)v(γ · (φ ◦ λ))
= βb(γ)
c̃φ(λ)µ̂a(λ)µ̂b(φ ◦ λ)v(φ ◦ λ)
= βb(γ) ◦ π(u(λ)), γ ∈ SL(2,Z), λ ∈ Λ(Ha),
it turns out that the ∗-isomorphism π preserves the SL(2,Z)-action.
CLASSIFICATION AND CENTRALIZERS 13
For all λ ∈ Λ(Ha) and k ∈ Z2, we have
π ◦ βa(k)(u(λ)) = π
χa(λ(l))
det(k,l)u(k · λ)
cφ((k · λ)(l))gcd(l)
χa(λ(l))
det(k,l)µ̂a(k · λ) µ̂b(φ ◦ (k · λ)) v(φ ◦ (k · λ)).
Since cφ(h)
det(k,l)cφ(h)
gcd(k+l) = cφ(h)
gcd(k)cφ(h)
gcd(l), by Lemma 2.1 (2), the unitary
π ◦ βa(k)(u(λ)) equals to
cφ(λ(l))
gcd(k+l)
cφ(λ(l))
det(k,l)χb(φ ◦ λ(l))det(k,l)
µ̂a(k · λ) µ̂b(φ ◦ (k · λ)) v(φ ◦ (k · λ))
cφ(λ(l))
gcd(k)
cφ(λ(l))
gcd(l)
χb(φ ◦ λ(l))det(k,l)
µ̂a(k · λ) µ̂b(φ ◦ (k · λ)) v(k · (φ ◦ λ))
= c̃φ(λ) µ̂a(k · λ) µ̂b(k · (φ ◦ λ))βb(k)(v(φ · λ))
= βb(k) ◦ π(u(λ)).
This means that the ∗-isomorphism π preserves the Z2-actions.
We get the ∗-isomorphism π = πφ fromN(Ha, µa) ontoN(Hb, µb) giving conjugacy
between βa and βb. �
Remark 4.3. The proof of the first half of Theorem 4.1 shows that any isomorphism
π giving conjugacy between βa and βb is of the form πφ. This means that an
isomorphism which gives conjugacy between two twisted Bernoulli shift actions must
be trace preserving.
This proof shows that an isomorphism giving conjugacy between the two actions
β(Ha, µa, χa), β(Hb, µb, χb) is of a very special form derived from a group isomor-
phism between Ha and Hb. Taking notice of this fact, we can describe the centralizer
of a twisted Bernoulli shift action. We define two topological groups before we state
Theorem 4.4.
Let β be a trace preserving action of some group Γ on a separable finite von
Neumann algebra (N, tr). We denote by Aut(N, β) the group of all automorphisms
which commute with the action β, that is,
{α ∈ Aut(N) | β(γ) ◦ α = α ◦ β(γ), γ ∈ Γ}.
We regard the group Aut(N, β) as a topological group equipped with the pointwise-
strong topology. When β is a twisted Bernoulli shift action on N , an automorphism
α commuting with β is necessarily trace preserving by Remark 4.3. We consider
that Aut(N, β) is equipped with the pointwise-2-norm topology.
Let Aut(H, µ, χ) be the group of all automorphisms of an abelian group H which
preserve its 2-cocycle µ and character χ, that is,
{φ ∈ Aut(H) | µ(g, h) = µ(φ(g), φ(h)), χ(g) = χ(φ(g)), g, h ∈ H}.
We define the topology of Aut(H, µ, χ) by pointwise convergence.
14 HIROKI SAKO
Theorem 4.4. For π ∈ Aut(N(H, µ), β(H, µ, χ)), there exists a unique element
φ = φπ ∈ Aut(H, µ∗µ, χ2) satisfying π(u(λ)) = u(φ ◦ λ) mod T for λ ∈ Λ(H). The
map π 7→ φπ gives an isomorphism between two topological groups
Aut(N(H, µ), β(H, µ, χ)) ∼= Aut(H, µ∗µ, χ2).
Proof. We use the notations in the proof of the previous theorem letting Ha = Hb =
H , µa = µb = µ and χa = χb = χ. Denote N = N(H, µ) and β = β(H, µ, χ). We
have already have shown the first claim. Let
Aut(H, µ∗µ, χ2) ∋ φ 7→ πφ ∈ Aut(N, β),
be the map given as in the proof of Theorem 4.1, that is,
µ̂(λ)u(λ)
= c̃φ(λ) µ̂(φ ◦ λ)u(φ ◦ λ), λ ∈ Λ(H),
where
c̃φ(λ) =
χ(λ(k))gcd(k) χ(φ ◦ λ(k))
gcd(k)
It is easy to prove that φπφ = φ by the definition. Thus the map π 7→ φπ is
surjective. This map is also injective. Let φ be an element of Aut(H, µ∗µ, χ2).
Suppose that π is an arbitrary element of Aut(N, β) satisfying φ = φπ. The set
{β(γ)(u(λh)) | h ∈ H, γ ∈ Z2 ⋊ SL(2,Z)} generates N , so we have only to prove
the uniqueness of c(h) ∈ T satisfying
µ(h,−h) u(λh)
= c(h)µ(φ(h),−φ(h))u(λφ(h)),
for all h ∈ H . In the proof of the first half of the previous theorem (equation
(Eq2)), we have already shown that c(h) = χ(h)χ(φ(h)). Thus the ∗-isomorphism
π is uniquely determined and the map π 7→ φπ is injective.
We prove the two maps φ 7→ πφ and π 7→ φπ are continuous. Let (φi) be a net in
Aut(H, µ∗µ, χ2) converging to φ. For all h ∈ H , we have
χ(h)µ(h,−h) u(λh)
= χ(φi(h))µ(φi(h),−φi(h))u(λφi(h)).
The right side of the equation converges to
χ(φ(h))µ(φ(h),−φ(h))u(λφ(h)) = πφ
χ(h)µ(h,−h) u(λh)
This proves that πφi converges to πφ in pointwise 2-norm topology on the generating
set {β(γ)(u(λh)) | h ∈ H, γ ∈ Z2 ⋊SL(2,Z)} of N . Thus πφi converges to πφ on N .
Conversely, let (πi) be a net in Aut(N, β) converging to π. For all h ∈ H ,
we get πi(u(λh)) = u(λφπi(h)) mod T. The left side of the equation converges to
π(u(λh)) = u(λφπ(h)). If φπi(h) 6= φπ(h), then the distance between Tu(λφπ(h)) and
Tu(λφπi (h)) is
2 in the 2-norm. Thus φπi(h) = φπ(h) for large enough i. This
means that (φπi) converges to φπ.
As a consequence, the two maps φ 7→ πφ and π 7→ φπ are continuous group
homomorphisms and inverse maps of each other. �
CLASSIFICATION AND CENTRALIZERS 15
5. Examples
5.1. Twisted Bernoulli shift actions on L∞(X). In this subsection, we consider
the case of µ = 1 and H 6= {0}. Then the algebra N(H, 1) is abelian and has a
faithful normal state, so it is isomorphic to L∞(X), whereX is a standard probability
space. The measure of X is determined by the trace on N(H, 1). Furthermore, X is
non-atomic, since N(H, 1) is infinite dimensional and the action β(H, 1, χ) is ergodic.
As corollaries of Theorems 4.1 and 4.4, we get trace preserving Z2⋊SL(2,Z)-actions
on L∞(X) whose centralizers are isomorphic to some prescribed groups.
Remark 5.1. The Z2 ⋊ SL(2,Z)-action on X defined by β = β(H, 1, χ) is free.
An automorphism α ∈ Aut(L∞(X), β) is free or the identity map for any twisted
Bernoulli shift action β on L∞(X). This is proved as follows. We identify β(γ)
(γ ∈ Z2 ⋊ SL(2,Z)) and α with measure preserving Borel isomorphisms on X here.
Suppose that there exists a non-null Borel subset Y ⊂ X whose elements are fixed
under α. All elements in Ỹ = ∪{β(γ)(Y ) | γ ∈ Z2⋊SL(2,Z)} are fixed under α. By
the ergodicity of β, the measure of Ỹ is 1. Then α is the identity map of L∞(X).
Corollary 5.2. For any abelian countable discrete group H 6= {0}, there exists a
trace preserving essentially free ergodic action β of Z2⋊SL(2,Z) on L∞(X) satisfying
Aut(L∞(X), β) ∼= Aut(H).
Proof. When we define β = β(H, 1, 1), we have the above relation by Theorem
4.4. �
In the next corollary we use the effect of twisting by a character χ.
Corollary 5.3. For every abelian countable discrete group H 6= {0}, there exist
continuously many trace preserving essentially free ergodic actions {βc} of Z2 ⋊
SL(2,Z) on L∞(X) which are mutually non-conjugate and satisfy
Aut(L∞(X), βc) ∼= H ⋊Aut(H).
Here the topology of H ⋊ Aut(H) is the product of the discrete topology on H and
the pointwise convergence topology on Aut(H).
Proof. Let c ∈ {eiπt | t ∈ (0, 1/2) \Q}. We put βc = β(H ⊕ Z, 1, 1× χc), where the
character χc of Z is defined as χc(n) = c
n. By Theorem 4.4, we get
Aut(L∞(X), βc) ∼= Aut(H ⊕ Z, 1, 1× χ2c).
Since the character χ2c is injective, a group automorphism α ∈ Aut(H⊕Z, 1, 1×χ2c)
preserves the second entry. For all α ∈ Aut(H⊕Z, 1, 1×χ2c), there exist φα ∈ Aut(H)
and hα ∈ H satisfying
α(h, n) = (φα(h) + nhα, n), (h, n) ∈ H ⊕ Z.
The map Aut(H ⊕ Z, 1, 1× χ2c) ∋ α 7→ (hα, φα) ∈ H ⋊ Aut(H) is a homeomorphic
group isomorphism.
If c1, c2 ∈ {eiπt | t ∈ (0, 1/2) \ Q} and c1 6= c2, then there exists no isomorphism
from H ⊕ Z to H ⊕ Z whose pull back of the character 1 × χ2c2 is equal to 1 × χ
The two actions βc1 and βc2 are not conjugate by Theorem 4.1. �
16 HIROKI SAKO
Corollary 5.4. There exist continuously many trace preserving essentially free er-
godic actions {βc} of Z2 ⋊ SL(2,Z) on L∞(X) which are mutually non-conjugate
and have the trivial centralizer Aut(L∞(X), βc) = {idL∞X}.
Proof. Let {χc | c = eiπt, t ∈ (0, 1/2)} be characters of Z such that χc(m) = cm.
Since χ2c(1) is in the upper half plane, the identity map is the only automorphism
of Z preserving χ2c . By Theorem 4.4, we get Aut(β(Z, 1, χc)) = {id}.
If β(Z, 1, χc1), β(Z, 1, χc2) are conjugate, then there exists a group isomorphism
on Z whose pull back of χ2c2 is χ
by Theorem 4.1. This means c1 = c2. Thus the
actions {β(Z, 1, χc)} are mutually non-conjugate. �
5.2. Twisted Bernoulli shift actions on the AFD factor of type II1. Firstly,
we find a condition that the finite von Neumann algebra N(H, µ) is the AFD factor
of type II1.
Lemma 5.5. For an abelian countable discrete group H 6= {0} and its normalized
scalar 2-cocycle µ, the following statements are equivalent:
(1) The algebra N(H, µ) is the AFD factor of type II1.
(2) The group von Neumann algebra Lµ(H) twisted by the scalar 2-cocycle µ is
a factor (of type II1 or In).
(3) For all g ∈ H \ {0}, there exists h ∈ H such that µ(g, h) 6= µ(h, g).
Proof. The amenability of the group Λ(H) leads the injectivity for N(H, µ). The
injectivity for N(H, µ) implies that N(H, µ) is approximately finite dimensional
([Co]). We have only to show the equivalence of conditions (2), (3) and
(1)′ The algebra N(H, µ) is a factor.
By using Fourier expansion it is easy to see that condition (2) holds true if and only
if for any g ∈ H \ {0} there exists h ∈ H satisfying uguh 6= uhug. This is equivalent
to condition (3). Similarly, condition (1)′ is equivalent to
(1)′′ For any λ1 ∈ Λ(H) \ {0}, there exists λ2 ∈ Λ(H) satisfying
µ̃(λ2, λ1)µ̃(λ1, λ2) 6= 1.
Suppose condition (3). For any λ1, choose element k, l ∈ Z2 so that k ∈ supp(λ1)
and l /∈ supp(λ1). By condition (3), there exists h ∈ H satisfying µ∗µ(λ1(k), h) 6= 1.
Let λ2 be the element in Λ(H) which takes h at k, −h at l and 0 for the other
places. The element λ2 satisfies µ̃(λ2, λ1)µ̃(λ1, λ2) = µ
∗µ(λ1(k), h) 6= 1. Here we get
condition (1)′′. The implication from (1)′′ to (3) is easily shown. �
Remark 5.6. The twisted Bernoulli shift action β = β(H, µ, χ) is an outer action
of Z2 ⋊ SL(2,Z). Any non-trivial automorphism in Aut(R, β(H, µ, χ)) is also outer.
This is proved by the weak mixing property of the action β(H, µ, χ) as follows. If
α ∈ Aut(R, β(H, µ, χ)) is an inner automorphism Ad(u), then we have
Ad(β(γ)(u))(x) = β(γ)(uβ(γ)−1(x)u∗) = β(γ) ◦ α ◦ β(γ)−1(x)
= α(x) = Ad(u)(x),
for all x ∈ R and γ ∈ Z2 ⋊ SL(2,Z). Since Ad(β(γ)(u)u∗) = id, Cu ⊂ R is an
invariant subspace of the action β. The only subspace invariant under the weakly
mixing action β is C1 (Proposition 2.6), thus we get α = id.
CLASSIFICATION AND CENTRALIZERS 17
Using Theorems 4.1 and 4.4, we give continuously many actions of Z2 ⋊ SL(2,Z)
on R such that there exists no commuting automorphism except for trivial one.
Corollary 5.7. There exist continuously many ergodic outer actions {βc} of Z2 ⋊
SL(2,Z) on the AFD factor R of type II1 which are mutually non-conjugate and
have the trivial centralizer Aut(R, βc) = {idR}.
Proof. We can choose and fix a character χ on Z2 such that χ2 is injective. Let
{µc | c = eiπt, t ∈ (0, 1/2) \Q} be scalar 2-cocycles for Z2 defined by
= cs1t2−t1s2, s1, t1, s2, t2 ∈ Z.
We put βc = β(Z
2, µc, χ). The 2-cocycle µc satisfies condition (3) in Lemma 5.5.
Thus βc defines a Z
2 ⋊ SL(2,Z)-action on R. By Theorem 4.4, we get the following
isomorphism between topological groups:
Aut(R, βc) ∼= Aut(Z2, µ∗cµc, χ2) = Aut(Z2, µc2, χ2).
Since the character χ2 of Z2 is injective, so the group of the right side is {id|Z2}.
This means that the action βc has trivial centralizers.
Finally, we prove that the actions {βc | c = eiπt, t ∈ (0, 1/2) \ Q} are mutually
non-conjugate. Suppose that actions βc1 and βc2 are conjugate. By Theorem 4.1,
there exists a group isomorphism φ of Z2 satisfying
(g, h) = µc2
(φ(g), φ(h)), g, h ∈ Z2.
A group isomorphism of Z2 is given by an element of GL(2,Z). If the automorphism
φ is given by an element of SL(2,Z), we get c22 = c
1. If φ is given by an element
of GL(2,Z) \ SL(2,Z), then we get c22 = −c21. Since both c1 and c2 have the form
eiπt, t ∈ (0, 1/2), we get c1 = c2. �
Any cyclic group of an odd order can be realized as the centralizer of a twisted
Bernoulli shift actions on R.
Corollary 5.8. Let q be an odd natural number ≥ 3 and denote by Hq the abelian
group (Z/qZ)2. We define the 2-cocycle µq and the character χq on Hq as
= exp (2πis1t2/q), χq
= exp (2πis1/q).
Then the algebra N(Hq, µq) is the AFD factor R of type II1 and the centralizer of
the twisted Bernoulli shift action βa = β(Hq, µq, χq) is isomorphic to Z/qZ.
Proof. By Lemma 5.5, it is shown that the algebra N(HQ, µQ) is the AFD factor of
type II1. Using Theorem 4.4, we have only to prove that Aut(Hq, µ
qµq, χ
∼= Z/qZ.
Let φ be in Aut(Hq, µ
qµq, χ
q). The automorphism φ of Hq is given by a 2 × 2
matrix A of Z/qZ. Since φ preserves µ∗qµq, the determinant of A must be 1. Since
q is odd, the value of χ2q determines the first entry of (Z/qZ)
2 and φ preserves χ2q .
The matrix A is of the form(
, tφ ∈ Z/qZ.
The map φ 7→ tφ is an isomorphism. In turn, if the matrix A is of this form, it
defines an element in Aut(Hq, µ
qµq, χ
q). �
18 HIROKI SAKO
Corollary 5.9. For a set Q consisting of odd prime numbers, let βQ be the tensor
product
q∈Q βq of the actions βq on the AFD factor of type II1. The centralizer
of βQ is isomorphic to
q∈Q Z/qZ.
Proof. The action βQ is the twisted Bernoulli shift action β(HQ, µQ, χQ), where HQ
is the abelian group ⊕q∈QHq and the scalar 2-cocycle µq and a character χQ on HQ
are given by
µQ((sq), (tq)) =
µq(sq, tq),
χQ((sq)) =
χq(sq), (sq), (tq) ∈ HQ, sq, tq ∈ Hq.
Using Theorem 4.4, we have only to prove
Aut(HQ, µ
QµQ, χ
Z/qZ.
A group automorphism φ of HQ = ⊕q∈QHq has a form φ((kq)) = (φq(kq)), for some
{φq ∈ Aut(Hq)}. Thus we get
Aut(HQ, µ
QµQ, χ
Aut(Hq, µ
qµq, χ
Together with the previous corollary, we get the conclusion. �
Remark 5.10. If Q1 6= Q2, then the two groups
Z/qZ and
Z/qZ are not
isomorphic. The continuously many outer actions {βQ} are distinguished in view of
conjugacy only by using the centralizers {Aut(R, βQ)}.
6. Malleability and rigidity arguments
In this section, we give malleability and rigidity type arguments invented by S.
Popa, in order to examine weak 1-cocycles for actions. See Popa [Po2], [Po3], [Po4]
and Popa–Sasyk [PoSa] for the references. S. Popa in [Po3] showed that every
1-cocycle for a Connes-Størmer Bernoulli shift by property (T) group (or w-rigid
group like Z2 ⋊ SL(2,Z)) vanishes modulo scalars. As a consequence, two such
actions are cocycle conjugate if and only if they are conjugate. In our case, 1-
cocycles do not vanish modulo scalars but they are still in the situation that cocycle
(outer) conjugacy implies conjugacy. We need the following notion to examine outer
conjugacy of two group actions.
Definition 6.1. Let α be an action of discrete group Γ on a von Neumann algebra
M. A weak 1-cocycle for α is a map w : Γ → U(M) satisfying
wgh = wgαg(wh) mod T, g, h ∈ Γ.
The weak 1-cocycle w is called a weak 1-coboundary if there exists a unitary v ∈
U(M) satisfying wg = vαg(v)∗ mod T. Two weak 1-cocycles w and w′ are said to be
equivalent when w′g = vwgαg(v)
∗ mod T for some v ∈ U(M).
Let N be a finite von Neumann algebra with a faithful normal trace. The following
is directly obtained by combining Lemmas 2.4 and 2.5 in [PoSa], although these
Lemmas were proved for Bernoulli shift actions on standard probability space. The
CLASSIFICATION AND CENTRALIZERS 19
following can be also regarded as a weak 1-cocycle version of Proposition 3.2 in
[Po4].
Proposition 6.2. Let G be a countable discrete group. Let β be a trace preserving
weakly mixing action of G on N . A weak 1-cocycle {wg}g∈G ⊂ N for β is a weak
1-coboundary if only if there exists a non-zero element x̃0 ∈ N ⊗N satisfying
(wg ⊗ 1)(βg ⊗ βg)(x̃0)(1⊗ w∗g) = x̃0, g ∈ G.
The following is a weak 1-cocycle version of Proposition 3.6.3◦ in [Po4].
Proposition 6.3. Let Γ be a countable discrete group and G be a normal subgroup
of Γ. The group Γ acts on a finite von Neumann algebra N in a trace-preserving
way by β. Suppose that the restriction of β to G is weakly mixing. Let {wγ}γ∈Γ be a
weak 1-cocycle for β. If w|G is a weak 1-coboundary, then w is a weak 1-coboundary
for the Γ-action.
Proof. Suppose that w|G is a weak 1-coboundary, that is, there exists a unitary
element v in N such that wg = vβg(v
∗) mod T for g ∈ G. It suffices to show
that {w′γ} = {v∗wγβγ(v)} is in T for all γ ∈ Γ. Take arbitrary γ ∈ Γ, g ∈ G. Write
h = γ−1gγ ∈ G. Let πγ be the unitary on L2(N) induced from βγ . Since w′h, w′g ∈ T,
we get
w′γπgw
= (w′γπγ)(w
hπh)(w
∗ = w′γhγ−1πγhγ−1 = πg mod T,
By applying these operators to 1̂ ∈ N̂ ⊂ L2(N), it follows that w′γβg(w′γ
) ∈ T.
Since the G-action is weakly mixing, we have w′γ ∈ T. �
By using the above propositions, we will “untwist” some weak 1-cocycles later.
We require some ergodicity assumption on the weak 1-cocycles.
Definition 6.4. Let Γ be a discrete group and G be a subgroup of Γ. Suppose that
its restriction to G is ergodic. Let β be a trace preserving action of Γ on N . A weak
1-cocycle w = {wg}g∈Γ for β is said to be ergodic on G, if the action βw of G is still
ergodic, where βw is defined by βwg = Adwg ◦ βg, g ∈ G.
Let β be a Γ-action onN . Suppose that the diagonal action β⊗β on (N⊗N, tr⊗tr)
has an extension β̃ on a finite von Neumann algebra (Ñ , τ). The algebra Ñ is not
necessarily identical with N⊗N . When the action β̃ is ergodic on a normal subgroup
G ⊂ Γ, we get the following:
Proposition 6.5. Let {wγ}γ∈Γ ⊂ N be a weak 1-cocycle for β. Let α be a trace
preserving continuous action of R on Ñ satisfying the following properties:
• α1(x⊗ 1) = 1⊗ x, for all x ∈ N .
• αt ◦ β̃(γ)(x̃) = β̃(γ) ◦ αt(x̃), for all t ∈ R, γ ∈ Γ and x̃ ∈ Ñ .
Suppose that the weak 1-cocycle {wγ⊗1} ⊂ Ñ is ergodic for the G-action β̃|G. If the
group inclusion G ⊂ Γ has the relative property (T) of Kazhdan, then there exists a
non-zero element x̃0 ∈ Ñ so that (wg ⊗ 1)β̃g(x̃0)(1⊗ w∗g) = x̃0, g ∈ G.
This is proved in the same way for Bernoulli shift actions on the infinite tensor
product of abelian von Neumann algebras ([PoSa], Lemma 3.5). Since we are in-
terested in actions on the AFD II1 factor, we require the ergodicity assumption on
20 HIROKI SAKO
weak 1-cocycle {wγ ⊗ 1}. For the self-containedness and in order to make it clear
where the ergodicity assumption works, we write down a complete proof.
Proof. For t ∈ (0, 1], let Kt be the convex weak closure of
{(wg ⊗ 1)αt(w∗g ⊗ 1) | g ∈ G} ⊂ Ñ
and x̃t ∈ Kt be the unique element whose 2-norm is minimum in Kt. Since
(wg ⊗ 1)β̃g((wg1 ⊗ 1)αt(w∗g1 ⊗ 1))αt(w
g ⊗ 1)
= (wgβg(wg1)⊗ 1)αt(βg(w∗g1)w
g ⊗ 1)
= (wgg1 ⊗ 1)αt(w∗gg1 ⊗ 1), g, g1 ∈ G,
we have (wg ⊗ 1)β̃g(Kt)αt(w∗g ⊗ 1) = Kt, for g ∈ G. By the uniqueness of x̃t, we get
(wg ⊗ 1)β̃g(x̃t)αt(w∗g ⊗ 1) = x̃t, g ∈ G.(Eq3)
By the assumption, the action (Ad(wg ⊗ 1) ◦ β̃g)g∈G is ergodic on Ñ . By the calcu-
lation
(wg ⊗ 1)β̃g(x̃tx̃t∗)(w∗g ⊗ 1)
= (wg ⊗ 1)β̃g(x̃t)αt(w∗g ⊗ 1)αt(wg ⊗ 1)β̃g(x̃t
)(w∗g ⊗ 1)
= x̃tx̃t
, g ∈ G,
we get x̃tx̃t
∗ ∈ C1. The element x̃t is a scalar multiple of a unitary in Ñ .
We shall next prove that x̃1/n is not zero for some positive integer n. The pair
(Γ, G) has the relative property (T) of Kazhdan. By proposition 2.4, we can find a
positive number δ and a finite subset F ⊂ Γ satisfying the following condition: If a
unitary representation (π,H) of Γ and a unit vector ξ of H satisfy ‖π(γ)ξ − ξ‖ ≤
δ (γ ∈ F ), then ‖π(g)ξ− ξ‖ ≤ 1/2 (g ∈ G). By the continuity of the action α, there
exists n such that
‖(wγ ⊗ 1)α1/n(w∗γ ⊗ 1)− 1‖tr,2 ≤ δ, γ ∈ F,
The actions β and (αl1/n)l∈Z on Ñ give a Γ× Z action on Ñ . Let P be the crossed
product von Neumann algebra P = Ñ ⋊ (Γ × Z). Let (Uγ)γ∈Γ and W be the
implementing unitaries in P for Γ and 1 ∈ Z respectively. We put Vγ = (wγ ⊗
1)Uγ , γ ∈ Γ. We regard AdV· as a unitary representation of Γ on L2(P ). Since
‖AdVγ(W )−W‖L2(P ) = ‖(wγ ⊗ 1)W (w∗γ ⊗ 1)W ∗ − 1‖L2(P )
= ‖(wγ ⊗ 1)α1/n(w∗γ ⊗ 1)− 1‖L2(Ñ) ≤ δ, γ ∈ F,
we have the following inequality:
1/2 ≥ ‖AdVg(W )−W‖L2(P ) = ‖(wg ⊗ 1)α1/n(w∗g ⊗ 1)− 1‖L2(Ñ), g ∈ G.
We get 1/2 ≥ ‖x̃1/n − 1‖L2(Ñ) and x̃1/n 6= 0.
Let ũ1/n be the unitary of Ñ given by a scalar multiple of x̃1/n. By equation
(Eq3), the unitary satisfies
(wg ⊗ 1)β̃g(ũ1/n)α1/n(w∗g ⊗ 1) = ũ1/n, g ∈ G.
CLASSIFICATION AND CENTRALIZERS 21
Let x̃0 be the unitary defined by
x̃0 = ũ1/nα1/n(ũ1/n)α2/n(ũ1/n) . . . α(n−1)/n(ũ1/n).
By direct computations, we have the following desired equality:
(wg ⊗ 1)β̃g(x̃0)(1⊗ w∗g) = (wg ⊗ 1)β̃g(x̃0)α1(w∗g ⊗ 1) = x̃0, g ∈ G.
Theorem 6.6. Let β = β(H, µ, χ) be a twisted Bernoulli shift action on N(H, µ).
Suppose that N(H, µ) is the AFD factor of type II1 and that there exists a continuous
R-action (α
t )t∈R on Lµ(H)⊗ Lµ(H) satisfying the following properties:
• For any x ∈ Lµ(H), α(0)1 (x⊗ 1) = 1⊗ x,
• The automorphism α(0)t commutes with the diagonal action of Ĥ.
Let β(1) be another action of Z2 ⋊ SL(2,Z) on the AFD factor N (1) of type II1 and
suppose that its restriction to Z2 is ergodic. The action β(1) is outer conjugate to β,
if and only if β(1) is conjugate to β.
Proof. We deduce from outer conjugacy to conjugacy in the above situation. Let θ
be a ∗-isomorphism from N (1) onto N(H, µ) which gives the outer conjugacy of the
action β(1) and β = β(H, µ, χ). There exists a weak 1-cocycle {wγ}γ∈Z2⋊SL(2,Z) for β
satisfying
θ ◦ β(1)(γ) = Adwγ ◦ β(γ) ◦ θ, γ ∈ Z2 ⋊ SL(2,Z).
Since the action β(1) is ergodic on Z2, the weak 1-cocycle w is ergodic on Z2.
We use the notations Γ0, G0 given in Section 3. Let ρ̃ be the diagonal action ρ⊗ρ
of Γ0 on the tensor product algebra M̃ = Lµ̃(⊕Z2H)⊗ Lµ̃(⊕Z2H):
ρ̃(γ0)(a⊗ b) = ρ(γ0)(a)⊗ ρ(γ0)(b).
The fixed point algebra Ñ ⊂ M̃ of the diagonal Ĥ-action contains N(H, µ) ⊗
N(H, µ). Since Z2⋊SL(2,Z) = Γ0/Ĥ, the action ρ̃ gives a Z
2⋊SL(2,Z)-action β̃ on
Ñ . The action β̃ is the extension of the diagonal action β⊗β on N(H, µ)⊗N(H, µ).
We denote by αt the action on M̃ ∼=
(Lµ(H)⊗Lµ(H)) given by the infinite tensor
product of the R-action α
t . By the assumption on α
t , the R-action αt commutes
with the action ρ̃. It follows that the subalgebra Ñ is globally invariant under αt.
The set of unitary {Wγ = wγ ⊗ 1}γ∈Z2⋊SL(2,Z) ⊂ Ñ is a weak 1-cocycle for β̃. We
shall prove that this weak 1-cocycle is ergodic on Z2. Let a be an element in Ñ fixed
under β̃|Z2 . The element a can be written as a =
H aλ⊗u(λ) in L2M̃ , where
aλ ⊗ 1 = EM⊗C(a(1⊗ u(λ))∗). Since a is fixed under the action of Z2, we have
a = β̃W (k)(a) =
Adwk ◦ ρ(1, k)(aλ)⊗ ρ(1, k)(u(λ))
Adwk ◦ ρ(1, k)(aλ)⊗
χ(λ(l))det(k,l)u(k · λ).
Since Adwk ◦ ρ(1, k) preserves the 2-norm, we get ‖aλ‖2 = ‖ak−1·λ‖2. Since ‖a‖22 =∑
‖aλ‖22 < ∞ and the set {ak−1·λ | k ∈ Z2} is infinite for λ 6= 0, it turns out that
aλ = 0 for λ 6= 0 and thus a ∈ Ñ ∩ (M ⊗ C) = N(H, µ) ⊗ C. By the ergodicity of
22 HIROKI SAKO
the Z2-action {Adwk ◦ β(k)}, we get a ∈ C. We conclude that the weak 1-cocycle
{Wγ} ⊂ Ñ is ergodic on Z2.
By the relative property (T) for the inclusion Z2 ⊂ Z2⋊SL(2,Z) and Proposition
6.5, there exists a non-zero element x̃0 ∈ Ñ satisfying
(wk ⊗ 1)β̃(k)(x̃0)(1⊗ w∗k) = x̃0, k ∈ Z2.
The element x̃0 can be written as the following Fourier expansion:
x̃0 =
c(λ1, λ2)u(λ1)⊗ u(λ2) ∈ Ñ ⊂ Lµ̃(⊕Z2H)⊗ Lµ̃(⊕Z2H).
Here c(λ1, λ2) is a complex number and (λ1, λ2) ∈ (⊕Z2H)2 runs through all pairs
satisfying
k∈Z2(λ1(k) + λ2(k)) = 0. Choose and fix a pair (λ1, λ2) satisfying
λ1(k) = h =
λ2(k), c(λ1, λ2) 6= 0.
Let v′h ∈ M be the unitary written as v′h = uh ⊗ 1 ⊗ 1 ⊗ · · · , where uh ∈ Lµ(H) is
placed on 0 ∈ Z2. The following unitaries {w′γ} ⊂ N(H, µ) give a weak 1-cocycle
for β:
w′(k,γ0) = v
hw(k,γ0)ρ(1, k, γ0)(v
), (k, γ0) ∈ Z2 ⋊ SL(2,Z).
Letting ỹ = (v′h ⊗ 1)x̃0(1⊗ v′h)∗ ∈ M̃ , we get
ỹ = (w′k ⊗ 1)β̃(k)(ỹ)(1⊗ w′k
), k ∈ Z2.
Applying the trace preserving conditional expectation E = EN(H,µ)⊗N(H,µ), we get
E(ỹ) = (w′k ⊗ 1)E(β̃(k)(ỹ))(1⊗ w′k
= (w′k ⊗ 1)β̃(k)(E(ỹ))(1⊗ w′k
), k ∈ Z2.
Since the Fourier coefficient of x̃0 at (λ1, λ2) ∈ (⊕Z2H)2 is not zero, that of E(ỹ)
at (λ1 + δh,0, λ2 − δh,0) ∈ Λ(H)2 is also non-zero, where δh,0 ∈ ⊕Z2H is zero on
Z2 \ {0} and is h on 0 ∈ Z2. By Proposition 6.2, it follows that the weak 1-cocycle
{w′(k,e)}k∈Z2 ⊂ N(H, µ) is a weak 1-coboundary of β|Z2. Since the Z2-action β|Z2 is
weakly mixing, w′ is a weak 1-coboundary on Z2 ⋊ SL(2,Z), by Proposition 6.3. In
other words, there exists v ∈ N(H, µ) satisfying
w′γ = vβ(γ)(v
∗) mod T,
wγ = v
vρ(1, γ)(v∗v′h) mod T, γ ∈ Z2 ⋊ SL(2,Z).
Noting that u = v∗v′h ∈ M is a normalizer of N(H, µ), we get
(Ad(u) ◦ θ) ◦ β(1)(γ) = Ad(u) ◦ Ad(wγ) ◦ β(γ) ◦ θ
= Ad(ρ(1, γ)(u)) ◦ β(γ) ◦ θ
= ρ(1, γ) ◦ Ad(u) ◦ θ
= β(γ) ◦ (Ad(u) ◦ θ), γ ∈ Z2 ⋊ SL(2,Z).
Thus we get the conjugacy of two Z2 ⋊ SL(2,Z)-actions β(0) and β. �
We can always apply Theorem 6.6 if H is finite.
CLASSIFICATION AND CENTRALIZERS 23
Corollary 6.7. Let H be a finite abelian group and let β = β(H, µ, χ) be a twisted
Bernoulli shift action on N(H, µ). Suppose that N(H, µ) is the AFD factor of type
II1. Let β
(1) be an action of Z2 ⋊ SL(2,Z) on the AFD factor N (1) of type II1 and
suppose that its restriction to Z2 is ergodic. The action β(1) is outer conjugate to β,
if and only if β(1) is conjugate to β.
Proof. We have only to construct an R-action on Lµ(H) ⊗ Lµ(H) satisfying the
properties in Theorem 6.6. Let U be an element of Lµ(H)⊗ Lµ(H) defined by
|H|1/2
uh ⊗ u∗h.
We note that µ∗µ(g, ·) is a character of H and that it is not identically 1 provided
g 6= 0 by Lemma 5.5. The element U is self-adjoint and unitary, since
|H|1/2
u∗h ⊗ uh =
|H|1/2
µ(h,−h)u−h ⊗ µ(h,−h)u∗−h = U,
g,h∈H
uguh ⊗ u∗gu∗h =
g,h∈H
µ∗µ(g, h)ug+h ⊗ u∗g+h
µ∗µ(g, h− g)
ug ⊗ u∗g = 1.
The operator U is a fixed point under the action of Ĥ , so the projections P1 =
(1 + U)/2 and P−1 = (1 − U)/2 are also fixed points. Thus the R-action α(0)t =
Ad(P1+exp (iπt)P−1) commutes with the Ĥ-action. The automorphism α
1 satisfies
1 (ug ⊗ 1) = U(ug ⊗ 1)U∗ = (1⊗ ug)UU∗ = 1⊗ ug, g ∈ H.
This verifies the first condition for α(0). �
Corollary 6.8. Let Q be a set consisting of odd prime numbers and βQ be the twisted
Bernoulli shift action defined in Corollary 5.9. Let β be a Z2 ⋊ SL(2,Z)-action on
the AFD factor of type II1 whose restriction to Z
2 is ergodic. The actions βQ and
β are outer conjugate if and only if they are conjugate. In particular, {βQ} is an
uncountable family of Z2⋊SL(2,Z)-actions which are mutually non outer conjugate.
Proof. We will use the notation given in Corollary 5.8 and 5.9. Let α
t be the R-
action on Lµq (Hq)⊗Lµq(Hq) constructed as in the previous corollary. We define the
R-action α(Q) on LµQ(HQ)⊗LµQ(HQ) by α
t (⊗q∈Qxq) = ⊗q∈Qα
t (xq), where xq ∈
Lµq (Hq) ⊗ Lµq (Hq) and xq 6= 1 only for finitely many q. The R-action satisfies the
conditions in Theorem 6.6. By Corollary 5.9, {βQ} are mutually non conjugate and
their restriction to Z2 is ergodic. Thus they are mutually non outer conjugate. �
Acknowledgment . The author would like to thank Professor Yasuyuki Kawahigashi
for helpful conversations. He thanks the referee for careful reading and numerous
detailed comments. He is supported by JSPS Research Fellowships for Young Sci-
entists.
24 HIROKI SAKO
References
[Bu] M. Burger, Kazhdan constants for SL3(Z), J. Reine Angew. Math., 413 (1991), 36–67.
[Ch] M. Choda, A continuum of non-conjugate property T actions of SL(n,Z) on the hyperfinite
II1-factor, Math. Japon., 30 (1985), 133–150.
[Co] A. Connes, Classification of injective factors. Cases II1, II∞, IIIλ, λ 6= 1. Ann. of Math. (2)
104 (1976), no. 1, 73–115.
[Jol] P. Jolissaint, On property (T) for pairs of topological groups. Enseign. Math., (2) 51 (2005),
no. 1-2, 31E5.
[Jon] V. F. R. Jones, A converse to Ocneanu’s theorem, J. Operator Theory, 10 (1983), no. 1,
61–63.
[Ka] Y. Kawahigashi, Cohomology of actions of discrete groups on factors of type II1, Pac. J.
Math., 149 (1991), no. 1, 303–317.
[NPS] R. Nicoara, S. Popa, R. Sasyk, On II1 factors arising from 2-cocycles of w-rigid groups. J.
Funct. Anal. 242 (2007),no. 1, 230–246.
[Oc] A. Ocneanu, Actions of discrete amenable groups on von Neumann algebras, Lecture Notes
in Mathematics, 1138. Springer-Verlag, Berlin, 1985.
[OPT] D. Olesen, G. K. Pedersen, M. Takesaki, Ergodic actions of compact abelian groups, J.
Operator Theory, 3 (1980), no. 2, 237–269.
[Po1] S. Popa, On a class of type II1 factors with Betti numbers invariants, Ann. of Math., (2)
163 (2006), no. 3, 809–899.
[Po2] S. Popa, Some computations of 1-cohomology groups and construction of non orbit equivalent
actions, J. Inst. Math. Jussieu, 5 (2006), 309–332.
[Po3] S. Popa, Some rigidity results for non-commutative Bernoulli shifts, J. Funct. Anal., 230
(2006), no. 2, 273–328.
[Po4] S. Popa, Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups,
Invent. Math. 170 (2007), no. 2, 243–295.
[PoSa] S. Popa, R. Sasyk, On the cohomology of Bernoulli actions, Erg. Theory Dyn. Sys., 26
(2007), 1–11.
[Sh] Y. Shalom, Bounded generation and Kazhdan’s Property (T). Publ. Math. IHES, 90 (1999),
145–168.
[Vaes] S. Vaes, Rigidity results for Bernoulli actions and their von Neumann algebras (after Sorin
Popa), Sminaire Bourbaki, exp. no. 961, Astrisque 311 (2007) 237–294.
Department of Mathematical Sciences, University of Tokyo, Komaba, Tokyo,
153-8914, Japan
E-mail address : hiroki@ms.u-tokyo.ac.jp
	1. Introduction
	2. Preparations
	2.1. Functions det and gcd
	2.2.  Scalar 2-cocycles for abelian groups 
	2.3. Definition of a Z2 SL(2,Z)-action on (H)
	2.4. On the relative property (T) of Kazhdan
	2.5. Weakly mixing actions
	2.6. A remark on group von Neumann algebras
	3.  Definition of twisted Bernoulli shift actions
	4. Classification up to conjugacy
	5. Examples
	5.1. Twisted Bernoulli shift actions on L(X)
	5.2. Twisted Bernoulli shift actions on the AFD factor of type II1
	6. Malleability and rigidity arguments
	References
