Classification of actions of duals of finite groups on the AFD factor of type II_1

0.31 MB·English

by  Toshihiko MASUDA

Classification of actions of duals of finite groups
on the AFD factor of type II
1
6
MASUDA Toshihiko,
0 ∗
0
Graduate School of Mathematics, Kyushu University,
2
n 6-10-1 Hakozaki, Fukuoka, 812-8581, JAPAN
a
J

5
2
Abstract

]
A
We will show the uniqueness of outer coactions of finite groups on the AFD
O factor of type II along the arguments by Connes, Jones and Ocneanu. Namely, we
1
. construct the infinite tensor product type action, adopt it as the model action, and
h
t prove that any outer coaction is conjugate to the model action.
a
m
[
1 Introduction

1
v
1 In the theory of operator algebras, the study of automorphisms is one of the most impor-
0
tant topics. Especially, much progress has been made on classification of automorphisms
6
1 and group actions on injective factors since fundamental works of A. Connes. In [2]
0 and [3], A. Connes succeeded in classifying automorphisms of the approximately finite
6
dimensional (AFD) factor of type II up to outer conjugacy. The first generalization of
0 1
/ Connnes’ results was made by V. F. R. Jones in [5], where he classified actions of finite
h
t groups on the AFD factor of type II . Soon after Jones’ theory, A. Ocneanu classified
a 1
m actions of discrete amenable groups on the AFD factor of type II . One of extension of
1
: their results is analysis (or classification) of actions of dual object of groups, i.e., coaction
v
i of groups, which will be useful to understand actions of compact groups. (See [8] on basic
X
of coactions.)
r
a In this paper, we give the classification theorem for outer coactions of finite groups.
Here we have to remark that this follows indirectly from the works cited in above. In
fact, the uniqueness of outer coactions of finite groups follows from [5], since every outer
coaction of a finite group is dual to some outer (usual) action. Nowadays, this also follows
from the general theory of Popa’s classification of subfactors [10] (also see [12]). However
in these approach, one does not handle coactions directly. Hence in this paper, we present
the direct approach for classification theorem of outer coactions of finite groups on the
AFD factor of type II , which can be generalized to finite dimensional Kac algebras. Our
1
argument is similar to Connes-Jones-Ocneanu theory. We construct the model action on
the AFD factor of type II , prove several cohomology vanishing theorem, and compare a
1
given action to the model action. The main technique in this arguments is the ultraprod-
uct and the central sequence algebra. Unfortunately, coactions do not necessary induce
∗e-mail address [email protected]
1
coactions on the central sequence algebra unlike the usual group action case. Hence we
have to modify actions to apply the ultraproduct technique to handle with coactions, and
this is one of the important point in our theory.
Here we have another formulation to treat actions of duals of (finite) groups other
than coactions due to Roberts in [13]. His approach is essentially equivalent to coactions.
However it is convenient (at least for the author) to regard coactions as the Roberts type
actions, which we often call actions of of finite group duals. Hence in this paper, we
present our main theorem as the uniqueness of Roberts type actions of finite group duals.
This paper is organized as follows. In 2, we prepare notations used in this paper,
§
and discuss the Roberts type actions. In 3, we construct the infinite tensor product type
§
action, which we call the model action. In 4, we collect some technical lemmas, which
§
is necessary to treat actions on the ultraproduct algebra in 6. In 5, we show three
§ §
kinds of cohomology vanishing theorem, which are important tools for analysis of actions.
The contents in 6 and 7 are central in this paper. We discuss actions on the ultra
§ §
product algebra, and the central sequence algebra. By means of cohomology vanishing,
we construct the piece of the model action, and complete classification. In appendix, we
present the Roberts type action approach for (twisted) crossed product construction for
actions of finite group duals.
2 Preliminaries and Notations
2.1 Notations on duals of finite groups
Throughout this paper, we always assume that G is a finite group. Let Rep(G) and
Irr(G) be the collection of all finite dimensional unitary representations, and irreducible
unitary representations of G respectively. We denote the trivial representation by 1. We
fix representative elements of Irr(G)/ , where means a usual unitary equivalence, and
∼ ∼
denote by Gˆ, and assume 1 Gˆ.
∈
Let dπ := dimH be the dimension of π Rep(G). For π,ρ Rep(G), we denote the
π
∈ ∈
intertwiner space between σ and π by (σ,π) := T B(H ,H ) Tσ(g) = π(g)T,g G .
σ π
{ ∈ | ∈ }
If σ is irreducible, (σ,π) becomes a Hilbert space with an inner product T,S 1 := S T.
∗
Let π,ρ Gˆ, and π ρ = Nσ σ be the irreducible decompositionh, wheire Nσ is a
∈ ⊗ ∼ ⊕σ Gˆ πρ πρ
multiplicity. Fix an orthonormal∈basis Tσ,e Nπσρ (σ,π ρ). Then we have Tσ,e Tξ,f =
{ π,ρ}e=1 ⊂ ⊗ π,ρ∗ π,ρ
δ δ 1 , and Tσ,eTσ,e = 1 , where 1 (σ,σ) is an identity. Hence we have
πσ(,gξ)e,f ρσ(g) = σ,eTσπ,e,ρσ(πg,)ρT∗ σ,e eπs⊗pρecially. σ ∈
⊗ Pσ,e π,ρ π,ρ∗
Let v(π) with v(π) M (C). Then Tσ,ev(σ)Tσ,e does not depend on the
{ }πPGˆ ∈ dπ e π,ρ π,ρ∗
choice of Tσ,e ∈ (σ,π ρ).
{ π,ρ} ⊂ ⊗ P
In a similar way, one can easily see
(Tπη,,ρa ⊗1σ)Tηξ,,σbv(ξ)Tηξ,,σb∗(Tπη,,ρa∗ ⊗1σ) = (1π ⊗Tρζ,,σc)Tπξ,,ζdv(ξ)Tπξ,,ζd∗(1π ⊗Tρζ,,σc∗)
η,a,b ζ,c,d
X X
since both (Tη,a 1 )Tξ,b and (1 Tζ,c)Tξ,d are orthonormal basis for (ξ,π
{ π,ρ ⊗ σ η,σ}η,a,b { π⊗ ρ,σ π,ζ}ζ,c,d ⊗
ρ σ).
⊗
Remark. Assume that v(π) , v(π) A B(H ), is given for some vector space A.
{ }π Gˆ ∈ ⊗ π
∈
2
We can extend v(π) for a general π Rep(G) as follows. Let π = σi, σi Gˆ, be an
∼ i
∈ ⊕ ∈
irreducible decomposition, andfix Ti (σi,π) withTi Tj = δ and TiTi = 1. Define
∈ ∗ i,j i ∗
v(π) = Tiv(σi)Ti A B(H ). Then v(π) is well-defined, i.e., it is independent
i ∗ ∈ ⊗ π P
from the choice of Ti , and satisfies v(π)T = Tv(σ) for T (σ,π). In this notation,
P { } ∈
the contents of the previous paragraph is written as v((π ρ) σ) = v(π (ρ σ)), for
⊗ ⊗ ⊗ ⊗
example.
Let eπ dπ be an orthonormal basis for H , and fix it. Let us express Tσ,e = (Tσm,e)
{ i}i=1 π π,ρ πi,ρk
as a matrix form. Then we can write Tσ,e Tξ,f = δ δ 1 and Tσ,eTσ,e = 1 by
π,ρ∗ π,ρ σ,ξ e,f σ σ,e π,ρ π,ρ∗ π ρ
⊗
matrix coefficients as
Tσm,eTξn,f = δ δ δ , P
πi,ρk πi,ρk σ,ξ e,f m,n
i,k
X
and
Tσm,eTσm,e = δ δ .
πi,ρk πj,ρl i,j k,l
σ,m,e
X
Let T1 (1,π π¯) be an isometry given by T1 1 = 1 eπ eπ¯, and fix it. It is
π,π¯ ∈ ⊗ π,π¯ √dπ i i ⊗ i
easy to see Tπ1i,π¯j = √δid,jπ. Since Tπ1,∗π¯Tπρ,,π¯e = δ1,ρ, we have kTπρkPl,,eπ¯k = √dπδ1,ρ
Set T˜ρ,e := √dρdπ(1 Tσ,e )(T1 1 ) (ρ,π¯ σ). Then T˜ρ,e is an orthonormal
π¯,σ √dσ π¯ ⊗ π,ρ∗ π¯,π ⊗ ρ ∈ ⊗ P { π¯,σ}
basis for (ρ,π¯ σ). It is easy to see T˜ρk,e = dρTσm,e. As a consequence we have
⊗ π¯iσm dσ πi,ρk
q
dσ
Tσm,ev(σ) Tσn,e = Tρk,e v(σ) Tρl,e
πi,ρk m,n πj,ρl dρ π¯i,σm m,n π¯j,σn
σ,m,n,e σ,m,n,e
X X
for example.
2.2 Coactions and Roberts type actions
Let A,B be von Neumann algebras. We denote the set of unital -homomorphisms from
∗
A to B by Mor(A,B).
Letu bethe(right)regularrepresentationofG,andR(G) := u thegroupalgebra.
g g ′′
{ }
The coproduct ∆ of R(G) is given by ∆(u ) = u u .
g g g
⊗
For simplicity, we denote 1 T M B(H ,H ), T B(H ,H ), by T.
M π ρ π ρ
⊗ ∈ ⊗ ∈
Definition 2.1 (1) Let M be a von Neumann algebra. We say α = α is an
π π Rep(G)
{ } ∈
action of Rep(G) if α Mor(M,M B(H )), and following hold.
π π
∈ ⊗
(1a) α1 = idM.
(1b) α (x)T = Tα (x) for any T (σ,π).
π σ
∈
(1c) α id α = α .
π σ σ π σ
⊗ ◦ ⊗
(2) We say α = α is an action of Irr(G) if α Mor(M,M B(H )) and
π π Irr(G) π π
{ } ∈ ∈ ⊗
following holds.
(2a) α1 = idM.
(2b) α id α (x)T = Tα (x) for any T (σ,π ρ).
π ρ ρ σ
⊗ ◦ ∈ ⊗
(3) We say α = α an action of Gˆ if α Mor(M,M B(H )) and we have the
{ π}π Gˆ π ∈ ⊗ π
following. ∈
(3a) α1 = idM.
(3b) α id α (x)T = Tα (x) for any T (σ,π ρ).
π ρ ρ σ
⊗ ◦ ∈ ⊗
3
If an action α of Gˆ is given, then it is a routine work to extend α to those of Irr(G) and
Rep(G). Hence in this paper, we do not distinguish these notions. When M is properly
infinite, it is not difficult to see Definition 2.1 is reduced to that of Roberts action.
We remark that α is automatically injective. Suppose α (x) = 0. Then we have
π π
1 1 1 1
0 = Tπ¯,∗παπ¯ ⊗idπ ◦απ(x)Tπ¯,π = Tπ¯,∗πTπ¯,πα1(x) = x.
Let eπ beasystemofmatrixunitsforB(H ). Thenα (x)isdecomposed asα (x) =
{ ij} π π π
α (x) eπ. The -homomorphism property of α implies α (x) α (y) =
i,j π ij ⊗ ij ∗ π k π ik π kj
α (xy) and (α (x) ) = α(x ) .
π ij π ij ∗ ∗ ji
P P
The group algebra R(G) can be decomposed as R(G) = B(H ), and eπ =
π Gˆ π ij
dπ/ G π(g) u gives a matrix unit for B(H ). Then α Mor(M∈ ,M R(G)) can be
| | g ij g π ∈ L ⊗
decomposed as α(x) = α (x) (x) eπ, and we get α Mor(M,M B(H )). One
P π π ij ⊗ ij π ∈ ⊗ π
can verify that α is a coaction, i.e., α is injective and satisfies (α id) α = (id ∆) α.
if and only if α is anPaction of Gˆ in the sense of Definition 2.1⊗. ◦ ⊗ ◦
π
{ }
Definition 2.2 Let α be an action of Gˆ on M. The fixed point algebra Mα is defined as
Mα := a M α (a) = a 1 for any π Gˆ .
π π
{ ∈ | ⊗ ∈ }
If K Mα, then we say α is trivial on K, and often write as α = id on K. Let
π
⊂
K M be a von Neumann subalgebra, on which α acts trivially. Then it is easily seen
⊂
that α (K M) (K M) B(H ), and α is an action on K M. Note that even if
π ′ ′ π ′
∩ ⊂ ∩ ⊗ ∩
we have α (K) K B(H ), α does not induce an action on K M in general unlike
π π ′
⊂ ⊗ ∩
the usual group action case.
Let α be an action of Gˆ on M, and N be another von Neumann algebra. Then
α (x) := α (x) 1 eπ is an action of Gˆ on M N, which we denote by α id
π′ i,j π ij⊗ N ⊗ ij ⊗ ⊗ N
for simplicity.
P
2.3 Crossed product construction by Roberts type action
Let α be a coaction of G on M. The crossed product M ⋊ Gˆ is defined as α(M) C
α
∨ ⊗
ℓ (G) M B(ℓ2(G)). We discuss the crossed product construction from the point of
∞
⊂ ⊗
view of the Roberts type action. (Also see Appendix.)
We begin with the following definition.
Definition 2.3 Let M be a von Neumann algebra. We say U is a (unitary)
π π Irr(G)
{ } ∈
representation of Irr(G) in M if we have the following.
(1) Uπ U(M B(Hπ)), U1 = 1.
∈ ⊗
(2) Let F B(H H ,H H ) be a flip map. Set U12 := U 1 , and U13 :=
π,σ ∈ π ⊗ ρ ρ ⊗ π π π ⊗ ρ ρ
F (U 1 )F . Then U12U13T = TU for any T (σ,π ρ).
ρ,π ρ ⊗ π π,ρ π ρ σ ∈ ⊗
If werepresent U andT asU = (U ) andT = (Tm)1 m dσ respectively
π π πij 1 i,j dπ i,k 1≤i ≤dπ,1 k dρ
by matrix elements, then Definition 2.3(2) ≤is w≤ritten as U U≤≤ Tn≤=≤ TmU .
j,l πij ρkl j,k m i,k σmn
P P
Lemma 2.4 Let U be a unitary representation of Gˆ. Then we have U = U , and
{ π} π∗ij π¯ij
[U ,U ] = 0.
πij ρkl
1 1
Proof. Sincewehave j,lUπijUπ¯klTπj,π¯l = Tπi,π¯kU1, jUπijUπ¯kj = δik holds. Thisimplies
U tU = 1, and hence U = tU . Thus we get U = U .
π π¯ Pπ∗ π¯ π∗ij Pπ¯ij
4
We will verify the second statement. Since U is a representation, we have U12U13 =
π π ρ
Tσ,eU Tσ,e . Then we get
σ,e π,ρ σ π,ρ∗
P F (U12U13)F = U U eρ eπ = (F Tσ,e)U (F Tσ,e) .
π,ρ π ρ ρ,π πij ρkl ⊗ kl ⊗ ij π,ρ π,ρ σ π,ρ π,ρ ∗
i,j,k,l σ,e
X X
On the other hand, U12U13 = F Tσ,eU (F Tσ,e) holds, since F Tσ,e
ρ π σ,e π,ρ π,ρ σ π,ρ π,ρ ∗ { π,ρ π,ρ} ⊂
(σ,ρ π) is an orthonormal basis. (Note that we use π ρ ρ π here.) By comparing
these⊗, we get [U ,U ] = 0. P ⊗ ∼ ⊗ 2
πij ρkl
Lemma 2.4 shows that U behave like matrix coefficients π(g) . Let U be a
{ πij} { ij} π
representation of Gˆ, then it follows immediately that so is U , since [U ,U ] = 0.
π∗ πij ρkl
Remark. One can see that U is a conjugate representation ofGˆ, i.e., (U )12(U )13T =
{ π¯∗} π¯∗ ρ¯∗
TU for T (σ,π ρ), without using the commutativity of Gˆ.
σ¯∗ ∈ ⊗
Let π(g) be a matrix coefficient for π Rep(G). We regard π(g) as an element π
ij ij ij
∈
in ℓ (G) and set λ := 1 π . Then λ = λ eπ is the unitary representation
∞ πij M ⊗ ij π i,j πij ⊗ ij
ˆ
of G in the sense of Definition 2.3.
P
Since ℓ (G) = π , we have M ⋊ Gˆ = α(M) λ . The relation of generators
∞ { ij} α ∨{ πij}
are λ xλ = α (x) , or equivalently λ (x 1 )λ = α (x). Here we identify α(x)
k πik ∗πjk Wπ ij π ⊗ π ∗π π
and x as in the usual way. A unitary λ plays a roll of the implementing unitary in the
π
P
usual crossed product construction. Hence we also call λ the implementing unitary in
π
M ⋊ Gˆ. We can expand a M ⋊ Gˆ as a λ , a M, uniquely.
α ∈ α π,i,j π,i,j πij π,i,j ∈
Definition 2.5 Let α be an action of Gˆ oPn M. We say α is free if there exists no non-
zero a M B(H ), 1 = π Gˆ, so that α (x)a = a(x 1 ) for every x M. When α
π π π
∈ ⊗ 6 ∈ ⊗ ∈
is an action of a factor M, then we also say α is outer if α is free.
In usual, freeness of a coaction α on a factor M is defined by the relative commutant
condition M M ⋊ Gˆ = Z(M). We see that the usual definition and ours coincide in
′ α
∩
the following proposition.
Proposition 2.6 Let α be an action of Gˆ on M. Then α is free if and only if M M⋊
′ α
Gˆ = Z(M). Especially, M ⋊ Gˆ is a factor when α is free, and M is a factor. ∩
α
Proof. Let a = a λ M ⋊ Gˆ. Set a := a eπ M B(H ). Then
π,i,j π,i,j πij ∈ α π i,j π,j,i⊗ ij ∈ ⊗ π
it is easy to see a M M ⋊ Gˆ if and only if (x 1 )a = a α (x) for any x M,
π Gˆ. Then it iPs e∈asily′ s∩hown tαhat α is free if and o⊗nPlyπif Mπ Mπ ⋊π Gˆ = Z(M).∈ 2
′ α
∈ ∩
In the end of this subsection, we explain the dual action of G on the crossed product.
LetαbeanactionofGˆ onM. Thenthedualactionαˆ ofGonM⋊ Gˆ isgivenbyαˆ (a) = a
α g
for a M, and αˆ id (λ ) = λ π(g), or equivalently αˆ (λ ) = λ π(g) . Then it
∈ g ⊗ π π π g πij k πik kj
is shown that αˆ an action of G, and the fixed point algebra is (M ⋊ Gˆ)αˆ = M.
Pα
2.4 Quantum double construction for finite group duals
In this subsection, we collect definitions and basic properties for quantum double con-
struction (also known as the symmetric enveloping algebra [11], or the Longo-Rehren
5
construction [6]) arising from actions of group duals. We will use them in 6. We refer
§
[4, Chapter 12.8, 15.5], or [7, Appendix A] for details of this topic.
Forπ,ρ Rep(G),letπˆρarepresentationofG Ggivenbyπˆρ(g,h) := π(g) ρ(h).
Let α be an∈action of Gˆ Gˆ⊗on M. Set P := M⋊ (×Gˆ Gˆ). Let λ⊗ be an implem⊗enting
× α × π⊗ˆρ
unitary for α.
Lemma 2.7 Set w := λ . Then w = (w ) is a unitary representation of
πij k πik⊗ˆπ¯jk π πij
Gˆ.
P
Proof. Set v := λ , u := λ . Obviously we have w = v u and
πij πijˆ1 πij 1ˆπ¯ji πij k πik πkj
⊗ ⊗
[v ,u ] = 0. Since Tσ,e (σ¯,π¯ ρ¯) is an orthonormal basis, u = (u ) becomes a
πij ρkl { π,ρ} ⊂ ⊗ π Pπij ij
ˆ
unitary representation of G (also see Remark after Lemma 2.4). Hence
w w Tσm,e = v v u u Tσm,e
πij ρkl πj,ρl πin ρka πnj ρaj πj,ρl
j,l j,l,n,a
X X
= Tξb,p v Tξc,p Tηd,q u Tηf,qTσm,e
πi,ρk ξbc πn,ρa πn,ρa ηdf πj,ρl πj,ρl
j,l,n,a,
ξ,b,cX,pη,d,f,q
= Tξb,p Tξc,p Tηd,q v u Tηf,qTσm,e
πi,ρk πn,ρa πn,ρa ξbc ηdf πj,ρl πj,ρl
! !
ξ,b,c,pη,d,f,q n,a j,l
X X X
= Tσb,e v u
πi,ρk σbc σcm
!
b c
X X
= Tσb,e w
πi,ρk σbm
b
X
2
holds.
Definition 2.8 Set N := M w . We call M N is the quantum double for α.
∨{ πij} ⊂
Remark. Intheabove definition, we consider anactionofGˆ Gˆ onM directly. However,
×
usual quantum double construction is given as follows. Let M be a von Neumann algebra,
and α be an action of Gˆ on M. By the commutativity of Gˆ, (α )opp becomes an action
π¯
of Gˆ on Mopp. Hence we have an action of Gˆ Gˆ on M Mopp. The rest of construction
× ⊗
is same as above.
We embed G into G G by g (g,g). Let β := αˆ be the dual action of G G on P.
Then it is shown that N×= (M ⋊→(Gˆ Gˆ))G. ×
α
×
For example, we have
β (w ) = β (λ )
g,g πij g,g πik⊗ˆπ¯jk
k
X
= λ π(g) π(g)
πil⊗ˆπ¯jm lk mk
k,l,m
X
= λ
πil⊗ˆπ¯jl
l,m
X
= w .
πij
6
If we expand a P as a = a λ , then N = (M ⋊ (Gˆ Gˆ))G is verified
∈ πij,ρkl πij⊗ˆρkl α ×
in a similar way as above. We leave the proof to the reader. We remark that a N
P ∈
can be expand uniquely as a = a w , a M, and there exists the canonical
π,i,j π,i,j πij π,i,j ∈
conditional expectation E : N M given by E(a) = a1.
→P
2.5 Main result
Definition 2.9 Let α be an action of Gˆ. We say w a (unitary) 1-cocycle for α if
{ π}π Gˆ
wπ U(M B(Hπ)), normalized as w1 = 1, and followi∈ng holds.
∈ ⊗
(w 1 )α id (w )T = Tw , T (σ,π ρ).
π ρ π ρ ρ σ
⊗ ⊗ ∈ ⊗
A 1-cocycle w for α is called a coboundary if there exists a unitary v U(M) such
π
{ } ∈
that w = (v 1 )α (v).
π ∗ π π
⊗
If we extend v for ξ Rep(G) as in the remark in 2.1, then we have (v 1 )α (v ) =
ξ ξ η ξ η
∈ § ⊗
ˆ
v . It is easy to see that Adw α is an action of G for a 1-cocycle w .
ξ η π π π
⊗
Definition 2.10 Let α and β be actions of Gˆ on M.
(1) We say α and β are conjugate if there exists θ Aut(M) with θ id α θ 1 = β
π π − π
for every π Gˆ. ∈ ⊗ ◦ ◦
∈
(2) We say α and β are cocycle conjugate if there exists a 1-cocycle w for α, and
π
{ }
Adw α and β are conjugate.
π π π
Our main purpose is to show the following theorem by the traditional Connes-Jones-
Ocneanu type approach.
Theorem 2.11 Let R be the AFD factor of type II . Let α and β be outer actions of Gˆ
1
on R. Then α and β are conjugate.
3 Model action
In this section, we construct an infinite tensor product type action of Gˆ on R, which we
adopt as the model action.
It is easy to see the following lemma.
Lemma 3.1 Let M, N be von Neumann algebras, and U , V unitary representation of
π π
Gˆ in M and N respectively. We regard U and V as representations of Gˆ in M N in
π π
the canonical way. Then U V is also a representation of Gˆ. ⊗
π π
Toconstructthemodelaction,wefirstconstruct(thecanonical)unitaryrepresentation
of Irr(G) on M (C). Although we already discussed it in 2.3, we give a slightly different
G
| | §
approach, which will be useful for our argument.
Let φ be the Haar functional for R(G), i.e., φ(u ) = G δ . For v R(G), we
g e,g
| | ∈
denote by v = v(π), v(π) B(H ), via the decomposition R(G) = B(H ). Then
⊕ ∈ π ∼ π Gˆ π
we have φ(v) = dπTr (v(π)), where Tr be the canonical (non-no∈rmalized) trace
π π π L
on B(H ). We regard R(G) as a Hilbert space equipped with an inner product arising
π
from φ, and denotPe by ℓ2(Gˆ). Namely, an inner product on ℓ2(Gˆ) is given by v,w =
h i
7
dπ v(π),w(π) for v = v(π), w = w(π). Here v(π),w(π) = Tr (w(π) v(π))
π Gˆ h iπ ⊕ ⊕ h iπ π ∗
It is∈easy to see dπ 1/2eπ ℓ2(Gˆ) forms an orthonormal basis with respect to this inner
P { − ij} ⊂
product.
Set Tσ,e B(H ,H ) by (Tσ,e) = Tσk,e.
ρ,πi ∈ σ ρ ρ,πi ρj,σk ρj,πi
Lemma 3.2 Define λ B(ℓ2(Gˆ)) = M (C) by
πij ∈ |G|
(λπijv)(ρ) := Tρ¯σ¯,π,eiv(σ)Tρ¯σ¯,π,ej∗
σ,e
X
and λ := λ eπ M (C) B(H ). Then λ is a unitary representation of
π i,j πij ⊗ ij ∈ |G| ⊗ π { π}
Irr(G) on M (C).
G
P| |
Proof. We freely use notations and results in 2.1. We first show λ12λ13Tσ,a = Tσ,aλ ,
§ π ρ π,ρ π,ρ σ
Tσ,a = (Tσm,a) (σ,π ρ), equivalently λ λ Tσn,a = Tσm,aλ .
π,ρ πi,ρk ∈ ⊗ πij ρkl πj,ρl m πi,ρk σmn
P
λπijλρklTπσjn,ρ,alv!(ξ) = Tξ¯η¯,,πei(λρk,lTπσjn,ρ,alv)(η)Tξ¯η¯,,πej∗
j,l j,l,η,e
X X
= Tξ¯η¯,,πeiTη¯ζ¯,,ρfkTπσjn,ρ,alv(ζ)Tη¯ζ¯,,ρfl∗Tξ¯η¯,,πej∗
j,l,η,e,ζ,f
X
= Tπηi,,eρkTξ¯ζ¯,,ηfTπσjn,ρ,alv(ζ)Tξ¯ζ¯,,ηf∗Tπη¯j,e,ρ∗l
j,l,η,e,ζ,f
X
= Tπηi,,eρkTξ¯ζ¯,,ηfv(ζ)Tξ¯ζ¯,,ηf∗( Tπσjn,ρ,alTπηj∗,,ρel)
η,ζ,e,f j,l
X X
= Tπσi,,aρkTξ¯ζ¯,,σfv(ζ)Tξ¯ζ¯,,σfn∗
ζ,f
X
= Tπσim,ρ,kaTξ¯ζ¯,,σfmv(ζ)Tξ¯ζ,¯,σfn∗
m,ζ,f
X
= (Tσm,aλ v)(ξ).
πi,ρk σm,n
m
X
It is easy to see that λ v,λ w = δ . Hence we have λ λ = δ and
kh πki πkj i i,j k ∗πki πkj i,j
consequently λ λ = 1. Thus it suffices to show λ λ = 1.
∗π π π ∗π
P P
Here we have
λ λ = Tρl,e λ Tρm,e
πik π¯jk πi,π¯j ρlm πk,π¯k
k k,ρ,l,m,e
X X
= √dπT1 λ1
πi,π¯j
= δ .
i,j
Hence we have λ tλ = 1, and tλ = λ . It follows that λ = λ and λ λ = 1. 2
π π¯ π¯ ∗π ∗πij π¯ij π ∗π
Let E = e be a system of matrix units for B(ℓ2(Gˆ)) = M (C), that is, e
{ πij,ρkl} ∼ |G| πij,ρkl
is a partial isometry which sends dρ 1/2eρ ℓ2(Gˆ) to dπ 1/2eπ. It is not difficult to see
− kl ∈ − ij
dρ
λ = Tσm,eTσn,ee .
πij dσ πi,ρk πj,ρl σmn,ρkl
r
ρ,k,l,σ,m,n,e
X
8
It follows that √dπdρλπije1,1λρ¯kl = eπij,ρkl from the above expression of λπ.
Let M be a von Neumann algebra, and E = e M a system of matrix units
{ πij,ρkl} ⊂
for B(ℓ2(Gˆ)). Then we can construct a unitary representation λ of Gˆ in E by the above
π ′′
formula. In this case, we call λ a representation of Gˆ associated with E = e .
{ π} { πij,ρkl}
When we have to specify E, we denote the unitary representation of Gˆ associated with E
by λE.
π
We define the product type action of Gˆ on R. Express R = ∞n=1Kn, where Kn is
a copy of M (C). Let λn := λKn be a unitary representation of Gˆ on K , and regard
as one on R|.G|Define λ˜1 π:= λ1,πand λ˜n = λ˜n 1λn. Then λ˜n isNa represenntation of Gˆ
π π π π− π π
on K K by Lemma 3.1. Set mn(x) := Adλ˜n(x 1 ). Since λ˜n is a unitary
1 ⊗ ··· ⊗ n π π ⊗ π π
representation of Gˆ, mnπ is indeed an action of Gˆ on R. If x ∈ kn=−11Kk, then
Adλ˜n(x 1 ) = Adλ˜n 1λn(x 1 ) = Adλ˜n N1(x 1 )
π ⊗ π π− π ⊗ π π− ⊗ π
holds. Hence nlim mnπ(x) exists for x ∈ ∞n=1 nk=1Kk, and so does mπ(x) = nlim mnπ(x)
for every x R→.∞ →∞
∈ S N
Definition 3.3 We call m = m the model action for Gˆ.
π
{ }
Theorem 3.4 The model action m is outer.
Proof. Fix 1 = π Gˆ. Assume there exists non-zero a R B(H ) such that
π
m (x)a = a(x 6 1) ho∈lds for x R. If x n K , then (x∈ 1)⊗λ˜n a = λ˜n a(x 1)
π ⊗ ∈ ∈ k=1 n ⊗ π∗ π∗ ⊗
holds. Hence a is expressed as a = λ˜nπbn+1, bn+N1 = ijbnij+1 ⊗eπij ∈ ∞k=n+1Kk ⊗B(Hπ).
Since we assume a = 0, there exists c R B(H ) with τ Tr (ca) = 0. We may assume
π π
c is of the form c 6 eπ, c m K∈for⊗some m.PThen⊗ N6
1 ⊗ ij 1 ∈ k=1 k
N
τ Tr (ca) = τ(c λm+1bm+2) = τ(c λ˜m+1)τ(bm+2) = τ(c λ˜m )τ(λm+1)τ(bm+2) = 0
⊗ π 1 πij ji 1 πij ji 1 πil πlj ji
l
X
holds, and this is a contradiction. Hence a must be 0, and m is an outer action. 2
Definition 3.5 Let E = e M be a system of matrix units, and λE a represen-
{ πij,ρkl} ⊂ π
tation of Gˆ associated with E. Let α be an action of Gˆ on M. We say e is an
{ πij,ρkl}
α-equivariant system of matrix units if α (x) = AdλE(x 1) for x E.
π π ⊗ ∈
The following lemma is easily verified. We leave the proof to the reader.
Lemma 3.6 Let α be an action of Gˆ on M.
(1) Let E = e be an α-equivariant system of matrix units. Then λE is a 1-cocycle
{ πij,ρkl} π∗
for α, and AdλE α = id on E. Hence AdλE α induces an action on E M.
π∗ π π∗ π ′ ∩
(2) Let M K = M (C), and suppose α is trivial on K. Then λK is a 1-cocycle for α.
It follows t⊃hat A∼dλK|αG| is an action on M. π
π π
9
4 Technical results
In this section, we collect some technical lemmas, whose proof can be found in [2], [5], [9].
In the following, M is a factor of type II , and τ is the unique normalized trace on M.
1
Lemma 4.1 ([5, Lemma 3.2.7]) Let f M be such that f 1, f2 f < δ and
2
∈ k k ≤ k − k
f f < δ 1/4. Then there exists a projection p M such that f p < 6√4 δ
∗ 2 2
k − k ≤ ∈ k − k
and τ(p) = τ(f).
Lemma 4.2 ([5, Lemma 3.2.1]) Let u M be such that u u 1 < δ. Then there
∗ 2
∈ k − k
exists a unitary v M with u v < (3+ u )δ.
2
∈ k − k k k
Lemma 4.3 ([2, Proposition 1.1.3],[9, Proposition 7.1]) Let us fix a free ultrafilter
ω over N.
(1) Let A Mω be a unitary (resp. projection). Then there exists a representing sequence
∈
A = (a ) consisting of unitaries (resp. projections).
n
(2) Let V Mω be a partial isometry with V V = E and VV = F. Let E = (e ),F =
∗ ∗ n
∈
(f ) Mω be representing sequences consisting of projections such that e and f are
n n n
∈
equivalent for any n. Then there exist a representing sequence (v ) for V such that
n
v v = e , v v = f .
n∗ n n n n∗ n
(3) Let E Mω be a system of matrix units. Then there exists a representing
ij 1 i,j, m
sequence{E }=≤ e≤n ⊂such that en is a system of matrix units for every n.
ij { ij} { ij}1≤i,j≤m
5 Cohomology vanishing
ˆ
In this section, we mainly deal with actions of G on factors of type II . However many
1
parts of results in this section are valid for general factors (or von Neumann algebras).
We begin with the following lemma, which is known as the “push-down lemma” in
subfactor theory [4, Lemma 9.26].
Lemma 5.1 Let α be an action of Gˆ, and set e := 1/ G dπλ M ⋊ Gˆ. For any
| | π,i πii ∈ α
a M ⋊ Gˆ there exists b M such that ae = be.
∈ α ∈ P
Proof. Let a = a λ , a M be an expansion of a. Then
ρ,k,l ρkl ρkl πkl ∈
P
G ae = dπa λ λ
| | ρkl ρkl πii
π,i,ρ,k,l
X
= dπa Tσm,eλ Tσn,e
ρkl πi,ρk σmn πi,ρl
π,i,ρ,k,lσ,m,n,e
X X
= dσa Tπi,e Tπi,e λ
ρkl σm,ρ¯k σn,ρ¯l σmn
!
ρ,k,l,σ,m,n π,i,e
X X
= dσa λ
ρkk σmm
ρ,k,σ,m
X
= ( a ) G e
ρkk | |
ρ,k
X
holds. Set b := a , then we have ae = be and b M. 2
ρ,k ρkk ∈
P
10