AMENABILITY AND THE BICROSSED PRODUCT CONSTRUCTION 1. Introduction The theory of ...

19 Pages·2003·0.24 MB·English

IllinoisJournalofMathematics
Volume46,Number4,Winter2002,Pages1259–1277
S0019-2082
AMENABILITY AND THE BICROSSED PRODUCT
CONSTRUCTION
PIETERDESMEDT,JOHANQUAEGEBEUR,ANDSTEFAANVAES
Abstract. Westudy stabilityproperties of amenable locally compact
quantum groups under the bicrossed product construction. We obtain
asourmainresultanequivalencebetweenamenabilityofthebicrossed
productandamenabilityofthematchedquantumgroupsusedasbuild-
ing ingredients of the bicrossed product. Finally, we give examples of
non-amenablelocallycompactquantumgroupsobtainedbyabicrossed
productconstruction.
1. Introduction
The theory of locally compact quantum groups has been introduced by
J.Kustermansandthethirdauthor[9][10],unifyingcompactquantumgroups
and Kac algebras. As the example of the quantum SU (2)-group, developed
q
by Woronowicz, shows, the antipode of a compact quantum group need not
be bounded and it need not respect the ∗-operation. For this reason, com-
pact quantum groups are not always Kac algebras. The crucial difference
between Kac algebras and locally compact quantum groups is the possible
unboundedness of the antipode.
Taking into account the importance of amenable locally compact groups
within the category of all locally compact groups, it is natural to consider
amenability of locally compact quantum groups. In fact, the main results on
amenability of Kac algebras have been developed by Enock and Schwartz [6],
and their proofs can be repeated in the more general framework of locally
compact quantum groups. However, there is still one open problem. Recall
that there are many different characterizations of amenability of locally com-
pact groups. A first characterization deals with the existence of an invariant
mean on a suitable algebra of functions on the group G (bounded continuous
ReceivedMarch26,2002;receivedinfinalformJune14,2002.
2000 Mathematics Subject Classification. Primary46L89. Secondary43A07.
The third author is Research Assistant of the Fund for Scientific Research—Flanders
(Belgium)(F.W.O.).
(cid:13)c2002 University of Illinois
1259
1260 PIETERDESMEDT,JOHANQUAEGEBEUR,ANDSTEFAANVAES
functions or L∞(G)). Another characterization says that the trivial repre-
sentation of G is weakly contained in the left regular representation. Other
characterizationsaremostofthetimecloselyrelatedtooneofthesetwodefi-
nitions. Thesetwopropertiescanbeformulatedforlocallycompactquantum
groups;inthisway,onedefinesweaklyamenable andstronglyamenable locally
compact quantum groups. It is known that all strongly amenable quantum
groups are weakly amenable. The converse has been claimed by Enock and
Schwartz [6] on the level of Kac algebras. However, their proof contains a
gap as pointed out by Ruan [14]. In the latter paper, Ruan closed the gap
for discrete Kac algebras. Recently, E. Blanchard and the third author ex-
tended the result of Ruan to arbitrary discrete quantum groups. Even more
recently,Tomatsu[15]foundanalternativeproofofthesameresult. Also,for
locally compact groups the two notions of amenability coincide; see, e.g., [7].
However, in the general case this remains an open problem.
Having defined both notions of amenability for locally compact quantum
groups, one asks for examples. A systematic way of constructing examples of
locally compact quantum groups has been developed by Majid [11], Baaj and
Skandalis [2] and Vainerman andthe thirdauthor [19]. In this paper, we pre-
ciselycharacterizewhentheselocallycompactquantumgroupsareamenable.
We also give two examples of non-amenable locally compact quantum groups
obtained by this so-called bicrossed product construction.
In [19], one also defines bicrossed products of quantum groups, and one
makesthelinkwithshortexactsequencesoflocallycompactquantumgroups,
called extensions. In this paper, we will characterize in this full generality
when the bicrossed product is amenable; in fact, our result is a quantum
version of the well known result that a locally compact group G with normal
closed subgroup H is amenable if and only if H and G/H are amenable.
2. Preliminaries
We refer to [9] and [10] for the theory of locally compact quantum groups
in the C∗-algebra, as well as in the von Neumann algebra language. For the
non-specialists,[18]isagoodstartingpoint. Werecallfrom[10]thedefinition
ofavonNeumannalgebraicquantumgroup: (M,∆)iscalleda(vonNeumann
algebraic) locally compact quantum group when
• M is a von Neumann algebra and ∆ : M → M ⊗M is a normal
and unital ∗-homomorphism satisfying the coassociativity relation
(∆⊗ι)∆=(ι⊗∆)∆;
• thereexistnormal, semi-finite, faithful(n.s.f.) weightsϕandψ onM
such that
– ϕisleftinvariant,i.e.,ϕ (ω⊗ι)∆(x) =ϕ(x)ω(1)forallx∈M+
(cid:0) (cid:1) ϕ
and ω ∈M+,
∗
AMENABILITY AND THE BICROSSED PRODUCT CONSTRUCTION 1261
– ψ is right invariant, i.e., ψ (ι⊗ω)∆(x) = ψ(x)ω(1) for all x ∈
(cid:0) (cid:1)
M+ and ω ∈M+.
ψ ∗
Here we use the notation M+ = {x ∈ M+ | ϕ(x) < +∞}, and define analo-
ϕ
gously M+.
ψ
Fixaleftinvariantn.s.f.weightϕon(M,∆)andrepresentM ontheGNS-
spaceofϕsuchthat(H,ι,Λ)isaGNS-constructionforϕ. Thenwecandefine
a unitary W on H ⊗H by
W∗(Λ(a)⊗Λ(b))=(Λ⊗Λ)(∆(b)(a⊗1)) for alla,b∈N .
ϕ
HereΛ⊗ΛdenotesthecanonicalGNS-mapforthetensorproductweightϕ⊗ϕ.
OneprovesthatW satisfiesthepentagonalequation: W W W =W W .
12 13 23 23 12
We say that W is a multiplicative unitary. The comultiplication can be given
in terms of W by the formula ∆(x) = W∗(1⊗x)W for all x ∈ M. Also the
von Neumann algebra M can be written in terms of W as
M ={(ι⊗ω)(W)|ω ∈B(H)∗}−σ-strong∗ .
Next,thelocallycompactquantumgroup(M,∆)hasanantipode S,whichis
the uniqueσ-strong∗ closed linearmap fromM to M satisfying(ι⊗ω)(W)∈
D(S) for all ω ∈ B(H)∗, S(ι ⊗ω)(W) = (ι ⊗ω)(W∗), and such that the
elements (ι⊗ω)(W) form a σ-strong∗ core for S. The antipode S has a polar
decomposition S = Rτ−i/2, where R is an anti-automorphism of M and (τt)
is a strongly continuous one-parameter group of automorphisms of M. We
call R the unitary antipode and (τ ) the scaling group of (M,∆). It is known
t
that σ(R⊗R)∆=∆R, where σ denotes the flip map on M ⊗M.
We turn the predual M∗ into a Banach algebra with product ω ∗ µ =
(ω⊗µ)∆, for all ω,µ∈M∗.
We use the notation ∆op to denote the opposite comultiplication defined
by ∆op:=σ∆.
The dual locally compact quantum group (Mˆ,∆ˆ) is defined as follows. Its
von Neumann algebra Mˆ is
Mˆ ={(ω⊗ι)(W)|ω ∈B(H)∗}−σ-strong∗
and the comultiplication is given by ∆ˆ(x) = ΣW(x⊗1)W∗Σ for all x ∈ Mˆ,
where Σ denotes the flip map on the tensorproduct of Hilbert spaces. Set
Wˆ =ΣW∗Σ.
Since (Mˆ,∆ˆ) is again a locally compact quantum group, we can introduce
the antipode Sˆ, the unitary antipode Rˆ and the scaling group (τˆ) exactly as
t
wediditfor(M,∆). Also,wecanagainconstructthedualof(Mˆ,∆ˆ),starting
from the left invariant weight ϕˆ with GNS-construction (H,ι,Λˆ). From the
biduality theorem, we get that the bidual locally compact quantum group
(Mˆˆ,∆ˆˆ) is isomorphic to (M,∆).
1262 PIETERDESMEDT,JOHANQUAEGEBEUR,ANDSTEFAANVAES
Define M to be the norm closure of the space
c
{(ι⊗ω)(W)|ω ∈B(H)∗}
and ∆ to be the restriction of ∆ to M . It was proven in [10] that the
c c
pair (M ,∆ ) is a reduced C∗-algebraic locally compact quantum group. We
c c
know that there is a bijective correspondence between reduced C∗-algebraic
quantumgroupsandvonNeumannalgebraicquantumgroups. So, thechoice
for the von Neumann algebra language is not a restriction.
A ∗-homomorphism ε:M →C is called a co-unit of (M ,∆ ), if
c c c
(ε⊗ι)∆=(ι⊗ε)∆=ι.
Classical locally compact groups appear as M =L∞(G) with ∆(f)(p,q)=
f(pq). The invariant weights are defined by integrating with respect to the
left or the right Haar measure. The dual Mˆ can be identified with the group
von Neumann algebra L(G).
Working with tensor products with more than two factors, we will some-
timesusetheleg-numberingnotation. Forexample,ifH,K andLareHilbert
spacesandX ∈B(H⊗L), wedenotebyX (respectively, X , X )theop-
13 12 23
erator(1⊗Σ∗)(X⊗1)(1⊗Σ)(respectively,X⊗1, 1⊗X)definedonH⊗K⊗L.
If now H =H ⊗H is itself a tensor product of two Hilbert spaces, then we
1 2
sometimesswitchfromtheleg-numberingnotationwithrespecttoH⊗K⊗L
to the one with respect to the finer tensor product H ⊗H ⊗K⊗L, for ex-
1 2
ample, from X to X . There is no confusion here, because the number of
13 124
legs changes. Weak and σ-weak convergence are denoted by −w→, respectively
−σ→w.
3. Amenability
Let (M,∆) be a von Neumann algebraic locally compact quantum group.
A state m∈M∗ is said to be a left invariant mean (LIM) on (M,∆) if
m((ω⊗ι)∆(x))=m(x)ω(1),
for all ω ∈M∗ and x∈M. It is said to be a right invariant mean (RIM) if
m((ι⊗ω)∆(x))=m(x)ω(1),
for all ω ∈M∗ and x∈M. Finally, if m is both a LIM and a RIM, we call m
an invariant mean (IM).
Definition 1. We call (M,∆) weakly amenable if there exists a left in-
variant mean (LIM) on (M,∆). We say that (M,∆) is weakly coamenable if
(Mˆ,∆ˆ) is weakly amenable.
Definition2. Wecall(M,∆)stronglyamenable ifthereexistsabounded
co-unit on (Mˆ ,∆ˆ ). We say that (M,∆) is strongly coamenable if (Mˆ,∆ˆ) is
c c
strongly amenable.
AMENABILITY AND THE BICROSSED PRODUCT CONSTRUCTION 1263
Classicallocallycompactgroupsappearas(L∞(G),∆ )inthetheoryoflo-
G
cally compact quantum groups. For these groups, weak and strong amenabil-
ity coincide and agree with the usual notion of amenability of the group G.
Otherauthorssometimesusea“dual”terminology. Thisdifferenceoriginates
fromthechoiceofthequantumgroupassociatedwithalocallycompactgroup,
L∞(G) or L(G). Here we adopt the point of view of Enock and Schwartz [6]
and Ruan [14] (i.e., we take L∞(G) as the associated quantum group). The
“dual” convention is used, amongst others, by Banica [3], Baaj and Skandalis
[2]andNg[12]. Theyusecoamenable whereweusestronglyamenable. When-
everwecitearesultofoneofthepapersmentionedwithdifferentterminology,
it will be already translated to our setting.
M.EnockandJ.M.Schwartz[6]provedthat,forKacalgebras,thefollowing
statementsareequivalenttothefactthataKacalgebraisstronglyamenable:
(i) There exists a net (ξ ) of normalized vectors in H such that
j j
(ι⊗ω )(W)−w→1.
ξj
(ii) There exists a bounded left (resp., right) approximate unit in Mˆ∗.
It was proven in [6] that strong amenability implies weak amenability. The
authors also claim that the opposite implication is true, but, as mentioned
by Ruan [14], there is a gap in their proof. It is an important open question
whether or not weak amenability implies strong amenability. Until now, this
is only known to be true for locally compact groups (see, for example, [7]),
for discrete Kac algebras [14], and recently, also for discrete quantum groups
[15].
Enock and Schwartz further showed that the following statements are
equivalent:
(i) There exists a LIM on (M,∆) (resp., RIM).
(ii) Thereexistsanet(ωi)i ofstatesinM∗ suchthatω∗ωi−ωi converges
weakly to 0 (resp., ωi∗ω−ωi), for all ω ∈M∗ with ω(1)=1.
(iii) Thereexistsanet(ωi)iofstatesinM∗suchthat(cid:107)ω∗ωi−ωi(cid:107)converges
to 0 (resp., (cid:107)ωi∗ω−ωi(cid:107)), for all ω ∈M∗ with ω(1)=1.
All of these results are also true for locally compact quantum groups. Not
surprisingly, we can prove the following proposition.
Proposition 3. Let (M,∆) be a locally compact quantum group. There
exists a LIM on (M,∆) if and only if there exists an invariant mean on
(M,∆).
Proof. One implication is immediate.
Conversely, suppose there exists a LIM on (M,∆). From the result men-
tioned above, we know that there exists a net of states (ωi)i in M∗ such that
(cid:107)ω∗ωi−ωi(cid:107) converges to 0 for all ω ∈M∗ with ω(1)=1. It is obvious that
this is equivalent to the existence of a net of states (ωi◦)i in M∗ such that
1264 PIETERDESMEDT,JOHANQUAEGEBEUR,ANDSTEFAANVAES
(cid:107)ωi◦∗ω−ωi◦(cid:107) converges to 0 for all ω ∈M∗ with ω(1)=1; take ωi◦ =ωi◦R.
It is easy to prove that µ =(ω ∗ω◦) is a net of states such that
k i j (i,j)
(cid:107)µ ∗ω−µ (cid:107)→0 and (cid:107)ω∗µ −µ (cid:107)→0
k k k k
for all ω ∈ M∗ with ω(1) = 1. Let m be a weak-∗ limit point of (µk)k in the
unit ball of M∗. It is obvious that m will be an invariant mean. (cid:3)
4. Bicrossed products
In this section, we collect some results and definitions from [19].
Definition 4. Wecallapair(α,U)acocycle action ofalocallycompact
quantum group (M,∆) on a von Neumann algebra N if
α:N →M ⊗N
is a normal, injective and unital ∗-homomorphism,
U ∈M ⊗M ⊗N
is a unitary, and if α and U satisfy
(ι⊗α)α(x)=U(∆⊗ι)α(x)U∗ for all x∈N,
(ι⊗ι⊗α)(U)(∆⊗ι⊗ι)(U)=(1⊗U)(ι⊗∆⊗ι)(U).
If U is trivial, i.e., U =1, we call α an action.
Notation 5. If (α,U) is a cocycle action of (M,∆) on a von Neumann
algebra N, we introduce the notation
W˜ =(W ⊗1) U∗;
then, W˜ is a unitary in M ⊗B(H)⊗N.
Given a cocycle action (α,U) of a locally compact quantum group (M,∆)
on a von Neumann algebra N, we construct the crossed product Mα,U(cid:110)N.
This is the von Neumann subalgebra of B(H)⊗N generated by
α(N)and{(ω⊗ι⊗ι)((W ⊗1) U∗) | ω ∈M∗}.
When U is trivial, the crossed product is denoted by M (cid:110)N. There is a
α
unique action αˆ of (Mˆ,∆ˆop) on Mα,U(cid:110)N such that, for all x∈N,
(1) αˆ α(x) =1⊗α(x)and(ι⊗αˆ) (W ⊗1) U∗ =W W U∗ .
(cid:0) (cid:1) (cid:0) (cid:1) 12 13 134
We call this action αˆ the dual action. It was proven in [19] that the fixed
point algebra is
(Mα,U(cid:110)N)αˆ ={x∈Mα,U(cid:110)N | αˆ(x)=1⊗x}=α(N).
Definition 6. A pair (M ,∆ ),(M ,∆ ) is said to be a matched pair
1 1 2 2
of locally compact quantum groups if there exists a triple (τ,U,V) (called a
cocycle matching) satisfying the following conditions:
AMENABILITY AND THE BICROSSED PRODUCT CONSTRUCTION 1265
• U ∈M ⊗M ⊗M and V ∈M ⊗M ⊗M are both unitaries;
1 1 2 1 2 2
• τ :M ⊗M →M ⊗M is a faithful ∗-homomorphism;
1 2 1 2
• defining α(y)=τ(1⊗y) and β(x)=τ(x⊗1) we have:
– (α,U) is a cocycle action of (M ,∆ ) on M ,
1 1 2
– (σβ,V ) is a cocycle action of (M ,∆ ) on M ,
321 2 2 1
– (α,U) and (β,V) are matched in the following sense:
τ (α⊗ι)∆ (y)=V (ι⊗∆ )α(y)V∗ ,
13 2 132 2 132
τ σ (β⊗ι)∆ (x)=U(∆ ⊗ι)β(x) U∗,
23 23 1 1
(∆ ⊗ι⊗ι)(V)(ι⊗ι⊗∆op)(U∗)=
1 2
(U∗⊗1)(ι⊗τσ⊗ι) (β⊗ι⊗ι)(U∗)(ι⊗ι⊗α)(V) (1⊗V).
(cid:0) (cid:1)
Givenacocyclematching(τ,U,V)of(M ,∆ )and(M ,∆ ),oneisableto
1 1 2 2
construct the cocycle bicrossed product (M,∆). By definition M =
M1α,U(cid:110)M2 and ∆(x)=W∗(1⊗x)W with W =ΣWˆ∗Σ and
Wˆ =(β⊗ι⊗ι) (W ⊗1) U∗ (ι⊗ι⊗α) V(1⊗Wˆ )
1 2
(cid:0) (cid:1) (cid:0) (cid:1)
∈M ⊗B(H )⊗B(H )⊗M ,
1 2 1 2
with W and W the multiplicative unitaries of (M ,∆ ) and (M ,∆ ), re-
1 2 1 1 2 2
spectively. It was proven in [19] that (M,∆) is a locally compact quantum
group and that W is its multiplicative unitary.
In Section 5, (M ,∆ ) and (M ,∆ ) will always be two locally compact
1 1 2 2
quantum groups matched by (τ,U,V) and their cocycle bicrossed product
locally compact quantum group will be denoted by (M,∆). All the objects
associated with a quantum group (e.g., W, ∆,...) will be denoted with an
index, when they refer to (M ,∆ ) and (M ,∆ ), respectively, and without
1 1 2 2
an index when they refer to (M,∆). So we have that W (resp., W ) is the
1 2
multiplicative unitary of (M ,∆ ) (resp., (M ,∆ ) ) and
1 1 2 2
W˜ =(W ⊗1) U∗.
1 1
From Propositions 2.4 and 2.5 of [19] we know how the comultiplication ∆
works on the generators α(x) and (ω⊗ι⊗ι)(W˜ ) of (M,∆):
1
∆(α(x))=(α⊗α)∆ (x)
2
(2) (ι⊗∆op)(W˜ )=(W˜ ⊗1⊗1)((ι⊗α)β⊗ι⊗ι)(W˜ )(ι⊗α⊗α)(V)
1 1 1
DefineMˆ asthevonNeumannsubalgebraofM ⊗B(H )generatedbyβ(M )
1 2 1
and {(ι⊗ι⊗ω)(V(1⊗Wˆ2)) | ω ∈M2∗}. We define ∆ˆ(z)=Wˆ∗(1⊗z)Wˆ, for
all z ∈ Mˆ. In [19, Sec. 3.3.2] it was shown that (Mˆ,∆ˆ) is the dual locally
compact quantum group of (M,∆). So, if we interchange the roles of α and
β, M and M , respectively, then we find, as the cocycle bicrossed product,
1 2
the dual of the original cocycle bicrossed product.
1266 PIETERDESMEDT,JOHANQUAEGEBEUR,ANDSTEFAANVAES
Definition 7. A cocycle action (α,U) of (M,∆) on a von Neumann al-
gebra N is said to be stabilizable with a unitary X ∈M ⊗N if
(1⊗X)(ι⊗α)(X)=(∆⊗ι)(X)U∗ .
Proposition 8. Let (α,U) be a cocycle action of (M,∆) on N which is
stabilizable with a unitary X ∈M ⊗N. Then the formulas
β :N →M ⊗N :β(x)=Xα(x)X∗ and Φ:z (cid:55)→X∗zX
define, respectively, an action of (M,∆) on N and a ∗-isomorphism from
Mβ(cid:110)N onto Mα,U(cid:110)N satisfying
αˆ◦Φ=(ι⊗Φ)◦βˆ.
The next proposition shows that many cocycle actions are stabilizable.
Proposition 9. Let (α,U) be a cocycle action of (M,∆) on N. Then
(α⊗ι,U⊗1) is a cocycle action of (M,∆) on N⊗B(H) which is stabilizable.
5. Amenability and the bicrossed product construction
In this section, we investigate the relation between weak, respectively
strong, amenability of the cocycle bicrossed product quantum group (M,∆),
and of its building ingredients (M ,∆ ) and (M ,∆ ).
1 1 2 2
Westartwithatechnicalremarkaboutslicingwithnon-normalfunctionals.
Let N and L be von Neumann algebras, n∈N∗ and X ∈N ⊗L.
If n ∈ N∗, then it is obvious that (n ⊗ ι)(X) ∈ L. This remains true
for n ∈ N∗, even if n is not normal. Indeed, consider the map L∗ → C :
ω (cid:55)→ n((ι⊗ω)(X)). It is obvious that this is a bounded linear functional
and since L = (L∗)∗, we know that there exists a unique Y ∈ L such that
ω(Y)=n((ι⊗ω)(X) for all ω ∈L∗. Set Y =(n⊗ι)(X).
Suppose that Φ : L → K is a normal ∗-homomorphism of von Neumann
algebras. Since for all ω ∈K∗
ω(Φ(n⊗ι)(X))=n((ι⊗ω◦Φ)(X))
=n((ι⊗ω)(ι⊗Φ)(X))
=ω((n⊗ι)(ι⊗Φ)(X)),
we may conclude that for all Φ
Φ((n⊗ι)(X))=(n⊗ι)(ι⊗Φ)(X).
This will be used several times in the sequel, where n will be an invariant
mean and Φ=α,∆,....
Definition 10. If α is an action of (M,∆) on a von Neumann algebra
N, we define an α-invariant mean to be a state m∈N∗ such that
m((ω⊗ι)α(x))=m(x)ω(1)
AMENABILITY AND THE BICROSSED PRODUCT CONSTRUCTION 1267
for all ω ∈M∗ and x∈N.
Proposition 11. Let (α,U) be a cocycle action of (M,∆) on a von Neu-
mannalgebraN, Mα,U(cid:110)N thecocyclecrossedproductandαˆ thedualaction.
Then(Mˆ,∆ˆ)isweaklyamenableifandonlyifthereexistsaαˆ-invariantmean
on Mα,U(cid:110)N.
Proof. Suppose that mˆ is an invariant mean on (Mˆ,∆ˆ). Then we argue
that there exists an αˆ-invariant mean on Mα,U(cid:110)N. This can be done by
generalizing a result in [6] from the Kac algebra level to the general set-
ting. However, the proof in [6] is based on a non-constructive argument. We
construct explicitly an αˆ-invariant mean on Mα,U(cid:110)N. The dual weight con-
struction is the source of inspiration. Define T : Mα,U(cid:110)N → Mα,U(cid:110)N by
T(z) := (mˆ ⊗ι)αˆ(z). We prove that T(z) ∈ α(N) for all z ∈ Mα,U(cid:110)N.
Since α(N) is the fixed point algebra of αˆ, it is sufficient to show that
αˆ(T(z))=1⊗T(z).
Observe that, since αˆ is an action, αˆ(T(z))=(mˆ ⊗ι⊗ι)(∆ˆop⊗ι)αˆ(z). So
we have to prove that, for all ω ∈(Mˆ ⊗Mα,U(cid:110)N)∗,
ω((mˆ ⊗ι⊗ι)(∆ˆop⊗ι)αˆ(z))=ω(1⊗(mˆ ⊗ι)αˆ(z)).
But it is sufficient to check this for normal functionals of the form µ⊗ν with
µ∈Mˆ∗ and ν ∈(Mα,U(cid:110)N)∗. Using the fact that mˆ is an invariant mean on
(Mˆ,∆ˆ) and hence an invariant mean on (Mˆ,∆ˆop), we get
(µ⊗ν)((mˆ ⊗ι⊗ι)(∆ˆop⊗ι)αˆ(z))=mˆ((ι⊗µ⊗ν)(∆ˆop⊗ι)αˆ(z))
=µ(1)mˆ((ι⊗ν)(αˆ(z)))
=(µ⊗ν)(1⊗(mˆ ⊗ι)αˆ(z)).
So we may conclude that T(z)∈α(N) for all z ∈Mα,U(cid:110)N.
Choose a state η ∈ N∗. Define m(z) = η(α−1(T(z))). We will prove that
m is αˆ-invariant. For all ω ∈Mˆ∗ and z ∈Mα,U(cid:110)N, we get that
m((ω⊗ι)αˆ(z))=η(α−1((mˆ ⊗ι)αˆ((ω⊗ι)αˆ(z))))
=η(α−1((mˆ ⊗ι)(ω⊗ι⊗ι)((ι⊗αˆ)αˆ(z))))
=η(α−1((mˆ ⊗ι)(ω⊗ι⊗ι)((∆ˆop⊗ι)αˆ(z))))
=η(α−1((mˆ ⊗ι)αˆ(z)))ω(1)=m(z)ω(1).
Conversely, suppose that m is a αˆ-invariant mean on Mα,U(cid:110)N. We have
to prove that (Mˆ,∆ˆ) is weakly amenable. We consider three cases:
Case 1: U is trivial.
We know that M (cid:110)N is generated by α(N) and Mˆ ⊗C. Define mˆ(xˆ) :=
α
m(xˆ⊗1) forall xˆ∈Mˆ. Using the formula αˆ(xˆ⊗1)=∆ˆop(xˆ)⊗1, we get that
1268 PIETERDESMEDT,JOHANQUAEGEBEUR,ANDSTEFAANVAES
for all ω ∈Mˆ∗
mˆ((ω⊗ι)∆ˆop(xˆ))=m((ω⊗ι⊗ι)(∆ˆop(xˆ)⊗1))
=m((ω⊗ι⊗ι)αˆ(xˆ⊗1))
=m(xˆ⊗1)ω(1)=mˆ(xˆ)ω(1).
So we may conclude that mˆ is a left invariant mean on (Mˆ,∆ˆop).
Case 2: (α,U) is stabilizable.
We know from Proposition 8 that, in this case, there exists an action β of
(M,∆) on N and a ∗-isomorphism
Φ:Mβ(cid:110)N →Mα,U(cid:110)N,
such that αˆ◦Φ=(ι⊗Φ)◦βˆ.
Define m˜ := m◦Φ. Then it is easy to prove that m˜ is βˆ-invariant. Using
the first case, we may conclude that the restriction of m˜ to Mˆ will be a LIM
on (Mˆ,∆ˆop).
General case: Arbitrary (α,U).
In general, (α⊗ι,U ⊗1) will be a cocycle action of (M,∆) on N ⊗B(H),
and we know from Proposition 9 that it will be stabilizable. It is not too
difficult to show that its corresponding cocycle crossed product factorizes as
(Mα,U(cid:110)N)⊗B(H), as well as the dual action, which is given by αˆ⊗ι.
Chooseanormalizedvectorξ ∈H. Thenwehaveforallz ∈(Mα,U(cid:110)N)⊗
B(H) that
(ι⊗m⊗ω )((αˆ⊗ι)(z))=(ι⊗m)αˆ((ι⊗ω )(z))
ξ ξ
=m((ι⊗ω )(z))1
ξ
=(m⊗ω )(z)1.
ξ
We find that m⊗ω is (αˆ⊗ι)-invariant and from the second case, we may
ξ
conclude that (Mˆ,∆ˆ) is weakly amenable. (cid:3)
With this theorem in mind, we are going to prove our main results, gen-
eralizing a result of Ng [13]. Ng showed in [13] that the bicrossed product
with trivial cocycles of two locally compact groups G and G is strongly
1 2
amenable if G is amenable. Notice that for any group G , (L∞(G ),∆ ) is
2 1 1 1
always strongly coamenable.
Toproveourmaintheorem, weneedalemma. Wecangetthisresultfrom
Propositions 3.1 and 3.4 of [19], but we have chosen to give a straightforward
proof.
Lemma 12. Let(τ,U,V)beacocyclematchingof(M ,∆ )and(M ,∆ ),
1 1 2 2
and let (M,∆) be the cocycle bicrossed product. Then
(3) (ι⊗∆op)αˆ(z)=(αˆ⊗ι)∆op(z) for all z ∈M.