L^2-Betti numbers of coamenable quantum groups
L2-BETTI NUMBERS OF COAMENABLE QUANTUM
GROUPS
DAVID KYED
Abstra
t. We prove that a 
ompa
t quantum group is 
oamen-
able if and only if its 
orepresentation ring is amenable. We further
propose a Følner 
ondition for 
ompa
t quantum groups and prove
it to be equivalent to 
oamenability. Using this Følner 
ondition,
we prove that for a 
oamenable 
ompa
t quantum group with tra-

ial Haar state, the enveloping von Neumann algebra is dimension
�at over the Hopf algebra of matrix 
oe�
ients. This generalizes a
theorem of Lü
k from the group 
ase to the quantum group 
ase,
and provides examples of 
ompa
t quantum groups with vanishing
-Betti numbers.
Introdu
tion
The theory of L2-Betti numbers for dis
rete groups is originally due
to Atiyah and dates ba
k to the seventies [Ati76℄. These L2-Betti num-
bers are de�ned for those dis
rete groups that permit a free, proper
and 
o
ompa
t a
tion on some 
ontra
tible, Riemannian manifold X .
If Γ is su
h a group, the spa
e of square integrable p-forms on X
be
omes a �nitely generated Hilbert module for the group von Neu-
mann algebra L (Γ). As su
h it has a Murray-von Neumann dimen-
sion whi
h turns out to be independent of the 
hoi
e of X and is 
alled
the p-th L2-Betti number of Γ, denoted β
p (Γ). More re
ently, Lü
k
[Lü
97, Lü
98a, Lü
98b℄ transported the notion of Murray-von Neu-
mann dimension to the setting of �nitely generated proje
tive (alge-
brai
) L (Γ)-modules and extended thereafter the domain of de�nition
to the 
lass of all modules. With this extended dimension fun
tion,
dimL (Γ)(−), it is possible to extend the notion of L2-Betti numbers to

over all dis
rete groups Γ by setting
β(2)p (Γ) = dimL (Γ) Tor
p (L (Γ),C).
For more details on the relations between the di�erent de�nitions of L2-
Betti numbers and the extended dimension fun
tion we refer to Lü
k's
book [Lü
02℄.
2000 Mathemati
s Subje
t Classi�
ation. 16W30,43A07, 46L89, 16E30.
http://arxiv.org/abs/0704.1582v4
2 DAVID KYED
All the ingredients in the homologi
al algebrai
 de�nition above have
fully developed analogues in the world of 
ompa
t quantum groups, and
using this di
tionary the notion of L2-Betti numbers was generalized
to the quantum group setting in [Kye08℄. Sin
e this generalization is

entral for the work in the present paper, we shall now explain it in
greater detail. Consider a 
ompa
t quantum group G = (A,∆) and
assume that its Haar state h is a tra
e. If we denote by A0 the unique
dense Hopf ∗-algebra and by M the enveloping von Neumann algebra
of A in the GNS representation arising from h, then the p-th L2-Betti
number of G is de�ned as
β(2)p (G) = dimM Tor
p (M,C).
Here C is 
onsidered an A0-module via the 
ounit ε : A0 → C and
dimM(−) is Lü
k's extended dimension fun
tion arising from (the ex-
tension of) the tra
e-state h. This de�nition extends the 
lassi
al one
[Kye08, 1.3℄ in the sense that
β(2)p (G) = β
p (Γ)
when G = (C∗red(Γ),∆red).
The aim of this paper is to investigate the L2-Betti numbers of the 
lass
of 
oamenable, 
ompa
t quantum groups. In the 
lassi
al 
ase we have
that β
p (Γ) = 0 for all p ≥ 1 whenever Γ is an amenable group. This

an be seen as a spe
ial 
ase of [Lü
98a, 5.1℄ where it is proved that
the von Neumann algebra L (Γ) is dimension �at over CΓ, meaning
dimL (Γ) Tor
p (L (Γ), Z) = 0 (p ≥ 1)
for any CΓ-module Z � provided, of 
ourse, that Γ is still assumed
amenable. We generalize this result to the quantum group setting
in Theorem 6.1. More pre
isely, we prove that if G = (A,∆) is a

ompa
t, 
oamenable quantum group with tra
ial Haar state and Z is
any module for the algebra of matrix 
oe�
ients A0 then
dimM Tor
p (M,Z) = 0. (p ≥ 1)
Here M is again the enveloping von Neumann algebra in the GNS rep-
resentation arising from the Haar state. In order to prove this result
we need a Følner 
ondition for 
ompa
t quantum groups. The 
lassi
al
Følner 
ondition for groups [Føl55℄ is a geometri
al 
ondition, on the
a
tion of the group on itself, whi
h is equivalent to amenability of the
group. In order to obtain a quantum analogue of Følner's 
ondition a
detailed study of the ring of 
orepresentations, asso
iated to a 
ompa
t
quantum group, is needed. The ring of 
orepresentations is a spe
ial
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 3

ase of a so-
alled fusion algebra and we have therefore devoted a sub-
stantial part of this paper to the study of abstra
t fusion algebras and
their amenability. Amenability for (�nitely generated) fusion algebras
was introdu
ed by Hiai and Izumi in [HI98℄ where they also gave two
equivalent Følner-type 
onditions for fusion algebras. We generalize
their results to the non-�nitely generated 
ase and prove that a 
om-
pa
t quantum group is 
oamenable if and only if its 
orepresentation
ring is amenable. From this we obtain a Følner 
ondition for 
ompa
t
quantum groups whi
h is equivalent to 
oamenability. Using this Følner

ondition we prove our main result, Theorem 6.1, whi
h implies that

oamenable 
ompa
t quantum groups have vanishing L2-Betti numbers
in all positive degrees.
Stru
ture. The paper is organized as follows. In the �rst se
tion we re-

apitulate (parts of) Woronowi
z's theory of 
ompa
t quantum groups.
The se
ond and third se
tion is devoted to the study of abstra
t fu-
sion algebras and amenability of su
h. In the fourth se
tion we dis
uss

oamenability of 
ompa
t quantum groups and investigate the relation
between 
oamenability of a 
ompa
t quantum group and amenability of
its 
orepresentation ring. The �fth se
tion is an interlude in whi
h the
ne
essary notation 
on
erning von Neumann algebrai
 
ompa
t quan-
tum groups and their dis
rete duals is introdu
ed. The sixth se
tion is
devoted to the proof of our main theorem (6.1) and the seventh, and
�nal, se
tion 
onsists of examples.
A
knowledgements. I wish to thank my supervisor Ryszard Nest for
the many dis
ussions about quantum groups and their (
o)amenability,
and Andreas Thom for pointing out to me that the bi
rossed produ
t

onstru
tion 
ould be used to generate examples of quantum groups
satisfying Følner's 
ondition.
Notation. Throughout the paper, the symbol ⊙ will be used to denote
algebrai
 tensor produ
ts while the symbol ⊗̄ will be used to denote
tensor produ
ts in the 
ategory of Hilbert spa
es or the 
ategory of
von Neumann algebras. All tensor produ
ts between C∗-algebras are
assumed minimal/spatial and these will be denoted by the symbol ⊗.
1. Preliminaries on 
ompa
t quantum groups
In this se
tion we brie�y re
all Woronowi
z's theory of 
ompa
t
quantum groups. Detailed treatments, and proofs of the results stated,

an be found in [Wor98℄, [MVD98℄ and [KT99℄.
4 DAVID KYED
A 
ompa
t quantum group G is a pair (A,∆) where A is a unital C∗-
algebra and ∆: A −→ A ⊗ A is a unital ∗-homomorphism from A to
the minimal tensor produ
t of A with itself satisfying:
(id⊗∆)∆ = (∆⊗ id)∆ (
oasso
iativity)
∆(A)(1⊗ A) = ∆(A)(A⊗ 1) = A⊗A (non-degenera
y)
For su
h a 
ompa
t quantum group G = (A,∆), there exists a unique
state h : A → C, 
alled the Haar state, whi
h is invariant in the sense
(h⊗ id)∆(a) = (id⊗h)∆(a) = h(a)1,
for all a ∈ A. Let H be a Hilbert spa
e and let u ∈ M(K(H)⊗ A) be
an invertible multiplier. Then u is 
alled a 
orepresentation if
(id⊗∆)u = u(12)u(13),
where we use the standard leg numbering 
onvention; for instan
e
u(12) = u⊗1. Intertwiners, dire
t sums and equivalen
es between 
orep-
resentations as well as irredu
ibility are de�ned in a straight forward
manner. See e.g. [MVD98℄ for details. We shall denote by Mor(u, v)
the set of intertwiners from u to v. It is a fa
t that ea
h irredu
ible

orepresentation is �nite dimensional and equivalent to a unitary 
o-
representation. Moreover, every unitary 
orepresentation is unitarily
equivalent to a dire
t sum of irredu
ible 
orepresentations. For two
�nite dimensional unitary 
orepresentations u, v their tensor produ
t is
de�ned as
u T©v = u(13)v(23).
This is again a unitary 
orepresentation of G. The algebra A0 gener-
ated by all matrix 
oe�
ients arising from irredu
ible 
orepresentations
be
omes a Hopf ∗-algebra (with the restri
ted 
omultipli
ation) whi
h
is dense in A. We denote its antipode by S and its 
ounit by ε. We also
re
all that the restri
tion of the Haar state to the ∗-algebra A0 is always
faithful. The quantum group G is 
alled a 
ompa
t matrix quantum
group if there exists a fundamental unitary 
orepresentation; i.e. a �nite
dimensional, unitary 
orepresentation whose matrix 
oe�
ients gener-
ate A0 as a ∗-algebra. Ea
h �nite dimensional, unitary 
orepresenta-
tion u de�nes a 
ontragredient 
orepresentation uc on the dual Hilbert
spa
e; if u ∈ B(H) ⊙ A0 for some �nite dimensional Hilbert spa
e H
then uc ∈ B(H ′)⊙A0 is given by uc = (( · )′⊗S)u, where for T ∈ B(H)
the operator T ′ ∈ B(H ′) is the natural dual (T ′(y′))(x) = y′(Tx). In
general uc is not a unitary, but it is a 
orepresentation; i.e. it is invert-
ible and satis�es (id⊗∆)uc = uc
and is therefore equivalent to a
unitary 
orepresentation. By 
hoosing an orthonormal basis e1, . . . , en
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 5
for H we get an identi�
ation of B(H) ⊙ A0 with Mn(A0). If, under
this identi�
ation, u be
omes the matrix (uij) then u
is identi�ed with
the matrix ū = (u∗ij), where we identify B(H
′)⊙A0 with Mn(A0) using
the dual basis e′1, . . . , e
n. From this it follows that u
is equivalent to
u. Note also that one has (u⊕ v)c = uc ⊕ vc and (u T©v)c = vc T©uc for
unitary 
orepresentations u and v (see e.g. [Wor87℄). If u ∈ B(H)⊙A0
is a �nite dimensional 
orepresentation its 
hara
ter is de�ned as
χ(u) = (Tr⊗ id)u ∈ A0,
where Tr is the unnormalized tra
e on B(H). The 
hara
ter map has
the following properties.
Proposition 1.1 ([Wor87℄). If u and v are �nite dimensional, unitary

orepresentations then
χ(u T©v) = χ(u)χ(v), χ(u⊕v) = χ(u)+χ(v) and χ(uc) = χ(u)∗.
Moreover, if u and v are equivalent then χ(u) = χ(v).
We end this se
tion with the two basi
 examples of 
ompa
t quantum
groups arising from a
tual groups.
Example 1.2. If G is a 
ompa
t, Hausdor� topologi
al group then the
Gelfand dual C(G) be
omes a 
ompa
t quantum group with 
omulti-
pli
ation ∆c : C(G) −→ C(G)⊗ C(G) = C(G×G) given by
∆c(f)(s, t) = f(st).
The Haar state is in this 
ase given by integration against the Haar
probability measure on G, and the �nite dimensional unitary 
orepre-
sentations of C(G) are exa
tly the �nite dimensional unitary represen-
tations of G.
Example 1.3. If Γ is a dis
rete, 
ountable group then the redu
ed
group C∗-algebra C∗red(Γ) be
omes a 
ompa
t quantum group when
endowed with 
omultipli
ation given by
∆red(λγ) = λγ ⊗ λγ .
Here λ denotes the left regular representation of Γ. In this 
ase, the
Haar state is just the natural tra
e on C∗red(Γ), and a 
omplete family of
irredu
ible, unitary 
orepresentations is given by the set {λγ | γ ∈ Γ}.
Remark 1.4. All 
ompa
t quantum groups to be 
onsidered in the
following are assumed to have a separable underlying C∗-algebra. The
quantum Peter-Weyl theorem [KT99, 3.2.3℄ then implies that the GNS
spa
e arising from the Haar state is separable and, in parti
ular, that
there are at most 
ountable many (pairwise inequivalent) irredu
ible

orepresentations.
6 DAVID KYED
2. Fusion Algebras
In this se
tion we introdu
e the notion of fusion algebras and amen-
ability of su
h obje
ts. This topi
 was treated by Hiai and Izumi in
[HI98℄ and we will follow this referen
e 
losely throughout this se
-
tion. Other referen
es on the subje
t are [Yam99℄, [HY00℄ and [Sun92℄.
Throughout the se
tion, N0 will denote the non-negative integers.
De�nition 2.1 ([HI98℄). Let R be a unital ring and assume that R is
free as Z-module with basis I. Then R is 
alled a fusion algebra if the
unit e is an element of I and the following holds:
(i) The abelian monoid N0[I] is stable under multipli
ation. That
is, for all ξ, η ∈ I the unique family (Nαξ,η)α∈I of integers satis-
fying
Nαξ,ηα,

onsists of non-negative numbers.
(ii) The ring R has a Z-linear, anti-multipli
ative involution x 7→ x̄
preserving the basis I globally.
(iii) Frobenius re
ipro
ity holds, i.e. for ξ, η, α ∈ I we have
Nαξ,η = N
α,η̄.
(iv) There exists a Z-linear multipli
ative fun
tion d : R → [1,∞[
su
h that d(ξ) = d(ξ̄) for all ξ ∈ I. This fun
tion is 
alled the
dimension fun
tion.
Note that the distinguished basis, involution and dimension fun
tion
are all in
luded in the data de�ning a fusion algebra. Ea
h fusion
algebra 
omes with a natural tra
e τ given by
τ7−→ ke.
We shall use this tra
e later to de�ne a C∗-envelope of a fusion algebra.
Note also that the multipli
ativity of d implies
d(ξ)d(η)
Nαξ,η,
for all ξ, η ∈ I. For an element r =
α∈I kαα ∈ R, the set {α ∈ I |
kα 6= 0} is 
alled the support of r and denoted supp(r). We shall also

onsider the 
omplexi�ed fusion algebra C⊗ZZ[I] whi
h will be denoted
C[I] in the following. Note that this be
omes a 
omplex ∗-algebra with
the indu
ed algebrai
 stru
tures.
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 7
Example 2.2. For any dis
rete group Γ the integral group ring Z[Γ] be-

omes a fusion algebra when endowed with (the Z-linear extension of)
inversion as involution and trivial dimension fun
tion given by d(γ) = 1
for all γ ∈ Γ.
The irredu
ible representations of a 
ompa
t group 
onstitute the
basis in a fusion algebra where the tensor produ
t of representations is
the produ
t. We shall not go into details with this 
onstru
tion sin
e
it will be 
ontained in the following more general example.
Example 2.3. If G = (A,∆) is a 
ompa
t quantum group its irre-
du
ible 
orepresentations 
onstitute the basis of a fusion algebra with
tensor produ
t as multipli
ation. Sin
e this example will play a promi-
nent role later, we shall now elaborate on the 
onstru
tion. Denote
by Irred(G) = (uα)α∈I a 
omplete family of representatives for the
equivalen
e 
lasses of irredu
ible, unitary 
orepresentations of G. As
explained in Se
tion 1, for all uα, uβ ∈ Irred(G) there exists a �nite
subset I0 ⊆ I and a family (Nγα,β)γ∈I0 of positive integers su
h that
uα T©uβ is equivalent to
uγ ⊕ · · · ⊕ uγ
︸ ︷︷ ︸
times
Thus, a produ
t 
an be de�ned on the free Z-module Z[Irred(G)] by
setting
uα · uβ =
and the trivial 
orepresentation e = 1A ∈ Irred(G) is a unit for this
produ
t. If we denote by uᾱ ∈ Irred(G) the unique representative
equivalent to (uα)c, then the map uα 7→ uᾱ extends to a 
onjugation on
the ring Z[Irred(G)] and sin
e ea
h uα is an element ofMnα(A) for some
nα ∈ N we 
an also de�ne a dimension fun
tion d : Z[Irred(G)] → [1,∞[
by d(uα) = nα. When endowed with this multipli
ation, 
onjugation
and dimension fun
tion Z[Irred(G)] be
omes a fusion algebra. The only
thing that is not 
lear at this moment is that Frobenius re
ipro
ity
holds. To see this, we �rst note that for any α ∈ I and any �nite
dimensional 
orepresentation v we have (by S
hur's Lemma [MVD98,
6.6℄) that uα o

urs exa
tly
dimC Mor(u
α, v)
8 DAVID KYED
times in the de
omposition of v. Moreover, we have for any two unitary

orepresentations v and w that
dimC Mor(v, w) = dimC((Vw ⊗ V ′v)w T#v
dimCMor(v
cc, w) = dimC((V
v ⊗ Vw)v
Here the right hand side denotes the linear dimension of the spa
e of in-
variant ve
tors under the relevant 
oa
tion. These formulas are proved
in [Wor87, 3.4℄ for 
ompa
t matrix quantum groups, but the same proof

arries over to the 
ase where the 
ompa
t quantum group in question
does not ne
essarily possess a fundamental 
orepresentation. Using the
�rst formula, we get for α, β, γ ∈ I that
α,β = dimC Mor(u
γ, uα T©uβ)
= dimC(Vα ⊗ Vβ ⊗ V ′γ)u
T#uβ T#(uγ)c
= dimC(Vγ ⊗ V ′β ⊗ V ′α)u
T#(uβ)c T#(uα)c
= dimC Mor(u
α, uγ T©(uβ)c)
The remaining identity in Frobenius re
ipro
ity follows similarly us-
ing the se
ond formula. The fusion algebra Z[Irred(G)] is 
alled the

orepresentation ring (or fusion ring) of G and is denoted R(G).
Re
all that the 
hara
ter of a 
orepresentation u ∈ Mn(A) is de�ned
as χ(u) =
i=1 uii. It follows from Proposition 1.1 that the Z-linear
extension
χ : Z[Irred(G)] −→ A0
is an inje
tive homomorphism of ∗-rings. I.e. χ is additive and mul-
tipli
ative with χ(uᾱ) = (χ(uα))∗. This gives a link between the two
∗-algebras R(G) and A0 whi
h will be of importan
e later.
Other interesting examples of fusion algebras arise from in
lusions
of II1-fa
tors. See [HI98℄ for details.
Remark 2.4. In the following we shall only 
onsider fusion algebras
with an at most 
ountable basis. This will therefore be assumed with-
out further noti
e throughout the paper. Sin
e we will primarily be
interested in 
orepresentation rings of 
ompa
t quantum groups, this
is not very restri
tive sin
e the standing separability assumption (Re-
mark 1.4) ensures that the 
orepresentation rings always have a 
ount-
able basis.
Consider again an abstra
t fusion algebra R = Z[I]. For ξ, η ∈ I we
de�ne the (weighted) 
onvolution of the 
orresponding Dira
 measures,
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 9
δξ and δη, as
δξ ∗ δη =
d(ξ)d(η)
Nαξ,ηδα ∈ ℓ1(I).
This extends linearly and 
ontinuously to a submultipli
ative produ
t
on ℓ1(I). For f ∈ ℓ∞(I) and ξ ∈ I we de�ne λξ(f), ρξ(f) : I → C by
λξ(f)(η) =
f(α)(δξ̄ ∗ δη)(α)
ρξ(f)(η) =
f(α)(δη ∗ δξ)(α)
Denote by σ the 
ounting measure on I s
aled with d2; that is σ(ξ) =
d(ξ)2. Combining Proposition 1.3, Remark 1.4 and Theorem 1.5 in
[HI98℄ we get
Proposition 2.5 ([HI98℄). For ea
h f ∈ ℓ∞(I) we have λξ(f), ρξ(f) ∈
ℓ∞(I) and for ea
h p ∈ N ∪ {∞} the maps λξ, ρξ : ℓ∞(I) → ℓ∞(I) re-
stri
t to bounded operators on ℓp(I, σ) denoted λp,ξ and ρp,ξ respe
tively.
By linear extension, we therefore obtain a map λp,− : Z[I] → B(ℓp(I, σ))
and this map respe
ts the weighted 
onvolution produ
t. Moreover, for
p = 2 the operator U : ℓ2(I) → ℓ2(I, σ) given by U(δη) = 1d(η)δη is
unitary and intertwines λ2,ξ with the operator
lξ : δη 7−→
Nαξ,ηδα.
Remark 2.6. Under the natural identi�
ation of ℓ2(I) with the GNS
spa
e L2(C[I], τ), we see that πτ (ξ) = d(ξ)lξ. In parti
ular the GNS
representation 
onsists of bounded operators. Here τ is the natural
tra
e de�ned just after De�nition 2.1.
3. Amenability for Fusion Algebras
The notion of amenability for fusion algebras was introdu
ed in
[HI98℄, but only in the slightly restri
ted setting of �nitely generated
fusion algebras; a fusion algebra R = Z[I] is 
alled �nitely generated if
there exists a �nitely supported probability measure µ on I su
h that
supp(µ∗n) and µ(ξ̄) = µ(ξ) for all ξ ∈ I.
That is, if the union of the supports of all powers of µ, with respe
t
to 
onvolution, is I and µ is invariant under the involution. The �rst

ondition is referred to as non-degenera
y of µ and the se
ond 
ondition
is referred to as symmetry of µ.
10 DAVID KYED
In [HI98℄, amenability is de�ned, for a �nitely generated fusion al-
gebra, by requiring that ‖λp,µ‖ = 1 for some 1 < p < ∞ and some
�nitely supported, symmetri
, non-degenerate probability measure µ.
It is then proved that this is independent of the 
hoi
e of µ and p,
using the non-degenera
y property of the measure. If we 
onsider a

ompa
t quantum group G = (A,∆) it is not di�
ult to prove that its

orepresentation ring R(G) is �nitely generated exa
tly when G is a

ompa
t matrix quantum group. Sin
e we are also interested in quan-
tum groups without a fundamental 
orepresentation we will 
hoose the
following de�nition of amenability.
De�nition 3.1. A fusion algebra R = Z[I] is 
alled amenable if 1 ∈
σ(λ2,µ) for every �nitely supported, symmetri
 probability measure µ
on I.
Here σ(λ2,µ) denotes the spe
trum of the operator λ2,µ. From Propo-
sition 1.3 and Corollary 4.4 in [HI98℄ it follows that our de�nition agrees
with the one in [HI98℄ on the 
lass of �nitely generated fusion algebras.
The relation between amenability for fusion algebras and the 
lassi
al
notion of amenability for groups will be explained later. See e.g. Re-
mark 3.8 and Corollary 4.7.
De�nition 3.2. Let R = Z[I] be a fusion algebra. For two �nite
subsets S, F ⊆ I we de�ne the boundary of F relative to S as the set
∂S(F ) = {α ∈ F | ∃ ξ ∈ S : supp(αξ) * F}
∪ {α ∈ F c | ∃ ξ ∈ S : supp(αξ) * F c}.
Here, and in what follows, F c denotes the set I \ F .
The modi�ed de�nition of amenability allows the following extension
of [HI98, 4.6℄ from where we also adopt some notation.
Theorem 3.3. Let R = Z[I] be a fusion algebra with dimension fun
-
tion d. Then the following are equivalent:
(A) The fusion algebra is amenable.
(FC1) For every �nitely supported, symmetri
 probability measure µ
on I with e ∈ supp(µ) and every ε > 0 there exists a �nite
subset F ⊆ I su
h that
ξ∈supp(χF ∗µ)
d(ξ)2 < (1 + ε)
d(ξ)2.
(FC2) For every �nite, non-empty subset S ⊆ I and every ε > 0 there
exists a �nite subset F ⊆ I su
h that
∀ ξ ∈ S : ‖ρ1,ξ(χF )− χF‖1,σ < ε‖χF‖1,σ,
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 11
where ρ1,ξ ∈ B(ℓ1(I, σ)) is the operator from Proposition 2.5.
(FC3) For every �nite, non-empty subset S ⊆ I and every ε > 0 there
exists a �nite subset F ⊆ I su
h that
ξ∈∂S(F )
d(ξ)2 < ε
d(ξ)2.
The 
ondition (FC3) was not present in [HI98℄. It is to be 
onsidered
as a fusion algebra analogue of the Følner 
ondition for groups as it is
presented in [BP92, F.6℄. The strategy for the proof of Theorem 3.3 is
to prove the following impli
ations:
(A) ⇔ (FC2) ⇒ (FC3) ⇒ (FC1) ⇒ (FC2).
The proof of the impli
ations (A) ⇔ (FC2) and (FC1) ⇒ (FC2) are
small modi�
ations of the 
orresponding proof in [HI98℄. We �rst set
out to prove the 
ir
le of impli
ations
(FC2) ⇒ (FC3) ⇒ (FC1) ⇒ (FC2).
For the proof we will need the following simple lemma.
Lemma 3.4. If Nαξ,η > 0 for some ξ, η, α ∈ I then d(α)d(η) ≥ d(ξ).
Proof. By Frobenius re
ipro
ity, we have Nαξ,η = N
α,η̄ > 0 and hen
e
d(α)d(η) = d(α)d(η̄) =
α,η̄d(γ) ≥ N ξα,η̄d(ξ) ≥ d(ξ).
Proof of (FC2) ⇒ (FC3). We �rst note that (FC2), by the triangle in-
equality, implies the following 
ondition:
For every �nite, non-empty set S ⊆ I and every ε > 0 there exists a
�nite set F ⊆ I su
h that
‖ρ1,χS(χF )− |S|χF‖1,σ < ε‖χF‖1,σ. (†)
Here |S| denotes the 
ardinality of S. Let S and ε > 0 be given
and 
hoose F su
h that (†) is satis�ed. De�ne a map ϕ : I → R by
12 DAVID KYED
ϕ(ξ) = ρ1,χS(χF )(ξ)− |S|χF (ξ). We note that
ϕ(ξ) =
χF (α)(δξ ∗ χS)(α)
− |S|χF (ξ)
(δξ ∗ δη)(α)
− |S|χF (ξ)
d(ξ)d(η)
Nαξ,η − |S|χF (ξ).
We now divide into four 
ases.
(i) If ξ ∈ F ∩∂S(F )c then supp(ξη) ⊆ F for all η ∈ S and hen
e we
get the relation
d(ξ)d(η)
Nαξ,η = 1. This implies ϕ(ξ) = 0.
(ii) If ξ ∈ F c ∩ ∂S(F )c we see that Nαξ,η = 0 for all α ∈ F and all
η ∈ S and hen
e ϕ(ξ) = 0.
(iii) If ξ ∈ F c ∩ ∂S(F ) we have χF (ξ) = 0 and there exist α0 ∈ F
and η0 ∈ S su
h that Nα0ξ,η0 6= 0. Using Lemma 3.4, we now get
ϕ(ξ) ≥ d(α0)
d(ξ)d(η0)
Nα0ξ,η0 ≥
d(η0)2
Nα0ξ,η0 ≥
d(η0)2
where M = max{d(η)2 | η ∈ S}.
(iv) If ξ ∈ F ∩ ∂S(F ) we have
ϕ(ξ) =
d(ξ)d(η)
Nαξ,η − |S|
= (−1)
d(ξ)d(η)
Nαξ,η
= (−1)
d(ξ)d(η)
Nαξ,η,
and be
ause ξ ∈ ∂S(F )∩ F there exist η0 ∈ S and α0 /∈ F su
h
that Nα0ξ,η0 6= 0. Using Lemma 3.4 again we 
on
lude, as in (iii),
that |ϕ(ξ)| ≥ 1
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 13
We now get
d(ξ)2 = ε‖χF‖1,σ
> ‖ρ1,χS(χF )− |S|χF‖1,σ (by (†))
|ϕ(ξ)|d(ξ)2
ξ∈∂S(F )
|ϕ(ξ)|d(ξ)2 (by (i) and (ii))
ξ∈∂S(F )
d(ξ)2, (by (iii) and (iv))
and sin
e ε was arbitrary the 
laim follows. �
Proof of (FC3) ⇒ (FC1). Given a �nitely supported, symmetri
 pro-
bability measure µ, with µ(e) > 0, and ε > 0 we put S = supp(µ) and

hoose F ⊆ I su
h that (FC3) is ful�lled with respe
t to ε. We have
(χF ∗ µ)(ξ) =
α∈F,β∈S
d(α)d(β)
α,β ,
(χF ∗ µ)(ξ) = 0 ⇔ ∀α ∈ F ∀β ∈ S : N ξα,β = 0
⇔ ∀α ∈ F ∀β ∈ S : Nα
= 0 (Frobenius)
⇔ ∀α ∈ F ∀β ∈ S : Nαξ,β = 0 (S symmetri
)
⇔ ξ ∈ F c ∩ ∂S(F )c. (e ∈ S)
Hen
e supp(χF ∗ µ) = (F c ∩ ∂S(F )c)c = F ∪ ∂S(F ) and we get
ξ∈supp(χF ∗µ)
d(ξ)2 −
d(ξ)2 =
ξ∈F∪∂S(F )
d(ξ)2 −
d(ξ)2
ξ∈∂S(F )∩F
d(ξ)2
ξ∈∂S(F )
d(ξ)2
d(ξ)2. (by (FC3))
Proof of (FC1) ⇒ (FC2). Given ε > 0 and S ⊆ I we de�ne S̃ = S ∪
S̄ ∪{e} and µ = 1
χS̃. Choose F ⊆ I su
h that µ and F satisfy (FC1)
14 DAVID KYED
with respe
t to
. We aim to prove that (FC2) is satis�ed for all ξ ∈ S̃.
For arbitrary ξ ∈ I we have
‖ρ1,ξ(χF )− χF‖1,σ =
|ρ1,ξ(χF )(α)− χF (α)|d(α)2
d(α)d(ξ)
α,ξ)− χF (α)|d(α)2
d(α)d(ξ)
d(α)2
d(α)d(ξ)
d(α)2
d(η)d(α)
α,ξ +
d(η)d(α)
d(η)d(α)
α,ξ +N
d(η)d(α)
+Nαη,ξ). (†)
For ξ ∈ supp(µ) = S̃ and α /∈ F , it is easy to 
he
k that (χF ∗µ)(α) > 0
if there exists an η ∈ F su
h that Nα
+Nαη,ξ > 0. Hen
e the 
al
ulation
(†) implies that
‖ρ1,ξ(χF )− χF‖1,σ ≤
α∈supp(χF ∗µ)\F
d(η)d(α)
+Nαη,ξ)
α∈supp(χF ∗µ)\F
d(η)d(α)
+Nαη,ξ)
α∈supp(χF ∗µ)\F
d(α)2
α∈supp(χF ∗µ)
d(α)2 −
d(α)2
< ε‖χF‖1,σ,
where the last estimate follows from (FC1). Note that the 
ondition e ∈
supp(µ) was used to get the fourth step in the 
al
ulation above. �
We now set out to prove the remaining equivalen
e in Theorem 3.3.
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 15
Proof of (A) ⇔ (FC2). At the end of this se
tion four formulas are
gathered; these will be used during the proof and referred to as (F1) -
(F4). For the a
tual proof we also need the following de�nitions. Con-
sider a �nitely supported, symmetri
 probability measure µ on I and
de�ne pµ : I × I → R by
pµ(ξ, η) = (δξ ∗ µ)(η) =
d(ξ)d(ω)
Note that the fun
tion pµ satis�es the reversibility 
ondition:
σ(ξ)pµ(ξ, η) = σ(η)pµ(η, ξ).
For a �nitely supported fun
tion f ∈ c0(I) and r ∈ N we also de�ne
‖f‖Dµ(r) =
σ(ξ)pµ(ξ, η)|f(ξ)− f(η)|r
Although this is referred to as the generalized Diri
hlet r-norm of f ,
one should keep in mind that the fun
tion ‖·‖Dµ(r) is only a semi norm.
We shall now 
onsider the following 
ondition:
For all �nitely supported, symmetri
, probability measures µ we have
{‖f‖Dµ(r)
‖f‖r,σ
| f ∈ c0(I) \ {0}
= 0. (NWr)
The reason for the name (NWr), whi
h appeared in [HI98℄, is that
the 
ondition is the negation of a so-
alled Wirtinger inequality. See
[HI98℄ for more details. To prove (A) ⇔ (FC2) we will a
tually prove
the following equivalen
es
(FC2) ⇔ (NW1) and ∀r : (NW1) ⇔ (NWr) and (A) ⇔ (NW2).
For the latter of these equivalen
es the following lemma will be useful.
Lemma 3.5. For all f ∈ c0(I) we have
‖f‖2Dµ(2) = 〈f |f〉2,σ − 〈ρ2,µ(f)|f〉2,σ,
where 〈·|·〉2,σ denotes the inner produ
t on ℓ2(I, σ).
Proof. This is proven by a dire
t 
al
ulation using the reversibility

ondition and the formula (F4) from the end of this se
tion. �
Proof of (A)⇔ (NW2). Let µ be a �nitely supported, symmetri
 prob-
ability measure on I. By [HI98, 1.3,1.5℄, we have that ρ2,µ is self-adjoint
16 DAVID KYED
and ‖ρ2,µ‖ ≤ ‖µ‖1 = 1 so that 1− ρ2,µ ≥ 0. We now get
1 ∈ σ(λ2,µ) ⇔ 1 ∈ σ(ρ2,µ) ([HI98, 1.5℄)
⇔ 0 ∈ σ(1− ρ2,µ)
⇔ 0 ∈ σ(
1− ρ2,µ)
⇔ ∃xn ∈ (ℓ2(I, σ))1 : ‖(
1− ρ2,µ)xn‖2,σ −→ 0
⇔ ∃fn ∈ (c0(I))1 : ‖(
1− ρ2,µ)fn‖2,σ −→ 0
⇔ ∃fn ∈ (c0(I))1 : 〈(1− ρ2,µ)fn |fn〉2,σ −→ 0
⇔ ∃fn ∈ (c0(I))1 : ‖fn‖Dµ(2) −→ 0 (Lem. 3.5)
⇔ inf
{‖f‖Dµ(2)
‖f‖2,σ
| f ∈ c0(I) \ {0}
Hen
e (A) ⇔ (NW2) as desired. �
Proof of (NW1) ⇒ (FC2). Given ε > 0 and ξ1, . . . , ξn ∈ I, we 
hoose a
�nitely supported, symmetri
 probability measure µ with ξ1, . . . , ξn ∈
supp(µ). De�ne
min{µ(ξ) | ξ ∈ I},
and 
hoose, a

ording to (NW1), an f ∈ c0(I) su
h that
‖f‖Dµ(1) < ε′‖f‖1,σ. (∗)
Sin
e ‖|f |‖Dµ(1) ≤ ‖f‖Dµ(1) and ‖|f |‖1,σ = ‖f‖1,σ we may assume that f
is positive. Sin
e f 
an be approximated by a rational fun
tion we may
a
tually assume that f has integer values. Put N = max{f(ξ) | ξ ∈ I}
and de�ne, for k = 1, . . . , N , Fk = {ξ | f(ξ) ≥ k}. Then f =
k=1 χFk
and the following formulas hold.
‖f‖Dµ(1) =
‖χFk‖Dµ(1) and ‖f‖1,σ =
‖χFk‖1,σ.
The �rst formula is proved by indu
tion on the integer N and the
se
ond follows from a dire
t 
al
ulation using only the reversibility
property of pµ. Be
ause of (∗), there must therefore exist some j ∈
{1, . . . , N} su
h that
‖χFj‖Dµ(1) < ε′‖χFj‖1,σ. (∗∗)
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 17
For the sake of simpli
ity we denote this Fj by F in the following. We
now get
‖χF‖Dµ(1) =
σ(ξ)pµ(ξ, η)|χF (ξ)− χF (η)|
ξ∈F,η/∈F
σ(ξ)pµ(ξ, η) (reversibility)
ξ∈F,η/∈F
d(ξ)d(ω)
ξ∈F,η/∈F
d(ξ)d(η)
ξ∈F,η/∈F
d(ξ)d(η)
ξ,ω +N
ξ,ω̄)
µ(ω)‖ρ1,ω(χF )− χF‖1,σ. (‡)
Here the last equality follows from the 
omputation (†) in the proof of
(FC1) ⇒ (FC2). The inequality (∗∗) therefore reads
µ(ω)‖ρ1,ω(χF )− χF‖1,σ < ε′‖χF‖1,σ.
For every ω ∈ I we therefore 
on
lude, sin
e ε′ = ε
min(µ), that
µ(ω)‖ρ1,ω(χF )− χF‖1,σ < min(µ)ε‖χF‖1,σ.
Sin
e ea
h of the given ξi's are in supp(µ) we get for all i that
‖ρ1,ξi(χF )− χF‖1,σ < ε‖χF‖1,σ,
as desired. �
Proof of (FC2) ⇒ (NW1). Assume now (FC2) and let µ and ε be given.
Choose F su
h that
‖ρ1,ξ(χF )− χF‖1,σ < ε‖χF‖1,σ
18 DAVID KYED
for all ξ ∈ supp(µ). Using the 
al
ulation (‡), from the proof of opposite
impli
ation, we get
‖χF‖Dµ(1) =
µ(ω)‖ρ1,ω(χF )− χF‖1,σ
µ(ω)ε‖χF‖1,σ
‖χF‖1,σ
< ε‖χF‖1,σ.
For the proof of the statement (NW1) ⇔ (NWr) we will need the
following lemma.
Lemma 3.6 ([Ger88℄). For r ≥ 2 and f ∈ c0(I)+ we have
‖f r‖Dµ(1) ≤ 2r‖f‖r−1r,σ ‖f‖Dµ(r).
Proof. First note that
‖f r‖Dµ(1) =
σ(ξ)pµ(ξ, η)|f(ξ)r − f(η)r|
σ(ξ)pµ(ξ, η)(f(ξ)
r−1 + f(η)r−1)|f(ξ)− f(η)|,
where the inequality follows from (F1). De�ne a measure ν on I × I
by ν(ξ, η) = 1
σ(ξ)pµ(ξ, η) and 
onsider the fun
tions ϕ, ψ : I × I → R
given by
ϕ(ξ, η) = f(ξ)r−1 + f(η)r−1 and ψ(ξ, η) = |f(ξ)− f(η)|.
De�ne s > 1 by the equation 1
= 1. Then the inequality above

an be written as ‖f r‖Dµ(1) ≤ r‖ϕψ‖1,ν and using Hölder's inequality
we therefore get
‖f r‖Dµ(1) ≤ r‖ϕψ‖1,ν
≤ r‖ϕ‖s,ν‖ψ‖r,ν
σ(ξ)pµ(ξ, η)(f(ξ)
r−1 + f(η)r−1)s
σ(ξ)pµ(ξ, η)|f(ξ)− f(η)|r
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 19
σ(ξ)pµ(ξ, η)(f(ξ)
(r−1)s + f(η)(r−1)s)
s‖f‖Dµ(r)
σ(ξ)pµ(ξ, η)f(ξ)
(r−1)s
s‖f‖Dµ(r)
pµ(ξ, η)
f(ξ)(r−1)s
s‖f‖Dµ(r)
σ(ξ)f(ξ)(r−1)s
s‖f‖Dµ(r)
σ(ξ)f(ξ)r
] r−1
r ‖f‖Dµ(r)
= 2r‖f‖r−1r,σ ‖f‖Dµ(r).
Also the following observation will be useful.
Observation 3.7. Under the assumptions of Lemma 3.6 we have
‖f‖Dµ(r) =
σ(ξ)pµ(ξ, η)|f(ξ)− f(η)|r
σ(ξ)pµ(ξ, η)|f(ξ)r − f(η)r|
(by (F3))
= ‖f r‖
Dµ(1)
Having these results, we are now able to prove (NW1) ⇔ (NWr).
Proof of (NW1) ⇒ (NWr). Assume (NW1) and let µ and ε > 0 be
given. Put ε′ = εr and 
hoose non-zero f ∈ c0(I)+ su
h that
‖f‖Dµ(1)
‖f‖1,σ
< ε′.
Using Observation 3.7 we get
f‖Dµ(r)
f‖r,σ
Dµ(1)
< (ε′)
r = ε.
by (F2)
by reversibility
20 DAVID KYED
Proof of (NWr) ⇒ (NW1). Given µ and ε > 0 and put ε′ = 12rε. Then

hoose non-zero f ∈ c0(I)+ with
‖f‖Dµ(r)
‖f‖r,σ
< ε′.
Using Lemma 3.6, we get
‖f r‖Dµ(1)
‖f r‖1,σ
2r‖f‖r−1r,σ ‖f‖Dµ(r)
‖f‖rr,σ
< 2rε′ = ε.
Gathering all the results just proven we get (A) ⇔ (FC2). �
This 
on
ludes the proof of Theorem 3.3.
Remark 3.8. Consider a 
ountable, dis
rete group Γ and the 
orre-
sponding fusion algebra Z[Γ]. It is not di�
ult to prove that Z[Γ]
satis�es (FC3) from Theorem 3.3 if and only if Γ satis�es Følner's

ondition (for groups) as presented in [BP92, F.6℄. Sin
e a group is
amenable if and only if it satis�es Følner's 
ondition, we see from this
that Γ is amenable if and only if the 
orresponding fusion algebra Z[Γ]
is amenable.
3.1. Formulas used in the proof of Theorem 3.3. We 
olle
t here
four formulas used in the proof of Theorem 3.3. Let r, s > 1 and assume
= 1. Then for all z, w ∈ C, a, b ≥ 0 and n ∈ N we have
|ar − br| ≤ r(ar−1 + br−1)|a− b| (F1)
(a+ b)r ≤ 2r−1(ar + br) (F2)
|a− b|n ≤ |an − bn| (F3)
|z − w|2 + |w − z|2 = 2(|z|2 − zw̄) + 2(|w|2 − wz̄) (F4)
Proof. The inequality (F1) 
an be proved using the mean value theorem
on the fun
tion f(x) = xr and the interval between a and b. To prove
(F2), 
onsider a two-point set endowed with 
ounting measure. Using
Hölder's inequality, we then get
a + b = 1 · a + 1 · b ≤ (1s + 1s) 1s (ar + br) 1r .
From this the desired inequality follows using the fa
t that
= r−1
The inequality (F3) follows using the binomial theorem. If, for instan
e,
a = b+ k for some k ≥ 0 we have
(a− b)n = kn ≤ (b+ k)n − bn = an − bn.
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 21
The formula (F4) follows by splitting w and z into real and imaginary
parts and 
al
ulating both sides of the equation. �
4. Coamenable Compa
t Quantum Groups
In this se
tion we introdu
e the notion of 
oamenability for 
ompa
t
quantum groups and dis
uss the relationship between 
oamenability of
a 
ompa
t quantum group and amenability of its 
orepresentation ring.
The notion of (
o-)amenability has been treated in di�erent quantum
group settings by numerous people. A number of referen
es for this
subje
t are [BMT01℄, [Voi79℄, [Rua96℄, [Ban99a℄, [Ban99b℄, [ES92℄ and
[BS93℄. For our purposes, the approa
h of Bédos, Murphy and Tuset in
[BMT01℄ is the most natural and we are therefore going to follow this
referen
e throughout this se
tion. We will assume that the reader is
familiar with the basi
s on Woronowi
z's theory of 
ompa
t quantum
groups. De�nitions, notation and some basi
 properties 
an be found
in Se
tion 1 and detailed treatments 
an be found in [Wor98℄, [MVD98℄
and [KT99℄.
De�nition 4.1 ([BMT01℄). Let G = (A,∆) be a 
ompa
t quantum
group and let Ared be the image of A under the GNS representation πh
arising from the Haar state h. Then G is said to be 
oamenable if the

ounit ε : A0 → C extends 
ontinuously to Ared.
Remark 4.2. It is well known that a dis
rete group Γ is amenable
if and only if the trivial representation of C∗full(Γ) fa
torizes through
C∗red(Γ). This amounts to saying that (C
red(Γ),∆red) is 
oamenable if
and only if Γ is amenable. Note also that the abelian 
ompa
t quan-
tum groups (C(G),∆c) are automati
ally 
oamenable sin
e the 
ounit
is given by evaluation at the identity and therefore already globally
de�ned and bounded.
In the following theorem we 
olle
t some fa
ts on 
oamenable 
om-
pa
t quantum groups. For more 
oamenability 
riteria and a proof of
the theorem below we refer to [BMT01℄.
Theorem 4.3 ([BMT01℄). For a 
ompa
t quantum group G = (A,∆)
the following are equivalent.
(i) G is 
oamenable.
(ii) The Haar state h is faithful and the 
ounit is bounded with re-
spe
t to the norm on A.
(iii) The natural map from the universal representation A
to the
redu
ed representation Ared is an isomorphism.
22 DAVID KYED
If G is a 
ompa
t matrix quantum group with fundamental 
orepre-
sentation u ∈ Mn(A) the above 
onditions are also equivalent to the
following.
(iv) The number n is in σ(πh(Re(χ(u))) where χ(u) =
i=1 uii is
the 
hara
ter map from Se
tion 2.
Re
all that σ(T ) denotes the spe
trum of a given operator T . Thus,
when we are dealing with a 
oamenable quantum group the Haar state
is automati
ally faithful and hen
e the 
orresponding GNS representa-
tion πh is faithful. We therefore 
an, and will, identify A and Ared. The

ondition (iv) is Skandalis's quantum analogue of the so-
alled Kesten

ondition for groups (see [Kes59℄ and [Ban99a℄) whi
h is proved by
Bani
a in [Ban99b℄. The next result is a generalization of the Kesten

ondition to the 
ase where a fundamental 
orepresentation is not (ne-

essarily) present. The proof draws inspiration from the 
orresponding
proof in [BMT01℄.
Theorem 4.4. Let G = (A,∆) be a 
ompa
t quantum group. Then
the following are equivalent:
(i) G is 
oamenable.
(ii) For any �nite dimensional, unitary 
orepresentation u ∈ Mnu(A)
we have nu ∈ σ(πh(Re(χ(u)))).
Proof. AssumeG to be 
oamenable and let a �nite dimensional, unitary

orepresentation u ∈ Mnu(A) be given. Sin
e the 
ounit extends to a

hara
ter ε : Ared → C and sin
e
ε(Re(χ(u))) = ε(
uii + u
) = nu,
we must have nu ∈ σ(πh(Re(χ(u)))). Assume 
onversely that the prop-
erty (ii) is satis�ed and de�ne, for a �nite dimensional, unitary 
ore-
presentation u, the set
C(u) = {ϕ ∈ S (Ared) | ϕ(πh(Re(χ(u)))) = nu}.
Here S (Ared) denotes the state spa
e of Ared. It is 
lear that ea
h C(u)
is 
losed in the weak
-topology and we now prove that the family
F = {C(u) | u �nite dimensional, unitary 
orepresentation}
has the �nite interse
tion property. We �rst prove that ea
h C(u) is
non-empty. For given u, we put xij = uij − δij and x =
ijxij .
Then x is 
learly positive and a dire
t 
al
ulation reveals that
x = 2(nu − Re(χ(u))). (†)
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 23
Hen
e, nu ∈ σ(πh(Re(χ(u)))) if and only if there exists [KR83, 4.4.4℄ a
ϕ ∈ S (Ared) with
ϕ(πh(Re(χ(u)))) = nu.
Thus, C(u) 6= ∅. Let now u(1), . . . , u(k) be given and put u = ⊕ki=1u(i).
We aim at proving that
C(u) ⊆
C(u(i)).
Let ϕ ∈ C(u) be given and note that
nu(k) = ϕ(πh(Re(χ(u)))) =
u(i)∑
ϕ(πh(u
jj ) + πh(u
jj )).
Sin
e the matrix u is unitary, we have ‖πh(ust)‖ ≤ 1 for all s, t ∈
{1, . . . , nu} and hen
e
ϕ(πh(u
jj ) + πh(u
jj )) ∈ [−1, 1].
This for
es
ϕ(πh(u
jj )+πh(u
jj )) = 1 and hen
e ϕ(πh(Re(χ(u
(i))))) =
nu(i). Thus ϕ is in ea
h of the sets C(u
(1)), . . . , C(u(k)) and we 
on
lude
that F has the �nite interse
tion property. By 
ompa
tness of S (Ared),
we may therefore �nd a state ϕ su
h that ϕ(πh(Re(χ(u)))) = nu for ev-
ery unitary 
orepresentation u. Denote by H the GNS spa
e asso
iated
with this ϕ, by ξ0 the natural 
y
li
 ve
tor and by π the 
orresponding
GNS representation of Ared. Consider an arbitrary unitary 
orepresen-
tation u and form as before the elements xij and x. Then the equation
(†) shows that ϕ(x∗ijxij) = 0 and hen
e π(xij)ξ0 = 0 and
π(uij)ξ0 = δijξ0.
From the Cau
hy-S
hwarz inequality we get
|ϕ(xij)|2 ≤ ϕ(x∗ijxij)ϕ(1) = 0,
and hen
e ϕ(uij) = δij . We therefore have that π(uij)ξ0 = ϕ(uij)ξ0.
Sin
e the matrix 
oe�
ients span A0 linearly we get π(a)ξ0 = ϕ(a)ξ0
for all a ∈ A0. By density of A0 in Ared it follows that π(a)ξ0 = ϕ(a)ξ0
for all a ∈ Ared. From this we see that
H = π(Ared)ξ0
= Cξ0,
and it follows that ϕ : Ared → C is a bounded ∗-homomorphism 
oin-

iding with ε on A0. Thus, G is 
oamenable.
24 DAVID KYED
The following result was mentioned, without proof, in [HI98, p.692℄
in the restri
ted setting of 
ompa
t matrix quantum groups whose Haar
state is a tra
e.
Theorem 4.5. A 
ompa
t quantum group G = (A,∆) is a 
oamenable
if and only if the 
orepresentation ring R(G) is amenable.
For the proof we will need the following lemma. For this, re
all from
Se
tion 2 that the ∗-algebra C[Irred(G)] 
omes with a tra
e τ given by
u∈Irred(G)
zuu 7−→ ze,
where e ∈ Irred(G) denotes the identity in R(G). In what follows, we
denote by C∗red(R(G)) the enveloping C
-algebra of C[Irred(G)] on the
GNS spa
e L2(C[Irred(G)], τ) arising from τ .
Lemma 4.6. The 
hara
ter map χ : R(G) → A0 extends to an isome-
tri
 ∗-homomorphism χ : C∗red(R(G)) → Ared.
Proof. Put I = Irred(G). For an irredu
ible, �nite dimensional, unitary

orepresentation u we have h(uij) = 0 unless u is the trivial 
orepre-
sentation and therefore the following diagram 
ommutes
// A0
Hen
e χ extends to an isometri
 embedding
K = L2(C[I], τ) −֒→ L2(A0, h) = H.
Denote by S the algebra χ(R(G)) and by S̄ the 
losure of πh(S) inside
Ared. Sin
e S is a ∗-algebra that maps K into itself it also maps K⊥
into itself and hen
e πh(χ(a)) takes the form
πh(χ(a))
0 πh(χ(a))
‖πh(χ(a))‖ = max{‖πh(χ(a))
‖, ‖πh(χ(a))
≥ ‖πh(χ(a))
= ‖πτ (a)‖.
This proves that the map κ : πh(S) → πτ (C[I]) given by κ(πh(χ(a))) =
πτ (a) is bounded and it therefore extends to a 
ontra
tion κ̄ : S̄ →
C∗red(R(G)). We now prove that κ̄ is inje
tive. Sin
e h is faithful on
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 25
Ared and τ is faithful on C
red(R(G)) we get the following 
ommutative
diagram
πh(S)
πτ (C[I])�
C∗red(R(G))�
L2(S̄, h) // L2(C∗red(R(G)), τ)
One easily 
he
ks that κ indu
es an isometry L2(S̄, h) → L2(C∗red(G), τ)
and it therefore follows that κ̄ is inje
tive and hen
e an isometry. Thus,
for χ(a) ∈ S we have
‖πh(χ(a))‖ = ‖κ̄(πh(χ(a)))‖ = ‖πτ (a)‖,
as desired.
Proof of Theorem 4.5. Assume �rst that G is 
oamenable and put I =
Irred(G). Consider a �nitely supported, symmetri
 probability measure
µ on I. We aim to show that 1 ∈ σ(λ2,µ), where λ2,µ is the operator on
ℓ2(I, σ) de�ned in Se
tion 2. Write µ as
ξ∈I tξδξ and re
all (Lemma
4.6) that the 
hara
ter map χ : C[I] → A0 extends to an inje
tive ∗-
homomorphism χ : C∗red(R(G)) → Ared. Using this, and Proposition
2.5, we get that
σ(λ2,µ) = σ(lµ)
tξlξ)
πτ (ξ))
= σ(χ(
πτ (ξ)))
πh(ξii)).
Sin
e G is 
oamenable, the 
ounit extends to a 
hara
ter ε : Ared → C
and we have
ξii)) =
nξ = 1.
26 DAVID KYED
Hen
e 1 ∈ σ
i=1 πh(ξii))
= σ(λ2,µ) and we 
on
lude that
R(G) is amenable.
Assume, 
onversely, that R(G) is amenable. We aim at proving
that G ful�lls the Kesten 
ondition from Theorem 4.4. Let therefore
u ∈ Mn(A) be an arbitrary, �nite dimensional, unitary 
orepresenta-
tion. Denote by (uα)α∈S ⊆ Irred(G) the irredu
ible 
orepresentations
o

urring in the de
omposition of u and by kα the multipli
ity of uα in
u. Now de�ne
µu(uα) =
if α ∈ S;
0 if α /∈ S.
Putting µ = 1
µū we obtain a �nitely supported, symmetri
probability measure and by assumption we have that 1 ∈ σ(λ2,µ). Using
again that the 
hara
ter map extends to an inje
tive ∗-homomorphism
χ : C∗red(R(G)) → Ared we obtain
σ(λ2,µ) = σ
λ2,uα +
λ2,uᾱ
luα +
(Prop. 2.5)
πτ (uα) +
πτ (uᾱ)
(Rem. 2.6)
πh(χ(uα)) +
πh(χ(uᾱ))
πh(χ(u)) +
πh(χ(ū))
πh(Re(χ(u)))
1 ∈ σ(λ2,µ) if and only if n ∈ σ(Re(πh(χ(u)))),
and the result now follows from Theorem 4.4. �
In parti
ular we (re-)obtain the following.
Corollary 4.7. A dis
rete group is amenable if and only if the group
ring, 
onsidered as a fusion algebra, is amenable.
Corollary 4.8 ([Ban99b℄). The quantum groups SUq(2) are 
oamen-
able.
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 27
Proof. By Theorem 4.5, SUq(2) is 
oamenable if and only if R(SUq(2))
is amenable. But, R(SUq(2)) = R(SU(2)) (see e.g. [Wor88℄) and sin
e
(C(SU(2)),∆c) is a 
oamenable quantum group R(SU(2)) is amenable.
As seen from Theorem 4.5, the answer to the question of whether
a 
ompa
t quantum group is 
oamenable or not 
an be determined
using only information about its 
orepresentations � a fa
t noted by
Bani
a in the setting of 
ompa
t matrix quantum groups in [Ban99a℄
and [Ban99b℄. With this in mind, we now propose the following Følner

ondition for quantum groups.
De�nition 4.9. A 
ompa
t quantum group G = (A,∆) is said to satis-
fy Følner's 
ondition if for any �nite, non-empty subset S ⊆ Irred(G)
and any ε > 0 there exists a �nite subset F ⊆ Irred(G) su
h that
u∈∂S(F )
n2u < ε
Here nu denotes the dimension of the irredu
ible 
orepresentation u
and ∂S(F ) is the boundary of F relative to S as in De�nition 3.2.
We immediately obtain the following.
Corollary 4.10. A 
ompa
t quantum group is 
oamenable if and only
if it satis�es Følner's 
ondition.
Proof. By Theorem 4.5, the 
ompa
t quantum group G is 
oamenable
if and only if R(G) is amenable. By Theorem 3.3, R(G) is amenable if
and only if it satis�es (FC3) whi
h is exa
tly the same as saying that
G satis�es Følner's 
ondition. �
In Se
tion 6 we will use this Følner 
ondition to dedu
e a vanishing
result 
on
erning L2-Betti numbers of 
ompa
t, 
oamenable quantum
groups.
5. An Interlude
In this se
tion we gather various notation and minor results whi
h
will be used in the following se
tion to prove our main result, Theorem
6.1. Some generalities on von Neumann algebrai
 quantum groups are
stated without proofs; we refer to [KV03℄ for the details.
Consider again a 
ompa
t quantum group G = (A,∆) with tra
ial
Haar state h. Denote by {uα | α ∈ I} a 
omplete set of representatives
for the equivalen
e 
lasses of irredu
ible, unitary 
orepresentations of
G. Consider the dense Hopf ∗-algebra
A0 = spanC{uαij | α ∈ I},
28 DAVID KYED
and its dis
rete dual Hopf ∗-algebra Â0. Sin
e h is tra
ial, the dis
rete
quantum group Â0 is unimodular; i.e. the left and right invariant fun
-
tionals are the same. Denote by ϕ̂ the left and right invariant fun
tional
on Â0 normalized su
h ϕ̂(h) = 1. For a ∈ A0 we denote by â ∈ A′0 the
A0 ∋ x7−→h(ax) ∈ C.
Then, by de�nition, we have Â0 = {â | a ∈ A0}. The algebra Â0 is
∗-isomorphi
 to
Mnα(C),
and be
ause h is tra
ial the isomorphism has a simple des
ription; if
we denote by Eαij the standard matrix units in Mnα(C) then the map
Φ((̂uαij)
∗) = 1
Eαij ,
extends to a ∗-isomorphism [MVD98℄. Denote by λ the GNS represen-
tation of A on H = L2(A0, h), by η the 
anoni
al in
lusion A0 ⊆ H
and by M (or λ(M)) the enveloping von Neumann algebra λ(A0)
The map η̂ : Â0 → H given by â 7→ η(a) makes (H, η̂) a GNS pair for
(Â0, ϕ̂) and the 
orresponding GNS representation L is given by
L(â)η(x) = η̂(âx̂).
We denote by M̂ (or L(M̂)) the enveloping von Neumann algebra
L(Â0)
. This is a dis
rete von Neumann algebrai
 quantum group
and ϕ̂ gives rise to a left and right invariant, normal, semi�nite, faith-
ful weight on M̂ . Ea
h �nite subset E ⊆ I gives rise to a 
entral
proje
tion
PE = Φ
χE(α)1nα
∈ Â0,
where 1nα denotes the unit in Mnα(C) and χE is the 
hara
teristi
fun
tion for the set E. A dire
t 
omputation shows that L(PE) is the
orthogonal proje
tion onto the �nite dimensional subspa
e
{uᾱij | 1 ≤ i, j ≤ nα, α ∈ E}.
Re
all from Example 2.3 that uβ̄ is the element in {uα | α ∈ I} whi
h
is equivalent to (uβ)c. Be
ause h is tra
ial, the left invariant weight ϕ̂
on Â0 has the parti
ular simple form [VKV
, p.47℄
nαTrnα
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 29
where Trnα is the non-normalized tra
e on Mnα(C). In parti
ular
ϕ̂(PE) =
n2α = ϕ̂(PĒ),
for any �nite subset E ⊆ I. For any m ∈ M and any �nite subset
E ⊆ I we have [VVD03, 2.10℄ that
TrH(m
∗PEm) = h(m
∗m)ϕ̂(PE),
where TrH denotes the standard tra
e on B(H). Here, and in what fol-
lows, we suppress the representations λ and L ofM and M̂ respe
tively
on H . The 
ommutant M ′ is the underlying von Neumann algebra of
a 
ompa
t, von Neumann algebrai
 quantum group whose Haar state
is also given by the ve
tor state h and whose dis
rete dual is given by
(M̂, ∆̂)op; this quantum group has M̂ as its underlying von Neumann
algebra, but is endowed with 
omultipli
ation ∆̂op = σ∆̂ where σ de-
notes the �ip-automorphism on M̂⊗̄M̂ . Sin
e (M̂, ∆̂) is unimodular
we see that ϕ̂op = ϕ̂ and hen
e the tra
e-formula above extends in the
following way.
Lemma 5.1 ([VVD03℄). For any m ∈ M or m ∈ M ′ and any �nite
subset E ⊆ I we have TrH(m∗PEm) = h(m∗m)ϕ̂(PE).
With this lemma we 
on
lude the interlude and move towards an
appli
ation of the quantum Følner 
ondition.
6. A Vanishing Result
In this se
tion we investigate the L2-Betti numbers of 
oamenable
quantum groups. The notion of L2-Betti numbers for 
ompa
t quantum
groups was introdu
ed in [Kye08℄ and we refer to that paper (and
the introdu
tion) for the de�nitions and basi
 results. Throughout
the se
tion, we will freely use Lü
k's extended Murray-von Neumann
dimension, but whenever expli
it properties are used there will be a
referen
e. These referen
es will be to the original work [Lü
97℄ and
[Lü
98a℄, but for the reader who wants to learn the subje
t Lü
k's
book [Lü
02℄ is probably a better general referen
e.
Consider again a 
ompa
t quantum group G = (A,∆) with Haar
state h and denote by M the enveloping von Neumann algebra in the
GNS representation arising from h. As promised in the introdu
tion,
we will now prove the following theorem whi
h should be 
onsidered as
a quantum group analogue of Theorem 5.1 from [Lü
98a℄.
30 DAVID KYED
Theorem 6.1. If G is 
oamenable and h is tra
ial then for any left
A0-module Z and any k ≥ 1 we have
dimM Tor
k (M,Z) = 0,
where dimM(−) is Lü
k's extended dimension fun
tion arising from the
extension of the tra
e-state h.
If M were �at as a module over A0 we would have Tor
k (M,Z) = 0
for any Z and any k ≥ 1, and the property in Theorem 6.1 is therefore
referred to as dimension �atness of the von Neumann algebra over the
algebra of matrix 
oe�
ients. The proof of Theorem 6.1, whi
h is a
generalization of the 
orresponding proof of [Lü
98a, 5.1℄, is divided
into three parts. Part I 
onsists of redu
tions while part II 
ontains
the 
entral argument 
arried out in detail in a spe
ial 
ase. Part III
shows how to boost the argument from part II to the general 
ase.
Throughout the proof, we will use freely the quantum group notation
developed in the previous se
tions without further referen
e; in par-
ti
ular, {uα | α ∈ I} will denote a �xed, 
omplete set of pairwise
inequivalent, irredu
ible, unitary 
orepresentations of G.
Proof of Theorem 6.1.
Part I
We begin with some redu
tions. Let an arbitrary A0-module Z be
given and 
hoose a free module F that surje
ts onto Z. Then we have
a short exa
t sequen
e
0 −→ K −→ F −→ Z −→ 0,
and sin
e F is free (in parti
ular �at) the 
orresponding long exa
t
Tor-sequen
e gives an isomorphism
TorA0k+1(M,Z) ≃ Tor
k (M,K) for k ≥ 1.
It is therefore su�
ient to prove the theorem for arbitrary Z and k =
1. Moreover, we may assume that Z is �nitely generated sin
e Tor

ommutes with dire
t limits, every module is the dire
ted union of
its �nitely generated submodules and dimM(−) is well behaved with
respe
t to dire
t limits [Lü
98a, 2.9℄. A
tually, we 
an assume that Z
is �nitely presented sin
e any �nitely generated module Z is a dire
t
limit of �nitely presented modules. To see this, 
hoose a short exa
t
sequen
e
0 −→ K −→ F −→ Z −→ 0,
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 31
with F �nitely generated and free. Denote by (Kj)j∈J the dire
ted
system of �nitely generated submodules in K. Then F/Kj is �nitely
presented for ea
h j ∈ J and
Z = lim−→
F/Kj .
Be
ause of this and the dire
t limit formula for the dimension fun
tion
[Lü
98a, 2.9℄ we may, and will, therefore assume that Z is �nitely
presented. Choose a �nite presentation
f−→ Am0 −→ Z −→ 0.
Put H = L2(A, h), K = ker(f) ⊆ An0 ⊆ Hn and denote by f (2) : Hn →
Hm the 
ontinuous extension of f . Then we have
TorA01 (M,Z) =
ker(idM ⊗f)
and hen
e
dimM Tor
1 (M,Z) = dimM ker(idM ⊗f)− dimM M ⊗
= dimM ker(f
(2))− dimM K
where the se
ond equality follows from [CS05, 2.11℄. See also [Lü
98a,
p.158-159℄. So we need to prove that K
= ker(f (2)).
Part II
We �rst treat the 
ase m = n = 1. Then the map f has the form
Ra (right-multipli
ation by a) for some a ∈ A0. If a = 0 we have
= H = ker(f (2)) so we may assume a 6= 0. Sin
e the uαij's

onstitute a linear basis for A0, the element a ∈ A0 has a unique
expansion
i,j=1
tαiju
ij, (t
ij ∈ C)
and we may therefore 
onsider the non-empty, �nite set S ⊆ I given
S = {α ∈ I | ∃ 1 ≤ i, j ≤ nα : tαij 6= 0}.
Denote by H0 the kernel of f
and by q0 ∈ M ′ the proje
tion onto
it. Denote by q the proje
tion onto H0 ∩ K⊥; we need to prove that
this subspa
e is trivial and sin
e the ve
tor-state h is faithful on M ′
this is equivalent to proving h(q) = 0. Let ε > 0 be given. Sin
e G is
32 DAVID KYED
assumed 
oamenable, the Følner 
ondition provides the existen
e of a
�nite, non-empty subset F ⊆ I su
h that
α∈∂S(F )
n2α < ε
Here we identify a subset E ⊆ I with the 
orresponding set of 
orep-
resentations {uα | α ∈ E}. To simplify notation further we will write
∂ instead of ∂S(F ) in the following and moreover we will suppress the
GNS-representations λ : M → B(H) and L : M̂ → B(H) as in Se
-
tion 5. Sin
e h is tra
ial, Woronowi
z's quantum Peter-Weyl Theorem
[KT99, 3.2.3℄ takes a parti
ular simple form and states that the set
{√nαuαij | 1 ≤ i, j ≤ nα, α ∈ I}

onstitutes an orthonormal basis for H . Hen
e every x ∈ H has an
ℓ2-expansion
i,j=1
ij. (x
ij ∈ C)
Consider a ve
tor x ∈ H and assume that P∂̄(x) = 0 su
h that the
ℓ2-expansion of x has the form
i,j=1 x
ij. For γ ∈ S and
1 ≤ p, q ≤ nγ we then have
PF̄ (x) =
α/∈∂,α∈F
i,j=1
(x) = PF̄
i,j=1
Here R
denotes the L2-extension of Ruγpq . Sin
e u
pq is 
ontained
in the linear span of the matrix 
oe�
ients of uα T©uγ and sin
e α /∈
∂ = ∂S(F ) and γ ∈ S we see that the two expressions above are equal.
By linearity and 
ontinuity we obtain
f (2)PF̄ (x) = PF̄f
(2)(x).
This holds for all x ∈ ker(P∂̄), so if x ∈ H0 ∩ ker(P∂̄) we have
0 = f (2)PF̄ (x) = f(PF̄ (x)),
where the last equality is due to the fa
t that rg(PF̄ ) ⊆ A0 ⊆ H .
This proves that PF̄ (x) ∈ K = ker(f) and sin
e q was de�ned as the
proje
tion onto H0∩K⊥ we get qPF̄ (x) = 0. Sin
e this holds whenever
x ∈ H0 = q0(H) and P∂̄(x) = 0 we get
qPF̄ (q0 ∧ (1− P∂̄)) = 0.
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 33
Thus, the restri
tion qPF̄ : H0 → H fa
torizes through H0/H0∩ker(P∂̄)
and we have
dimC(qPF̄ (H0)) ≤ dimC(H0/H0 ∩ ker(P∂̄))
≤ dimC(H/ ker(P∂̄))
= dimC(rg(P∂̄))
= ϕ̂(P∂).
For any �nite rank operator T ∈ B(H) one has
|TrH(T )| ≤ ‖T‖ dimC(T (H))
and using this and Lemma 5.1 we now get
h(q)ϕ̂(PF ) = h(q)ϕ̂(PF̄ )
= TrH(qPF̄ q)
≤ ‖qPF̄ q‖ dimC(qPF̄ q(H))
≤ dimC(qPF̄ (H0))
≤ ϕ̂(P∂).
h(q) ≤ ϕ̂(P∂)
ϕ̂(PF )
and sin
e ε > 0 was arbitrary we 
on
lude that q = 0.
Part III
We now treat the general 
ase of a �nitely presented A0-module Z with
�nite presentation
f−→ Am0 −→ Z −→ 0.
In this 
ase f is given by right multipli
ation by an n × m matrix
T = (tij) with entries in A0. Ea
h tij has a unique linear expansion as
tij =
α,k,l t
(i,j)
α,k,lu
kl and we put
S = {α ∈ I | ∃ i, j, k, l, α : t(i,j)α,k,l 6= 0}.
As in Part II, we may assume that T 6= 0 so that S 6= ∅. Denote by
H0 the spa
e ker(f
(2)) ⊆ Hn, by q0 ∈ Mn(M ′) the proje
tion onto H0
and by q ∈ Mn(M ′) the proje
tion onto H0 ∩ K⊥. We need to show
that q = 0. Denote by Trn the non-normalized tra
e on Mn(C) and
put hn = h ⊗ Trn : B(H) ⊗ Mn(C) → C. We aim at proving that
34 DAVID KYED
hn(q) = 0, whi
h su�
es sin
e h is faithful on M
. For ea
h x ∈ M̂ we
denote by xn the diagonal operator on Hn whi
h has x in ea
h diagonal
entry. Under the identi�
ation B(H) ⊗ Mn(C) = B(Hn) we see that
TrH ⊗Trn 
orresponds to TrHn , and Lemma 5.1 together with a dire
t

omputation therefore gives
TrHn(A
∗P nEA) = hn(A
∗A)ϕ̂(PE), (†)
for any �nite subset E ⊆ I and any A in Mn(M) or Mn(M ′). Let ε > 0
be given and 
hoose a

ording to the Følner 
ondition a �nite subset
F ⊆ I su
h that
α∈∂S(F )
n2α <
and put ∂ = ∂S(F ) for simpli
ity. By repeating the argument from the
beginning of Part II we arrive at the equation
(q0 ∧ (1− P n∂̄ )) = 0,
whi
h in turn yields
dimC(qP
(H0)) ≤ dimC(rg(P n∂̄ )) = n dimC(rg(P∂̄)) = nϕ̂(P∂).
Using the tra
e-formula (†) we 
on
lude that
hn(q)ϕ̂(PF ) = TrHn(qP
F̄ q)
≤ ‖qP n
q‖ dimC(qP nF̄ q(H))
≤ dimC(qP nF̄ (H0))
≤ nϕ̂(P∂).
hn(q) ≤ n
ϕ̂(P∂)
ϕ̂(PF )
and sin
e ε > 0 was arbitrary we 
on
lude that hn(q) = 0 as desired.
By putting Z = C in Theorem 6.1, we immediately obtain the fol-
lowing 
orollary.
Corollary 6.2. Let G = (A,∆) be a 
ompa
t, 
oamenable quantum
group with tra
ial Haar state. Then β
n (G) = 0 for all n ≥ 1. Here
n (G) is the n-th L2-Betti number of G as de�ned in [Kye08℄.
In parti
ular we obtain the following extension of [Kye08, 3.3℄.
Corollary 6.3. For an abelian, 
ompa
t quantum group G we have
n (G) = 0 for n ≥ 1.
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 35
Proof. Sin
e G is abelian it is of the form (C(G),∆c) for some 
ompa
t
(se
ond 
ountable) group G. Sin
e the 
ounit, given by evaluation at
the identity, is already globally de�ned and bounded it is 
lear that G
is 
oamenable and the result now follows from Corollary 6.2. �
We also obtain the 
lassi
al result of Lü
k.
Corollary 6.4. [Lü
98a, 5.1℄ If Γ is an amenable, 
ountable, dis
rete
group then for all CΓ-modules Z and all n ≥ 1 we have
dimL (Γ) Tor
n (L (Γ), Z) = 0.
In parti
ular, β
n (Γ) = 0 for n ≥ 1.
Proof. Put G = (C∗red(Γ),∆red). Then G is 
oamenable if and only if Γ
is amenable and the result now follows from Theorem 6.1 and Corollary
6.2 �
Note, however, that this does not really give a new proof of Lü
k's
result sin
e the proof of Theorem 6.1 
oin
ides with Lü
k's proof of the
statement in Corollary 6.4 when G = (C∗red(Γ),∆red).
In [CS05℄, Connes and Shlyakhtenko introdu
ed a notion of L2-Betti
numbers for tra
ial ∗-algebras. From the above results we also obtain
vanishing of these Connes-Shlyakhtenko L2-Betti numbers for 
ertain
Hopf ∗-algebras. More pre
isely we get the following.
Corollary 6.5. Let G = (A,∆) be a 
ompa
t, 
oamenable quantum
group with tra
ial Haar state h. Then β
n (A0, h) = 0 for all n ≥ 1,
where β
n (A0, h) is the n-th Connes-Shlyakhtenko L
-Betti number of
the ∗-algebra A0 with respe
t to the tra
e h.
Proof. By [Kye08, 4.1℄ we have β
n (G) = β
n (A0, h) and the 
laim
therefore follows from Corollary 6.2. �
The knowledge of dimension �atness also gives genuine homologi
al
information about the ring extension A0 ⊆ M . More pre
isely, the
following holds.
Corollary 6.6. If G = (A,∆) is 
ompa
t and 
oamenable with tra
ial
Haar state then the indu
tion fun
torM⊙A0− is an exa
t fun
tor from
the 
ategory of �nitely generated, proje
tive A0-modules to the 
ategory
of �nitely generated, proje
tive M-modules.
Proof. Let X and Y be �nitely generated, proje
tive A0-modules and
let f : X → Y be an inje
tive homomorphism. Then
0 −→ X f−→ Y −→ Y/rg(f) −→ 0,
36 DAVID KYED
is a proje
tive resolution of Y/rg(f). Thus TorA01 (M,Y/rg(f)) =
ker(idM ⊗f) and from Theorem 6.1 we 
on
lude that
dimM(ker(idM ⊗f)) = 0.
Be
ause idM ⊗f is a map of �nitely generated proje
tive M-modules,
it is not di�
ult to prove that
ker(idM ⊗f) = ker(idM ⊗f)
where ker(idM ⊗f)
is de�ned (see [Lü
98a℄) as the interse
tion of all
kernels arising from homomorphisms fromM⊙A0X toM vanishing on
ker(idM ⊗f). By [Lü
98a, 0.6℄, we 
on
lude from this that ker(idM ⊗f)
is �nitely generated and proje
tive. But, sin
e the dimension fun
tion
is faithful on the 
ategory of �nitely generated, proje
tive M-modules
this for
es ker(idM ⊗f) = {0} and the 
laim follows. �
Corollary 6.6, in parti
ular, implies the following result whi
h was
pointed out to us by A. Thom.
Corollary 6.7. Let G = (A,∆) be 
ompa
t and 
oamenable with tra
ial
Haar state and let x ∈ A0 be a non-zero element su
h that there exists
a non-zero m ∈ M with mx = 0. Then there exists a non-zero y ∈ A0
with yx = 0.
Proof. This follows by using Corollary 6.6 on the map a 7−→ ax. �
An analogous statement about produ
ts in the opposite order follows
by using the involution in M . So, formulated in ring theoreti
al terms,
we obtain the following: Any regular element in A0 stays regular in the
over-ring M .
7. Examples
A 
on
rete example of a non-
ommutative, non-
o
ommutative, 
o-
amenable (matrix) quantum group with tra
ial Haar state is the or-
thogonal quantum group Ao(2) ≃ SU−1(2). It follows from [Ban99a,
5.1℄ that Ao(2) is 
oamenable. To see that the Haar state is tra
ial, one
observes that the orthogonality property of the 
anoni
al fundamental

orepresentation implies that the antipode has period two.
7.1. Examples arising from tensor produ
ts. If G1 = (A1,∆1)
and G2 = (A2,∆2) are 
ompa
t quantum groups then the (minimal)
tensor produ
t A = A1 ⊗ A2 may be turned into a quantum group G
by de�ning the 
omultipli
ation ∆: A −→ A⊗ A to be
∆(a) = (id⊗σ ⊗ id)(∆1 ⊗∆2)(a),
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 37
where σ denotes the �ip-isomorphism from A1 ⊗ A2 to A2 ⊗ A1. The
Haar state is the tensor produ
t of the two Haar states and the 
ounit
is the tensor produ
t of the 
ounits. Using these fa
ts, it is not di�
ult
to see [BMT01℄ that if both G1 and G2 are 
oamenable and have tra
ial
Haar states, then the same is true for G. See e.g. [KR86, 11.3.2℄.
7.2. Examples arising from bi
rossed produ
ts. Another way to
obtain examples of 
ompa
t, 
oamenable quantum groups is via bi-

rossed produ
ts. We therefore brie�y sket
h the bi
rossed produ
t

onstru
tion following [VV03℄ 
losely. In [VV03℄, Vaes and Vainerman

onsider the more general notion of 
o
y
le bi
rossed produ
ts, but
sin
e we will mainly be interested in the 
ase where the 
o
y
les are
trivial we will restri
t our attention to this 
ase in the following. The
more general situation will be dis
ussed brie�y in Remark 7.5. The
bi
rossed produ
t 
onstru
tion is de�ned using the language of von
Neumann algebrai
 quantum groups. We will use this language freely
in the following and refer to [KV03℄ for the ba
kground material.
Let (M1,∆1) and (M2,∆2) be lo
ally 
ompa
t (l.
.) von Neumann
algebrai
 quantum groups. Let τ : M1⊗̄M2 → M1⊗̄M2 be a faith-
ful ∗-homomorphism and denote by σ : M1⊗̄M2 → M2⊗̄M1 the �ip-
isomorphism. Then τ is 
alled a mat
hing from M1 to M2 if the fol-
lowing holds.
• The map α : M2 −→ M1⊗̄M2 given by α(y) = τ(1 ⊗ y) is a
(left) 
oa
tion of (M1,∆1) on the von Neumann algebra M2.
• De�ning β : M1 −→M1⊗̄M2 as β(x) = τ(x⊗ 1) the map σβ is
a (left) 
oa
tion of (M2,∆2) on the von Neumann algebra M1.
• The 
oa
tions satisfy the following two mat
hing 
onditions:
τ(13)(α⊗ 1)∆2 = (1⊗∆2)α (M1)
τ(23)σ(23)(β ⊗ 1)∆1 = (∆1 ⊗ 1)β (M2)
Here we use the standard leg numbering 
onvention (see e.g. [MVD98℄).
If τ : M1⊗̄M2 → M1⊗̄M2 is a mat
hing from M1 to M2 then it is easy
to see that στσ−1 is a mat
hing fromM2 to M1. We will therefore just
refer to the pair (M1,M2) as a mat
hed pair and to τ as a mat
hing
of the pair. Let (M1,∆1) and (M2,∆2) be su
h a mat
hed pair of
l.
. quantum groups and denote by τ the mat
hing. We denote by Hi
the GNS spa
e ofMi with respe
t to the left invariant weight ϕi and by
Wi and Ŵi the natural multipli
ative unitaries on Hi⊗̄Hi for Mi and
M̂i respe
tively. By H we denote H1⊗̄H2 and by Σ the �ip-unitary on
38 DAVID KYED
H⊗̄H . We may now form two 
rossed produ
ts:
M =M1 ⋉α M2 = vNa{α(M2), M̂1 ⊗ 1} ⊆ B(H1⊗̄H2)
M̃ =M2 ⋉σβ M1 = vNa{σβ(M1), M̂2 ⊗ 1} ⊆ B(H2⊗̄H1)
Some of the main results in [VV03℄ are summarized in the following:
Theorem 7.1 ([VV03℄). De�ne operators
Ŵ = (β ⊗ 1⊗ 1)(W1 ⊗ 1)(1⊗ 1⊗ α)(1⊗ Ŵ2)
and W = ΣŴ ∗Σ on H⊗̄H. Then W and Ŵ are multipli
ative uni-
taries and the map ∆: M → B(H⊗̄H) given by ∆(a) = W ∗(1⊗1⊗a)W
de�nes a 
omultipli
ation on M turning it into a l.
. quantum group.
Denoting by Σ12 the �ip-unitary fromH1⊗̄H2 to H2⊗̄H1, the dual quan-
tum group M̂ be
omes Σ∗12M̃Σ12 with 
omultipli
ation implemented by
Thus, up to a �ip the two 
rossed produ
ts above are in duality. In
[DQV02℄, Desmedt, Quaegebeur and Vaes studied (
o)amenability of
bi
rossed produ
ts. Combining their Theorem 15 with [VV03, 2.17℄
we obtain the following: If (M1,M2) is a mat
hed pair with M1 dis-

rete and M2 
ompa
t then the bi
rossed produ
t M is 
ompa
t, and
M is 
oamenable if and only if both M2 and M̂1 are. Here a von Neu-
mann algebrai
 
ompa
t quantum group is said to be 
oamenable if the

orresponding C∗-algebrai
 quantum group is. Colle
ting the results
dis
ussed above we obtain the following.
Proposition 7.2. If (M1,M2) is a mat
hed pair of l.
. quantum groups
in whi
h M̂1 and M2 are 
ompa
t and 
oamenable, then the bi
rossed
produ
t M = M1 ⋉α M2 is 
oamenable and 
ompa
t. So if the Haar
state on M is tra
ial the quantum group (M,∆) has vanishing L2-Betti
numbers in all positive degrees.
In order to produ
e more 
on
rete examples, we will now dis
uss a
spe
ial 
ase of the bi
rossed produ
t 
onstru
tion in whi
h one of the

oa
tions 
omes from an a
tual group a
tion. This part of the theory
is due to De Cannière [DC79℄ and is formulated using the language of
Ka
 algebras. We remind the reader that a 
ompa
t Ka
 algebra is
nothing but a von Neumann algebrai
, 
ompa
t quantum group with
tra
ial Haar state. A dis
rete, 
ountable group Γ a
ts on a 
ompa
t
Ka
 algebra (M,∆, S, h) if the group a
ts on the von Neumann algebra
M and the a
tion 
ommutes with both the 
oprodu
t and the antipode.
-BETTI NUMBERS OF COAMENABLE QUANTUM GROUPS 39
Denoting the a
tion by ρ, this means that
∆(ργ(x)) = ργ ⊗ ργ(∆(x)),
S(ργ(x)) = ργ(S(x)),
for all γ ∈ Γ and all x ∈ M . In this situation, the a
tion of Γ on M
indu
es a 
oa
tion α : M −→ ℓ∞(Γ)⊗̄M . Denoting by H the Hilbert
spa
e on whi
h M a
ts and identifying ℓ2(Γ)⊗̄H with ℓ2(Γ, H), this

oa
tion is given by the formula
α(x)(ξ)(γ) = ργ−1(x)(ξ(γ)),
for ξ ∈ ℓ2(Γ, H). The 
rossed produ
t, whi
h is de�ned as
Γ⋉ρ M = {α(M),L (Γ)⊗ 1}′′,
be
omes again a Ka
 algebra [DC79, Thm.1℄. One should note at this
point that De Cannière works with the right 
rossed produ
t a
ting
on H⊗̄ℓ2(Γ) where we work with the left 
rossed produ
t a
ting on
ℓ2(Γ)⊗̄H . But, one 
an 
ome from one to the other by 
onjugation
with the �ip-unitary and we may therefore freely transport all results
from [DC79℄ to the setting of left 
rossed produ
ts. We now prove that
De Cannière's 
rossed produ
t 
an also be 
onsidered as a bi
rossed
produ
t. This is probably well known to experts in the �eld, but we
were unable to �nd an expli
it referen
e.
Proposition 7.3. De�ning τ : ℓ∞(Γ)⊗̄M −→ ℓ∞(Γ)⊗̄M by
τ(δγ ⊗ x) = δγ ⊗ ργ−1(x)
we obtain a mat
hing with the above de�ned α as the 
orresponding

oa
tion of ℓ∞(Γ) on M and trivial 
oa
tion of (M,∆) on ℓ∞(Γ).
Proof. A dire
t 
al
ulation shows that α(x) = τ(1 ⊗ x) and β(f) =
τ(f ⊗ 1) = f ⊗ 1. Therefore the two maps x 7→ τ(1 ⊗ x) and f 7→
στ(f ⊗ 1) are 
oa
tions as required. We therefore just have to 
he
k
that the mat
hing 
onditions are ful�lled. Denote the 
oprodu
t on
ℓ∞(Γ) by ∆1 and 
hoose f ∈ ℓ∞(Γ) su
h that ∆1(f) ∈ ℓ∞(Γ)⊙ ℓ∞(Γ).
Writing ∆1(f) as f(1) ⊗ f(2) we now get
τ(23)σ(23)(β ⊗ 1)∆1f = τ(23)σ(23)(β ⊗ 1)(f(1) ⊗ f(2))
= τ(23)σ(23)(f(1) ⊗ 1⊗ f(2))
= τ(23)(f(1) ⊗ f(2) ⊗ 1)
= f(1) ⊗ f(2) ⊗ 1
= (∆1 ⊗ 1)β(f),
and hen
e (M2) is satis�ed. An analogous, but slightly more 
umber-
some, 
al
ulation proves that (M1) is also satis�ed. �
40 DAVID KYED
Thus, as von Neumann algebras, we have ℓ∞(Γ) ⋉α M = Γ ⋉ρ M .
Using the fa
t that β is trivial, one 
an prove that the elements λγ ⊗ 1
are group-like and it therefore follows from [DC79, 3.3℄ that also the

omultipli
ations agree. Hen
e the two 
rossed produ
t 
onstru
tions
are identi
al as l.
. quantum groups. In parti
ular, the the bi
rossed
produ
t ℓ∞(Γ) ⋉α M is a Ka
 algebra so if (M,∆) is 
ompa
t then
ℓ∞(Γ)⋉α M is also 
ompa
t [VV03, 2.7℄ and the Haar state is tra
ial.
We therefore have the following.
Proposition 7.4. If G = (M,∆, S, h) is a 
ompa
t, 
oamenable Ka
algebra and Γ is a 
ountable, dis
rete, amenable group a
ting on G then
the 
rossed produ
t Γ⋉M is again a 
ompa
t, 
oamenable Ka
 algebra.
Proof. That Γ⋉M is a Ka
 algebra follows from the dis
ussion above
and the 
oamenability of the 
rossed produ
t follows from [DQV02, 15℄
sin
e ℓ̂∞(Γ) = L (Γ) is 
oamenable if (and only if) Γ is amenable. �
Remark 7.5. It is also possibly to 
onstru
t examples using the more
general notion of 
o
y
le 
rossed produ
ts introdu
ed in [VV03, 2.1℄.
It is shown in [DQV02, 13℄ that weak amenability (i.e. the existen
e of
an invariant mean) is preserved under 
o
y
le bi
rossed produ
ts. In
general it is not known whether or not weak amenability is equivalent
to strong amenability, the latter being de�ned as the dual quantum
group being 
oamenable in the sense of De�nition 4.1. But for dis-

rete quantum groups this equivalen
e has been proven by Tomatsu in
[Tom06℄ and also by Blan
hard and Vaes in unpublished work. There-
fore, if (M1,M2) is a 
o
y
le mat
hed pair of l.
. quantum groups with
both M̂1 and M2 
ompa
t and 
oamenable, then the 
o
y
le 
rossed
produ
t is also 
ompa
t and 
oamenable.
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David Kyed, Mathematis
hes Institut, Georg-August-Universität,
Göttingen, Bunsenstraÿe 3-5, D-37073 Göttingen, Germany
E-mail address : kyed�uni-math.gwdg.de
URL: www.uni-math.gwdg.de/kyed
www.wis.kuleuven.be/analyse/stefaan
	Introduction
	1. Preliminaries on compact quantum groups
	2. Fusion Algebras
	3. Amenability for Fusion Algebras
	3.1. Formulas used in the proof of Theorem ??
	4. Coamenable Compact Quantum Groups
	5. An Interlude
	6. A Vanishing Result
	7. Examples
	7.1. Examples arising from tensor products
	7.2. Examples arising from bicrossed products
	References
