Table Of ContentA GALOIS CORRESPONDENCE FOR COMPACT QUANTUM
GROUP ACTIONS
REIJI TOMATSU
8
0 Abstract. We establish a Galois correspondence for a minimal action of a
0 compact quantum group G on a von Neumann factor M. This extends the
2
resultofIzumi,LongoandPopawhotreatedthecaseofaKacalgebra. Namely,
n
there exists a one-to-one correspondence between the lattice of left coideals of
a G and that of intermediate subfactors of MG ⊂M.
J
8
]
A
1. Introduction
O
. The main purpose of this paper is to present a Galois correspondence for com-
h
t pact quantum group actions. The theory of Galois correspondences for group
a
m actions on von Neumann algebras was initiated by M. Nakamura and Z. Takeda
[ [17, 18] and has been studied extensively in various settings by many researchers.
Roughly speaking, the Galois correspondence for a group action G on a von Neu-
1
v mann algebra M refers to a one-to-one correspondence between the subgroups
3 H of G and the intermediate von Neumann subalgebras MG ⊂ MH ⊂ M, where
3
MH denotes the fixed point algebra by the H-action.
2
1 In [18], the Galois correspondence was established for a minimal action of a
.
1 finite group on a II factor. For compact group actions, A. Kishimoto obtained
1
0
a Galois correspondence between normal closed subgroups and intermediate von
8
0 Neumann subalgebras that are globally invariant under the compact group, as-
v: suming a certain condition on actions which is satisfied by minimal ones [10].
i Another kind of Galoiscorrespondence was provided in [17] for crossed product
X
von Neumann algebras in the case of a discrete group G acting freely on a finite
r
a
factor M. Their result again shows a one-to-one correspondence between the
lattice of subgroups and that of intermediate subfactors of M ⊂ M ⋊G. In [23],
M. Takesaki studied a generalization of this result for a locally compact abelian
group action. Y. Nakagami strengthened the result to the case of general locally
compact group actions [16]. In [2], H. Choda investigated the crossed product
by free actions of discrete groups on a factor of arbitrary type and obtained the
Galois correspondence for intermediate von Neumann subalgebras which are the
ranges of normal conditional expectations.
M. Izumi, R. Longo and S. Popa [9] have further developed theory of Galois
correspondence to compact group minimal actions and discrete group free actions
on factors of arbitrary type. Moreover, they obtained the Galois correspondence
2000 Mathematics Subject Classification. Primary 46L65; Secondary 46L55.
Supported by JSPS.
1
forminimalactionsofcompactKacalgebras,whichunifiestheresultsforcompact
groups and discrete groups.
Therefore, it is naturalto explore a generalization of the Galois correspondence
of [9] to minimal actions of compact quantum groups. In [3], M. Enock focused
on this problem in a more general setting, but there is a flaw in his proof. We
point out the reason why the proof of [9] does not work in the quantum case. The
main step of their proof is to show the existence of a normal conditional expec-
tation from an ambient von Neumann algebra onto any intermediate subfactor.
However, this is no longer true in general for compact quantum group actions.
Indeed, we consider a minimal action of SU (2) (0 < q < 1) on a factor, whose
q
existence has been shown by Y. Ueda [26]. The intermediate subfactor associ-
ated with a Podle´s sphere S2 [19, 20] is not the range of a normal conditional
q,θ
expectation.
For our purpose, we proceed to study an irreducible inclusion of discrete type
in the sense of [9]. Let N ⊂ M be an irreducible inclusion of discrete type. In [9],
assuming a technical condition on the modular automorphism groups, it is shown
that an intermediate subfactor N ⊂ L ⊂ M is generated by N and Hilbert spaces
in L which implement irreducible endomorphisms on N. We will strengthen the
result by dropping assumptions on the modular automorphism groups. Our main
result is the following (Theorem 2.7):
Theorem 1. Let N ⊂ M be an irreducible inclusion of discrete type with the
faithful normal conditional expectation E: M → N. Let N ⊂ L ⊂ M be an
intermediate subfactor. Then the following statements hold.
(1) The subfactor N ⊂ L is discrete.
(2) Suppose N is infinite. Let γM: M →N and γL: L →N be the canon-
N N
ical endomorphisms for N ⊂ M and N ⊂ L, respectively. Then [γM| ]
N N
contains [γL| ] in Sect(N).
N N
The second statement of the above theorem is equivalent to saying that the
bimodule L2(M) contains L2(L) . This statement might sound trivial as
N N N N
in the case of finite M, but indeed we need a little more efforts to prove it (see
Remark 2.8).
Next we apply this result to minimal actions of compact quantum groups and
prove the following Galois correspondence (Theorem 3.8) which generalizes the
correspondence presented in [9].
Theorem 2. Let G be a compact quantum group and M a factor. Let α: M →
M ⊗L∞(G) be a minimal action. Then there exists a one-to-one correspondence
between the lattice of left coideals in L∞(G) and that of intermediate subfactors
of Mα ⊂ M.
2. Irreducible inclusions of discrete type
First, we fix notations. Throughout this paper, we always assume separability
of von Neumann algebras. Let M be a von Neumann algebra with predual M .
∗
2
For a weight φ on M, we set n = {x ∈ M | φ(x∗x) < ∞}. The GNS representa-
φ
tion of M with respect to a faithful normal semifinite weight φ is denoted by the
pair {H ,Λ }, where Λ : n →H is the canonical injection to the GNS Hilbert
φ φ φ φ φ
space H . We always regard M as a von Neumann subalgebra in B(H ).
φ φ
Let N ⊂ M be an inclusion of von Neumann algebras. We denote by P(M,N)
the set of faithful normal semifinite operator valued weights from M to N. For
theory of operator valued weights, readers are referred to [5, 6].
w
ForasubsetX ⊂ M, wedenotebyX , cow(X)andspanw(X)theweak closure
of X, the weak closure of the convex hull of X and the weak closure of the linear
space spanned by X, respectively.
We denote by ⊗ the minimal tensor product for C∗-algebras and the spatial
tensor product for von Neumann algebras.
We say that H is a Hilbert space in M if H ⊂ M is a σ-weakly closed subspace
and η∗ξ ∈ C for all ξ,η ∈ H [21]. The smallest projection e ∈ M with eH = H
is called the support of H.
We denote by End(M) and Sect(M) the set of endomorphisms and sectors on
M, that is, Sect(M) is the set of equivalence classes of endomorphisms on M by
unitary equivalence. For the sector theory, readers are referred to [8, 13, 14].
2.1. Irreducible inclusions of discrete type
In this subsection, we recall the notion of discreteness introduced in [9] for an
inclusion of factors and summarize the basic properties.
Let N ⊂ M be an inclusion of factors with a faithful normal conditional ex-
pectation EM: M →N. Fix a faithful state ω ∈ N . We set ϕ := ω ◦EM ∈ M .
N ∗ N ∗
Let {H ,Λ } be the GNS representation of M associated with the state ϕ. We
ϕ ϕ
define the Jones projection e ∈ B(H ) by e Λ (x) = Λ (EM(x)) for x ∈ M
N ϕ N ϕ ϕ N
and set the basic extension M := M ∨ {e }′′ ⊂ B(H ). The dual operator
1 N ϕ
valued weight of EM is denoted by EM1 which is an element of P(M ,M) [11].
N M 1
By definition, we have EM1(ae b) = ab for a,b ∈ M. Define the faithful normal
M N
semifinite weight ϕ := ϕ◦EM1 on Mb .
1 M 1
In [9, Definition 3.7],bthe discreteness of an inclusion of factors is introduced
as follows. b
Definition 2.1. An inclusion of factors N ⊂ M is said to be discrete when there
exists a faithful normal conditional expectation EM: M →N such that its dual
N
operator valued weight EM1 is semifinite on N′ ∩M .
M 1
Note that the discreteness is equivalent to saying that the N-N-bimodule
b
L2(M) is the direct sum of N-N-bimodules of finite index.
Let N ⊂ M be an irreducible inclusion of discrete type. Then N′∩M can be
1
decomposed into a direct sum of matrix algebras as [9, Proposition 2.8]:
N′ ∩M = A ,
1 ξ
ξ∈Ξ
M
where A is a type I factor for some n ∈ N.
ξ nξ ξ
3
Consider the case that N is infinite. Let γM: M → N be the canonical en-
N
domorphism of the inclusion N ⊂ M [13, 14, 15]. By definition, we have the
isomorphism of M-N-bimodules
L2(N) ∼= L2(M) ,
MγM N M N
N
where L2(M) and L2(N) denote the standard Hilbert space for M and N, re-
spectively. Since N′ ∩ M = End( L2(M) ), a minimal projection e in A
1 N N ξ ξ
corresponds to an irreducible endomorphism with finite index, ρ ∈ End(N),
ξ
that is, we have the following isomorphism of N-N-bimodules [9, Lemma 3.1]:
L2(N) ∼= e L2(M) .
Nρξ N N ξ N
By definition, the sector [γM| ] contains the sector [ρ ] with multiplicity n in
N N ξ ξ
Sect(N).
We define the subspace H ⊂ M by
ξ
H = {V ∈ M | Vx = ρ (x)V for all x ∈ N}.
ξ ξ
Then by the inner product hV|Wi1 = W∗V, H is a Hilbert space in M. We note
ξ
that the support projection of H may not be equal to 1. We prepare another
ξ
inner product defined by (V,W)1 = d(ξ)EM(VW∗) for V,W ∈ H , where d(ξ) is
N ξ
the square root of the minimal index of ρ (N) ⊂ N [7]. By [9, Theorem 3.3], we
ξ
have dimH = n and A = H∗e H .
ξ ξ ξ ξ N ξ
2.2. Intermediate subfactors
Let N ⊂ M be an irreducible inclusion of discrete type with infinite N. Let
N ⊂ L ⊂ M be an intermediate subfactor. We denote by EL the restriction of
N
EM on L. For ξ ∈ Ξ, we define the Hilbert space K in L by
N ξ
K := H ∩L.
ξ ξ
We set Ξ := {ξ ∈ Ξ | K 6= 0} and m := dim(K ) for ξ ∈ Ξ .
L ξ ξ ξ L
For ξ ∈ Ξ, we take a basis {V } in H such that (V ,V ) = d(ξ)δ . If
ξi i∈Iξ ξ ξi ξj i,j
ξ ∈ Ξ , we may assume that the family {V } contains a basis of K , which
L ξi i∈Iξ ξ
we denote by {V } . Then the family {V∗e V } is a system of matrix
ξi i∈IξL ξi N ξj i,j∈Iξ
units of A . We prepare the following projections in N′ ∩M ,
ξ 1
(z ) := V∗e V for all ξ ∈ Ξ , z := (z ) .
L ξ ξi N ξi L L L ξ
iX∈IξL ξX∈ΞL
In the following lemma, we determine the subfactor L ⊂ M at the GNS Hilbert
space level. Our argument is essentially same as the one given in [9, Theorem
3.9], but the assumption there is different from ours. We present a proof for the
completeness of our discussion.
Lemma 2.2. With the above settings, z H = Λ (L) holds. In particular, one
L ϕ ϕ
has z ∈ L′ ∩M and e ≤ z .
L 1 N L
4
Proof. First we note that the following holds:
z H = span{Λ (V∗N) | ξ ∈ Ξ ,i ∈ IL}. (2.1)
L ϕ ϕ ξi L ξ
Indeed, let x ∈ M. Then we have
z Λ (x) = V∗e V Λ (x)
L ϕ ξi N ξi ϕ
ξX∈ΞLiX∈IξL
= Λ (V∗EM(V x)).
ϕ ξi N ξi
ξX∈ΞLiX∈IξL
Hence the left hand side is contained in the right hand one of (2.1). The converse
inclusion follows from EM(V V∗) = δ δ for ξ,η ∈ Ξ and i ∈ I ,j ∈ I .
N ξi ηj ξη ij ξ η
In particular, this yields z H ⊂ Λ (L). We will prove the equality by using
L ϕ ϕ
the averaging technique presented in the proof of [9, Thorem 3.9] as shown below.
To prove it, we may and do assume that N,L and M are factors of type III by
tensoring with a type III factor. Assume that there would exist x ∈ L such that
Λ (x) ∈/ z H . By the following equality:
ϕ L ϕ
(1−z )Λ (x) = Λ (V∗EM(V x))+ Λ (V∗EM(V x)),
L ϕ ϕ ξi N ξi ϕ ξi N ξi
ξ∈XΞ\ΞLXi∈Iξ ξX∈ΞLi∈XIξ\IξL
the following two cases could occur: (I) there exists ξ ∈ Ξ \ Ξ such that
L
EM(V x) 6= 0 for some i ∈ I or (II) there exists ξ ∈ Ξ such that EM(V x) 6= 0
N ξi ξ L N ξi
for some i ∈ I \ IL. In case (I), we set IL = ∅ and then proceed as with case
ξ ξ ξ
(II). Assume that case (II) would occur. Take ξ ∈ Ξ and i ∈ I \ IL such that
ξ ξ
EM(V x) 6= 0. Let E : N →ρ (N) be the faithful normal conditional expecta-
N ξi ξ ξ
tion with respect to ρ . By using the equality EM(V axb) = ρ (a)EM(V x)b for
ξ N ξi ξ N ξi
a,b ∈ N, we may assume that E (EM(V x)) = 1 since N is of type III.
ξ N ξi
We take a hyperfinite subfactor R ⊂ N which is simple in the sense of [15].
Then consider the weakly closed convex set
C := cow{uxρ (u∗) | u ∈ U(R)} ⊂ L,
ξ
where U(R) is the set of all unitaries in R. The hyperfiniteness of R assures that
there exists a point w∗ ∈ C such that w satisfies wx = ρ (x)w for all x ∈ R and
ξ
hence for all x ∈ N by [9, Proposition 2.10]. This shows w ∈ L∩H = K . Since
ξ ξ
i ∈ I \ IL, V is orthogonal to K , that is, EM(V w∗) = d(ξ)−1(V ,w) = 0.
ξ ξ ξi ξ N ξi ξi
However E (EM(V C)) = {1}, and this is a contradiction. Therefore the cases
ξ N ξi
(I) and (II) never occur, and for any x ∈ L, (1−z )Λ (x) = 0. This implies that
L ϕ
Λ (L) ⊂ z H . (cid:3)
ϕ L ϕ
By the previous lemma, we can describe the corner subalgebra z M z in M .
L 1 L 1
Lemma 2.3. One has z M z = Le Lw = Lz ∨{e }′′.
L 1 L N L N
Proof. Recall that e M e = Ne . For ξ,η ∈ Ξ, i ∈ IL and j ∈ IL, we have
N 1 N N ξ η
V∗e V M V∗e V ⊂ V∗e M e V ⊂ V∗Ne V ⊂ Le L.
ξi N ξi 1 ηj N ηj ξi N 1 N ηj ξi N ηj N
5
w
This implies that z M z ⊂ Le L . By the previous lemma, z ∈ L′ ∩M and
L 1 L N L 1
z e = e . Since M contains L and e , we have z M z ⊃ Lz ∨ {e }′′ ⊃
L N N 1 N L 1 L L N
Le L. Hence we have z M z = Le Lw = Lz ∨{e }′′. (cid:3)
N L 1 L N L N
Next we will show that the two-step inclusion Nz ⊂ Lz ⊂ z M z is identi-
L L L 1 L
fied with the basic extension of N ⊂ L. One might be able to prove this by using
the abstract characterization of the basic extension [9, Lemma 2.4]. To apply
that result, we need to show that the restriction EM1 on z M z is an operator
M L 1 L
∼
valued weight from z M z to Lz = L, but we do not have a proof for such
L 1 L L
a statement yet. We avoid using this method anbd directly compare the basic
extension of N ⊂ L with Nz ⊂ Lz ⊂ z M z instead.
L L L 1 L
We set ψ := ω ◦ EL ∈ L . Then ϕ| = ψ| holds trivially. Let {H ,Λ } be
N ∗ L L ψ ψ
the GNS representation of L associated with the state ψ. Let f ∈ B(H ) be
N ψ
the Jones projection defined by f Λ (x) = Λ (EL(x)) for x ∈ L. We set L :=
N ψ ψ N 1
L ∨ {f }′′ ⊂ B(H ). Then we obtain the Jones’ basic extension N ⊂ L ⊂ L
N ψ 1
associated with EL. The dual operator valued weight of EL is denoted by EL1.
N N L
Note that we do not know whether EL1 is semifinite on N′ ∩ L or not. Set a
L 1
weight ψ := ψ ◦ EL1 ∈ P(L ,C). Let {H ,Λ } be the GNS representabtion
1 L 1 ψ1 ψ1
of L associated with the weight ψ .bRecall the weight ϕ = ϕ ◦ EM1 on M .
1 1 1 M 1
Let {H ,Λ } be bthe GNS representation of M associated with the weight ϕ .
ϕ1 ϕ1 1 1
Then the following holds ([9, Lemma 2.1]): b
H = Λ (Lf L), H = Λ (Me M).
ψ1 ψ1 N ϕ1 ϕ1 N
We introduce an isometry U: H →H satisfying
ψ1 ϕ1
UΛ (xf y) = Λ (xe y), for x,y ∈ L.
ψ1 N ϕ1 N
The well-definedness is verified as follows. For x,y,a,b ∈ L, we have
hΛ (xe y),Λ (ae b)i =ϕ (b∗e a∗xe y) = ϕ (b∗EM(a∗x)e y)
ϕ1 N ϕ1 N 1 N N 1 N N
=ϕ◦EM1(b∗EM(a∗x)e y) = ϕ(b∗EM(a∗x)y)
M N N N
=ψ(b∗EL(a∗x)y) = hΛ (xf y),Λ (af b)i.
N ψ1 N ψ1 N
b
Lemma 2.4. One has xU = Ux for x ∈ L and e U = Uf .
N N
Proof. Since the subspace Λ (Lf L) ⊂ H is dense, it suffices to show the
ψ1 N ψ1
equalities on Λ (Lf L). Let x,a,b ∈ L. Then we have
ψ1 N
xUΛ (af b) = xΛ (ae b) = Λ (xae b) = UΛ (xaf b) = UxΛ (af b).
ψ1 N ϕ1 N ϕ1 N ψ1 N ψ1 N
Hence xU = Ux. Next e U = Uf is verified as follows.
N N
e UΛ (af b) =e Λ (ae b) = Λ (e ae b)
N ψ1 N N ϕ1 N ϕ1 N N
=Λ (EM(a)e b) = Λ (EL(a)e b)
ϕ1 N N ϕ1 N N
=UΛ (EL(a)f b) = Uf Λ (af b).
ψ1 N N N ψ1 N
(cid:3)
6
Set the range projection p := UU∗ ∈ B(H ). It is clear that p H =
L ϕ1 L ϕ1
Λ (Le L). By the previous lemma or the definition of p , p commutes with L
ϕ1 N L L
and e . In particular, p ∈ (Lz )′ ∩{e }′ ⊂ B(H ).
N L L N ϕ1
The subspace p H plays a similar role to the GNS Hilbert space of z M z
L ϕ1 L 1 L
associated with the restricted weight ϕ | , but p H may not coincide with
1 zLM1zL L ϕ1
the closure of Λ (n ∩z M z ) because the function t ∈ R 7→ σENM◦EbMM1(z ) ∈
ϕ1 ϕ1 L 1 L t L
N′ ∩M may not extend to the bounded analytic function on the strip {z ∈ C |
1
0 ≤ Im(z) ≤ 1/2}. However, the following lemma is sufficient for our purpose.
Lemma 2.5. In B(H ), p ≤ z holds.
ϕ1 L L
Proof. Using z ∈ L′ ∩M and z e = e , we have
L 1 L N N
z p H = z Λ (Le L) = Λ (Lz e L) = Λ (Le L) = p H .
L L ϕ1 L ϕ1 N ϕ1 L N ϕ1 N L ϕ1
Hence p ≤ z . (cid:3)
L L
Lemma 2.6. There exists an isomorphism Ψ : z M z →L such that
L L 1 L 1
(1) Ψ (xz ) = x for x ∈ L.
L L
(2) Ψ (e ) = f .
L N N
In particular, the inclusions Nz ⊂ Lz ⊂ z M z and N ⊂ L ⊂ L are isomor-
L L L 1 L 1
phic.
Proof. We define the normal positive map Ψ : z M z →B(H ) by Ψ (x) =
L L 1 L ψ1 L
U∗xU for x ∈ z M z . Since p commutes with Lz ∨ {e }′′ = z M z as is
L 1 L L L N L 1 L
remarked after Lemma 2.4, we see that Ψ is multiplicative. By the previous
L
lemma, we have
Ψ (z ) = U∗z U = U∗z p U = U∗p U = 1,
L L L L L L
that is, Ψ is unital. Hence Ψ is a unital ∗-homomorphism. By Lemma 2.4,
L L
the range of Ψ is equal to U∗(Lz ∨{e }′′)U = L∨{f }′′ = L . Also we have
L L N N 1
Ψ (xz ) = x for x ∈ L and Ψ (e ) = f . Since z M z is a factor, Ψ is an
L L L N N L 1 L L
isomorphism onto L . (cid:3)
1
Now we state our main result in this section.
Theorem 2.7. Let N ⊂ M be an irreducible inclusion of discrete type. Let
N ⊂ L ⊂ M be an intermediate subfactor. Then one has the following:
(1) The inclusion N ⊂ L is discrete.
(2) Suppose that N is infinite. Let γM and γL be the canonical endomor-
N N
phisms for N ⊂ M and N ⊂ L, respectively. Then [γM| ] contains
N N
[γL| ] in Sect(N).
N N
(3) Suppose that N is infinite and [γL| ] has in Sect(N) the following decom-
N N
position into irreducible sectors [ρ ], ξ ∈ Ξ :
ξ L
[γL| ] = m [ρ ].
N N ξ ξ
ξM∈ΞL
7
Set K = {V ∈ L | Vx = ρ (x)V for all x ∈ N} for ξ ∈ Ξ . Then one
ξ ξ L
has m = dim(K ) and L is weakly spanned by K∗N, ξ ∈ Ξ.
ξ ξ ξ
Proof. (1). We may and do assume thatN is infinite by tensoring with an infinite
factor if necessary. For ξ ∈ Ξ , we define the matrix algebra B ⊂ N′ ∩M by
L ξ 1
B := K∗e K . Then it is easy to see that z A z = (z ) A (z ) = B . Hence
ξ ξ N ξ L ξ L L ξ ξ L ξ ξ
we have
z (N′ ∩M )z = B .
L 1 L ξ
ξM∈ΞL
Let Ψ : z M z →L be the isomorphism constructed in the previous lemma.
L L 1 L 1
Using the equalities Ψ (z (N′ ∩M )z ) = Ψ ((Nz )′ ∩z M z ) = N′ ∩L and
L L 1 L L L L 1 L 1
Ψ (B ) = K∗f K , we have
L ξ ξ N ξ
N′ ∩L = K∗f K .
1 ξ N ξ
ξM∈ΞL
Since EL1 is finite on each matrix algebra K∗f K , EL1 is semi-finite on N′∩L .
L ξ N ξ L 1
Therefore the inclusion N ⊂ L is discrete.
(2).bTake V ∈ K such that V∗f V is a minimal bprojection in K∗f K . Note
ξ N ξ N ξ
that EL(VV∗) = 1. The projection V∗f V corresponds to an irreducible sector
N N
in Sect(N). The sector is actually equal to [ρ ] ∈ Sect(N) as seen below. Set
ξ
W := f V ∈ L . Using WW∗ = f EL(VV∗)f = f and W∗W = V∗f V, we
N 1 N N N N N
have
f ρ (x) = WW∗ρ (x) = WV∗f ρ (x) = WV∗ρ (x)f = WxV∗f = WxW∗.
N ξ ξ N ξ ξ N N
By [9, Lemma 3.1], the minimal projection V∗f V corresponds to [ρ ], and the
N ξ
canonical endomorphism γL: L→N has the following decomposition in Sect(N)
N
[γL| ] = dim(K )[ρ ].
N N ξ ξ
ξM∈ΞL
From this, we see that [γL| ] is contained in [γM| ] because each irreducible is
N N N N
contained in [γM| ] and we trivially have dim(K ) ≤ dim(H ).
N N ξ ξ
(3). Apply [9, Lemma 3.8] to the discrete inclusion N ⊂ L. (cid:3)
Remark 2.8. In the above theorem, we have shown [γM| ] contains [γL| ].
N N N N
This means L2(M) contains L2(L) . Since M contains L, it might sound
N N N N
trivial by regarding L2(M) ⊃ L2(L) naturally. Indeed this inclusion implies that
of the left N-modules, but it does not directly derive the inclusion of the right
N-modules when M is infinite because the right module structures are given by
the different modular conjugations J and J . If there exists a faithful normal
M L
conditional expectation EM: M →L, then we can regard the restriction J on
L M
L2(L)asJ byusing Takesaki’s theorem[22, p.309]. Henceinthiscase, theabove
L
theorem is really trivial. The point is that the above theorem also holds for an
intermediate subfactor N ⊂ L ⊂ M such that there exist no faithful normal
conditional expectations from M onto L.
8
3. Minimal actions of compact quantum groups
In this section, we apply Theorem 2.7 to inclusions of factors coming from
minimal actions of compact quantum groups, and we obtain the Galois corre-
spondence (Theorem 3.8).
3.1. Compact quantum groups
We briefly explain compact quantum groups and their actions. We adopt the
definition of a compact quantum group that is introduced in [29, Definition 2.1]
as follows:
Definition 3.1. A compact quantum group G is a pair (C(G),δ) that satisfies
the following conditions:
(1) C(G) is a separable unital C∗-algebra.
(2) The map δ: C(G) → C(G) ⊗ C(G) is a coproduct, i.e. it is a faithful
unital ∗-homomorphism satisfying the coassociativity condition,
(δ ⊗id)◦δ = (id⊗δ)◦δ.
(3) The vector spaces δ(C(G))(C⊗C(G)) and δ(C(G))(C(G)⊗C) are dense
in C(G)⊗C(G).
Let h be the Haar state on C(G), which satisfies the invariance condition:
(id⊗h)(δ(a)) = h(a)1 = (h⊗id)(δ(a)) for all a ∈ C(G).
In this paper, we always assume that the Haar state is faithful. If the Haar state
is tracial, we say that the compact quantum group is of Kac type [4].
Let {L2(G),Λ } be the GNS representation of C(G) associated with h. We
h
define the von Neumann algebra L∞(G) = C(G)w ⊂ B(L2(G)). We can extend
the coproduct δ to L∞(G) by the standard procedure [12]. The extended coprod-
uct is also denoted by δ. Then the pair (L∞(G),δ) is a von Neumann algebraic
compact quantum group in the sense of [12].
Let H be a Hilbert space. We say that a unitary v ∈ L∞(G)⊗B(H) is a (left)
unitary representation on H when it satisfies (δ ⊗ id)(v) = v v . The unitary
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representation v issaidtobeirreducible if {T ∈ B(H) | (1⊗T)v = v(1⊗T)} = C.
The set of the equivalence classes of all irreducible unitary representations is
denoted by Irr(G). For π ∈ Irr(G), take a representative v ∈ L∞(G)⊗B(K ).
π π
Then it is well-known that K is finite dimensional. Set d := dim(K ). We
π π π
denote by L∞(G) the subspace of L∞(G) that is spanned by the entries of v .
π π
Then the subspace A(G) = span{L∞(G) | π ∈ Irr(G)} is a weakly dense unital
π
∗-subalgebra of L∞(G).
We also use the modular objects of L∞(G). Let {fz}z∈C be the Woronowicz
characters on A(G). For its characterization, readers are referred to [29, Theo-
rem2.4]. Then the modular automorphism group {σth}t∈R on A(G) is given by
σh(x) = (f ⊗id⊗f ) (δ ⊗id)(δ(x)) for all t ∈ R, x ∈ A(G).
t it it
9
(cid:0) (cid:1)
We define the following map τ : A(G)→A(G) by
t
τ (x) = (f ⊗id⊗f ) (δ ⊗id)(δ(x)) for all t ∈ R, x ∈ A(G).
t it −it
Then {τt}t∈R is a one-paramet(cid:0)er automorphi(cid:1)sm group on A(G) and it is called
the scaling automorphism group. Since the Haar state h is invariant under the
∗-preserving maps σh and τ , we can extend them to the maps on C(G), and on
t t
L∞(G). By definition, we have
σh(x) = (f ⊗τ )(δ(x)) for all x ∈ A(G). (3.1)
t 2it −t
We recall the notion of a left coideal von Neumann algebra introduced in [9,
Definition 4.1]. In this paper, we simply call it a left coideal.
Definition 3.2. Let B ⊂ L∞(G) be a von Neumann subalgebra. We say that B
is a left coideal if δ(B) ⊂ L∞(G)⊗B.
If G comes from an ordinary compact group, it is known that any left coideal
is of the form L∞(G/H) for a closed subgroup H ⊂ G [1]. Therefore a left coideal
can be considered as an object like a “closed subgroup” of G. Even in quantum
case, we can also introduce the notion of a closed quantum subgroup H ⊂ G as
[20]. However when G is not a compact group, not all the left coideals of G have
quotient forms asL∞(G/H) [19, 20, 24]. Fora compact quantum groupsatisfying
a certain condition, we have a necessary and sufficient condition so that a left
coideal is of the form L∞(G/H) [25, Theorem 3.18].
Now let B be a left coideal, and we put B := B ∩L∞(G) . Since B admits
π π
the left G-action δ, B is weakly spanned by B , π ∈ Irr(G).
π
Lemma 3.3. Let B ⊂ L∞(G) be a left coideal and π ∈ Irr(G). Then there exist
a unitary representation u = (u ) and a subset IB ⊂ I such that
π πi,j i,j∈Iπ π π
(1) u is equivalent to v ,
π π
(2) B = span{u | i ∈ I , j ∈ IB}.
π πi,j π π
Proof. Let HomG(Kπ,L∞(G)) be the set of G-linear maps from Kπ into L∞(G),
that is, it consists of linear maps S: K →L∞(G) such that δ◦S = (id⊗S)◦v ,
π π
where v is regarded as a map from K to L∞(G) ⊗ K . Similarly we define
π π π π
HomG(Kπ,B), which is a subspace of HomG(Kπ,L∞(G)). Let (εi)i∈Iπ be an
orthonormal basis of Kπ. We prepare the inner product of HomG(Kπ,L∞(G))
defined by
hS|Ti1 := T(ε )∗S(ε ).
i i
Xi∈Iπ
Then HomG(Kπ,L∞(G)) is a Hilbert space of dimension dπ. We take an or-
thonormal basis {Si}i∈Iπ of HomG(Kπ,L∞(G)) which contains an orthonormal
basis of HomG(Kπ,B) denoted by {Si}i∈IB.
π
We define the linear map T : K →L∞(G) by T (ε ) = v for j ∈ I . Then it
j π j i πij π
is easy to see that {Tj}j∈Iπ is an orthonormal basis of HomG(Kπ,L∞(G)). Hence
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