Graphs of quantum groups and K-amenability

39 Pages·2017·0.72 MB·English

Graphs of quantum groups and K-amenability
Pierre Fima, Amaury Freslon
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GRAPHS OF QUANTUM GROUPS AND K-AMENABILITY
PIERREFIMA AND AMAURYFRESLON
Abstract. Building on aconstruction of J-P.Serre, weassociate toany graph of C*-algebras
a maximal and a reduced fundamental C*-algebra and use this theory to construct the fun-
damental quantum group of a graph of discrete quantum groups. This construction naturally
gives rise to a quantum Bass-Serre tree which can be used to study the K-theory of the fun-
damental quantumgroup. To illustrate the properties of this construction, we prove that if all
the vertex qantum groups are amenable, then the fundamental quantumgroup is K-amenable.
This generalizes previous results of P.Julg, A. Valette,R. Vergnioux and thefirst author.
1. Introduction
One of the first striking application of K-theory to the theory of operator algebras was the proof
by M.V. Pimsner and D.V. Voiculescu in [PV82] that the reduced C*-algebras of free groups
with different number of generators are not isomorphic. It relies on an involved computation of
the K-theory of these C*-algebras, which appear not to be equal. From then on, K-theory of
group C*-algebras has beena very active field of research, in particular inrelation with algebraic
andgeometricproblems, culminating inthecelebrated Baum-Connes conjecture (seeforexample
[BCH94]).
Along the way J. Cuntz introduced the notion of K-amenability in [Cun83]. Noticing that the
maximal and reduced C*-algebras of free groups have the same K-theory, he endeavoured to give
a conceptual explanation of this fact based on a phenomenon quite similar to amenability, but
on a K-theoretical level. Combined with a short and elegant computation of the K-theory of
the maximal C*-algebras of free groups, he could therefore recover the result of M.V. Pimsner
and D.V. Voiculescu in more conceptual way. K-amenability implies in particular that the K-
theoryofany reducedcrossed-productby thegroupisequaltotheK-theory ofthecorresponding
full crossed-product, thus giving a powerful tool for computing K-theory of C*-algebras. The
original definition of J. Cuntz was restricted to discrete groups but was later generalized to
arbitrary locally compact groups by P. Julg and A. Valette in [JV84]. In this seminal paper,
they also proved that any group acting on a tree with amenable stabilizers is K-amenable. As
particular cases, one gets that free products and HNN extensions of amenable groups are K-
amenable. This result was later extended to groups acting on trees with K-amenable stabilizers
by M.V. Pimsner in [Pim86]. Let us also mention the notion of K-nuclearity developped by G.
Skandalisin[Ska88]andfurtherstudiedbyE.Germainwhoprovedin[Ger96]thatafreeproduct
of amenable C*-algebras is K-nuclear. However, we will concentrate in the present paper on the
Date: July 22, 2013.
2010 Mathematics Subject Classification. 46L09, 46L65.
Key words and phrases. Quantum groups, quantumBass-Serre tree, K-amenability.
The first author is partially supported byANR grants OSQPIand NEUMANN.
1
2 PIERREFIMAANDAMAURYFRESLON
Julg-Valette theorem and extend it to the setting of discrete quantum groups. Let us first recall
an algebraic description of groups acting on trees which will be better suited to our purpose.
A graph of groups is the data of a graph G together with groups attached to each vertex and
each edge in a way compatible with the graph structure. One can generalize the topological
construction of the fundamental group π (G) to include the additional data of the groups and
1
thus obtain the notion of fundamental group of a graph of groups. The core of Bass-Serre theory,
developped in [Ser77], is a powerful correspondance between this construction and the structure
of groups acting on trees. Two very important particular cases are amalgamated free products
and HNN extensions. In both cases, Bass-Serre theory provides us with a tree on which the
group acts in a canonical way.
In the context of locally compact quantum groups, K-amenability was defined and studied by
R. Vergnioux in [Ver04], building on the work of S. Baaj and G. Skandalis on equivariant KK-
theory for coactions of Hopf-C*-algebras [BS89]. In the discrete case, R. Vergnioux was able to
prove that several classical characterizations still hold in the quantum setting (some of them are
recalled in Theorem 2.8). He also proved the K-amenability of amalgamated free products of
amenable discrete quantum groups. His proof used the first example of a quantum Bass-Serre
tree. This is a pair of Hilbert C*-modules over the C*-algebra of a compact quantum group
endowed with actions which can be used as a "geometric object" for the study of K-theoretical
properties. Similar techniques where used by the first author to prove K-amenability of HNN
extensionsofamenablediscretequantumgroupsin[Fim13]. TheuseofquantumBass-Serretrees
also proved crucial in the study of the Baum-Connes conjecture for discrete quantum groups by
R. Vergnioux and C. Voigt in [VV13].
In the present paper, we generalize the construction of the fundamental group to graphs of
discrete quantum groups. As one could expect, this fundamental quantum group comes along
with a quantum Bass-Serre tree which can be used to construct a natural KK-element. Our
construction is in some sense the most general construction of a quantum Bass-Serre tree such
that the "quotient" by the action of the quantum group is a classical graph. We then use
techniques combining the ones of [Ver04] and [Fim13] to prove that if all the vertex groups
are amenable, then the resulting quantum group is K-amenable. In view of the Bass-Serre
equivalence, this generalizes the resultof[JV84]. Notethat this gives alargeclass ofK-amenable
discrete quantum groups and improves the aforementionned results in the quantum setting. For
example, it is known by [Fim13] that an HNN extension of amenable discrete quantum groups
is K-amenable, but it was not known that if we again take an HNN extension or a free product
with a third amenable discrete quantum group, then the resulting quantum group will still be
K-amenable.
Let us now outline the organization of the paper. In Section 2, we specify some notations and
conventions used all along the paper and we give some basic definitions and results concerning
quantum groups and K-amenability. In section 3, we associate to any graph of C*-algebras a full
andareducedfundamentalC*-algebraandgivesomestructureresults. Thissectionisratherlong
but contains most of the technical results of this paper. It ends with an "unscrewing" technique
which can be used to prove that some properties of the vertex C*-algebras are inherited by the
fundamental C*-algebras. In Section 4, we use the previous results to define the fundamental
quantum group of a graph of quantum groups and describe its Haar state and representation
GRAPHS OF QUANTUM GROUPS AND K-AMENABILITY 3
theory. Eventually, we prove in Section 5 that the fundamental quantum group of a graph of
amenable discrete quantum groups is K-amenable. Note that one could also define graphs of von
Neumann algebras and work out similar constructions. This is outlined in the Appendix.
2. Preliminaries
2.1. Notations and conventions. In this paper all the Hilbert spaces, Hilbert C*-modules
and C*-algebras are assumed to be separable. Moreover, all the C*-algebras are assumed to be
unital. The scalar products on Hilbert spaces or Hilbert C*-modules are denoted by h.,.i and
are supposed to be linear in the second variable. For two Hilbert spaces H and K, B(H,K) will
denote the set of bounded linear maps from H to K and B(H):= B(H,H). For a C*-algebra A
andHilbertA-modules HandK, wedenote by L (H,K) thesetofboundedadjointable A-linear
A
operators from H to K and L (H) = L (H,H).
A A
We will also use the following terminology : if H is a Hilbert A-module and ϕ ∈ A∗ is a state,
the GNS construction of (H,ϕ) is the triple (H,π,η), where H is the Hilbert space obtained
by separation and completion of H with respect to the scalar product hξ,ηiH = ϕ(hξ,ηiH),
η : H → H is the canonical linear map with dense range and π : L (H)→ B(H) is the induced
A
unital ∗-homomorphism. Note that π and η are faithful as soon as ϕ is. Observe also that if
K is another Hilbert A-module and if (K,ρ,ξ) denotes the GNS construction of (K,ϕ), then we
also have an obvious induced linear map L (H,K) → B(H,K) which respects the adjoint and
A
the composition (if we take a third Hilbert A-module).
IfG is a graph in the senseof [Ser77, Def 2.1], its vertex set will bedenoted V(G) and its edge set
will be denoted E(G). For e∈ E(G) we denote by s(e) and r(e) respectively the source and range
of e and by e the inverse edge of e. An orientation of G is a partition E(G) = E+(G)⊔E−(G)
such that e ∈ E+(G) if and only if e ∈E−(G).
Finally, we will always denote by ı the identity map.
2.2. Compact quantum groups. We briefly recall the main definitions and results of the
theory of compact quantum groups in order to fix notations. The reader is referred to [Wor98]
or [MVD98] for details and proofs.
Definition 2.1. A compact quantum group is a pair G = (C(G),∆) where C(G) is a unital
C*-algebra and ∆ : C(G)→ C(G)⊗C(G) is a unital ∗-homomorphism such that
(∆⊗ı)◦∆ = (ı⊗∆)◦∆
and the linear span of ∆(C(G))(1⊗C(G)) as well as the linear span of ∆(C(G))(C(G)⊗1) are
dense in C(G)⊗C(G).
Theorem 2.2 (Woronowicz). Let G be a compact quantum group. There is a unique state
h∈ C(G)∗, called the Haar state of G, such that for every x ∈ C(G),
(h⊗ı)◦∆(x)= h(x).1 and (ı⊗h)◦∆(x)= h(x).1.
The Haar state need not be faithful. Let C (G) be the C*-algebra obtained by the GNS
red
construction of the Haar state. C (G) is called the reduced C*-algebra of the compact quantum
red
group G. By the invariance properties of the Haar state, the coproduct ∆ induces a coproduct
on C (G) which turns it into a compact quantum group called the reduced form of G. The
red
Haar state on the reduced form of G is faithful by construction.
4 PIERREFIMAANDAMAURYFRESLON
Definition 2.3. Let G be a compact quantum group. A representation of G of dimension n is
a matrix (u )∈ M (C(G)) = M (C)⊗C(G) such that for all 1 6 i,j 6n,
ij n n
∆(u )= u ⊗u .
ij ik kj
k
X
A representation is called unitary if u = (u ) ∈ M (C) ⊗C(G) is a unitary. An intertwiner
ij n
between two representations u and v of dimension respectively n and m is a linear map T :
Cn → Cm such that (T ⊗ı)u = v(T ⊗ı). If there exists a unitary intertwiner between u and v,
they are said to be unitarily equivalent. A representation is said to be irreducible if its only self-
intertwinersarethescalarmultiplesoftheidentity. Thetensorproduct ofthetworepresentations
u and v is the representation
u⊗v = u v ∈M (C)⊗M (C)⊗C(G)≃ M (C)⊗C(G).
13 23 n m nm
Theorem 2.4 (Woronowicz). Every unitary representation of a compact quantum group is uni-
tarily equivalent to a direct sum of irreducible unitary representations.
Let Irr(G) be the set of equivalence classes of irreducible unitary representations of G and,
for α ∈ Irr(G), denote by uα a representative of α. The linear span of the elements uα for
ij
α ∈ Irr(G) forms a Hopf-∗-algebra Pol(G) which is dense in C(G). Its enveloping C*-algebra is
denoted C (G) and it has a natural quantum group structure called the maximal form of G.
max
By universality, there is a surjective ∗-homomorphism
λG : Cmax(G) → Cred(G)
which intertwines the coproducts.
Remark 2.5. The C*-algebras C (G) and C (G) should be thought of as the reduced and
red max
maximal C*-algebras of the dual discrete quantum group G. This point of view justifies the
terminology "discrete quantum groups" used in the paper.
b
C (G) admits a one-dimensional representation ε : C (G)→ C, called the trivial represen-
max max
tation (or the counit) and defined by ε(uα) = δ for all α ∈Irr(G) and every i,j. The counit is
ij ij
the unique unital ∗-homomorphism ε : C (G) → C such that (ε⊗ı)∆ = ı= (ı⊗ε)∆.
max
2.3. K-amenability.
Definition 2.6. Acompactquantum group Gissaidtobeco-amenable ifλG isanisomorphism.
We will equivalently say that G is amenable.
Like in the classical case, co-amenability has several equivalent characterizations, we only give
b
the one which will be needed in the sequel (see [BMT01, Thm 3.6] for a proof).
Proposition 2.7. A compact quantum group G is co-amenable if and only if the trivial repre-
sentation factors through λG.
K-amenability admits similar characterizations on the level of KK-theory, which were proved by
R. Vergnioux in [Ver04, Thm 1.4]. We refer the reader to [Bla98] for the basic definitions and
results concerning KK-theory.
Theorem 2.8 (Vergnioux). Let G be a compact quantum group. The following are equivalent
• There exists γ ∈ KK(Cred(G),C) such that λ∗G(γ) = [ε] in KK(Cmax(G),C).
GRAPHS OF QUANTUM GROUPS AND K-AMENABILITY 5
• The element [λG] in invertible in KK(Cmax(G),Cred(G)).
In any of those two equivalent situations, we will say that G is K-amenable.
3. Graphs of C*-algebras
b
In this section we give the general construction of a maximal and a reduced fundamental C*-
algebra associated to a graph of C*-algebras.
Definition 3.1. A graph of C*-algebras is a tuple
(G,(A ) ,(B ) ,(s ) )
q q∈V(G) e e∈E(G) e e∈E(G)
where
• G is a connected graph.
• For every q ∈ V(G) and every e ∈E(G), A and B are unital C*-algebras.
q e
• For every e∈ E(G), B = B .
e e
• For every e∈ E(G), s : B → A is a unital faithful ∗-homomorphism.
e e s(e)
For every e ∈ E(G), we set r = s :B → A , Bs = s (B ) and Br = r (B ).
e e e r(e) e e e e e e
The notation will always be simplified in (G,(A ) ,(B ) ).
q q e e
3.1. The maximal fundamental C*-algebra. Like in thecase of freeproducts, thedefinition
of the maximal fundamental C*-algebra is quite obvious and simple. However, it requires the
choice of a maximal subtree of the graph G (which is implicit in the case of free products since
the graph is already a tree, see Example 3.4).
Definition3.2. Let(G,(A ) ,(B ) )beagraphofC*-algebrasandletT beamaximalsubtreeof
q q e e
G. The maximal fundamental C*-algebra with respect to T is the universal C*-algebra generated
by the C*-algebras A for q ∈ V(G) and by unitaries u for e ∈ E(G) such that
q e
• For every e∈ E(G), u = u∗.
e e
• For every e∈ E(G) and every b ∈ B , u s (b)u = r (b).
e e e e e
• For every e∈ E(T), u = 1.
e
This C*-algebra will be denoted πmax(G,(A ) ,(B ) ,T).
1 q q e e
Remark 3.3. It is not obvious that this C*-algebra is not 0 (i.e. that the relations admit a non-
trivialrepresentation). Withnatural additional assumptions, the non-triviality will beproved by
the construction of the reduced fundamental C*-algebra and it will be clear that the inclusions
of A in the maximal fundamental C*-algebra are faithful.
q
Example 3.4. Let A and A be two C*-algebras and let B be a C*-algebra together with
0 1
injective ∗-homomorphisms i : B → A for k = 0,1. Let G be the graph with two vertices p
k k 0
and p and two edges e and e, where s(e) = p and r(e) = p . This graph is obviously a tree.
1 1 2
Setting B = B, s = i , r = i and A = A yields a graph of C*-algebras whose maximal
e e 0 e 1 pi i
fundamental C*-algebra with respect to G is the maximal free product A ∗maxA of A and A
0 B 1 0 1
amalgamated over B.
Example 3.5. Let A be a C*-algebra, B a C*-subalgebra of A and θ : B → A an injective
∗-homomorphism. Let G be a graph with one vertex p and two edges e and e, where e is a loop
from p to p. Obviously, the only maximal subtree of G is the graph with one vertex p and no
edge. Setting B = B, s = ı, r = θ and A = A yields a graph of C*-algebras whose maximal
e e e p
6 PIERREFIMAANDAMAURYFRESLON
fundamental C*-algebra with respect to {p} is the maximal HNN extension HNNmax(A,B,θ) as
defined in [Ued05, Rmk 7.3].
By construction, the maximal fundamental C*-algebra of (G,(A ) ,(B ) ) satisfies the following
q q e e
universal property.
Proposition 3.6. Let (G,(A ) ,(B ) ) be a graph of C*-algebras, let T be a maximal subtree of
q q e e
G and let H be a Hilbert space. Assume that for every q ∈ V(G), we have a representation ρ of
q
A on H and that for every e ∈ E(G), we have a unitary U ∈ B(H) such that U = U∗, U = 1
q e e e e
for all e∈ E(T) and for every b ∈ B ,
e
∗
U ρ (s (b))U = ρ (r (b)).
e s(e) e e r(e) e
max
Then, there is a unique representation ρ of π (G,(A ) ,(B ) ,T) on H such that for every
1 q q e e
e∈ E(G) and every q ∈ V(G),
ρ(u ) =U and ρ| = ρ .
e e Aq q
Remark 3.7. Let p ∈ V(G). Define A to be the linear span of A and elements of the form
0 p0
a u ...u a where (e ,...,e ) is a path in G from p to p , a ∈ A and a ∈ A for
0 e1 en n 1 n 0 0 0 p0 i r(ei)
16 i 6 n. Observe that A is a dense ∗-subalgebra of πmax(G,(A ) ,(B ) ,T). Indeed, it suffices
1 q q e e
to show that it contains A for every q ∈ V(G) and u for every e ∈ E(G). Let q ∈ V(G) and
q e
a ∈ A . Let w = (e ,...,e ) be the unique geodesic path in T from p to q. Since e ∈ E(T), we
q 1 n 0 i
haveu = 1forevery1 6 i6 n. Hence,a = u ...u au ...u ∈ A. Now,lete ∈ E(G)\E(T)
ei e1 en en e1
and let (e ,...,e ) (resp. (f ,...,f )) be the geodesic path in T from p to s(e) (resp. r(e)).
1 n 1 m 0
Then, u = u ...u u u ...u ∈ A.
e e1 en e fm f1
We will need in the sequel the following slightly more general version of the universal property.
Corollary 3.8. Let (G,(A ) ,(B ) ) be a graph of C*-algebras, let T be a maximal subtree of G
q q e e
and let p ∈ V(G). Assume that for every q ∈ V(G), there is a Hilbert space H together with a
0 q
representation ρ of A and that, for every e ∈ E(G), there is a unitary U : H → H such
q q e r(e) s(e)
that U∗ = U and, for every b ∈B ,
e e e
∗
U ρ (s (b))U = ρ (r (b)).
e s(e) e e r(e) e
Then, there exists a unique representation ρ of πmax(G,(A ) ,(B ) ,T) on H such that ρ| =
1 q q e e p0 Ap0
ρ and, for every path w = (e ,...,e ) from p to p in G and every a ∈ A ,a ∈ A ,
0 1 n 0 0 0 s(e1) i r(ei)
ρ(a u ...u a ) = ρ (a )U ...U ρ (a ).
0 e1 en n s(e1) 1 e1 en r(en) n
Proof. The proof amounts to a suitable application of Proposition 3.6. Let p,q ∈ V(G) and
let w = (e ,...,e ) be the unique geodesic path in T from p to q. Set U = U ...U and
1 n pq e1 en
observe that U∗ = U . For every q ∈ V(G) and every e ∈ E(G), we can define a representation
pq qp
π = U∗ ρ (.)U of A on H and a unitary V ∈ B(H ) by V = U U U . It is easily
q qp0 q qp0 q p0 e p0 e p0s(e) e r(e)p0
checked that these satisfy the hyptohesis of Proposition 3.6, yielding the result. (cid:3)
3.2. The reduced fundamental C*-algebra. Wenowturntotheconstruction ofthereduced
fundamental C*-algebra, which is more involved. The basic idea is to build a concrete repre-
sentation of the C*-algebras forming the graph together with unitaries satisfying the required
relations. To beabletocarry outthis construction, we will need anextraassumption. From now
on, we assume that for every e ∈ E(G), there exists a conditional expectation Es : A → Bs
e s(e) e
and we set Er = Es : A → Br.
e e r(e) e
GRAPHS OF QUANTUM GROUPS AND K-AMENABILITY 7
3.2.1. Path Hilbert modules. For every e ∈ E(G) let (Hs,πs,ηs) be the GNS construction associ-
e e e
ated tothe completely positive map s−1◦Es. This means that Hs is the right Hilbert B -module
e e e e
obtained by separation and completion of A with respect to the B -valued inner product
s(e) e
hx,yi = s−1◦Es(x∗y) for x,y ∈ A .
e e s(e)
The right action of an element b ∈ B is given by right multiplication by s (b) and the represen-
e e
tation
πs : A → L (Hs)
e s(e) Be e
is induced by the left multiplication. Finally, ηs : A → Hs is the standard linear map with
e s(e) e
dense range. Let ξs denote the image of 1 in Hs. The triple (Hs,πs,ηs) will be denoted
e A e e e e
(Hr,πr,ηr). Although it is not necessary, we will assume, for convenience and simplicity of
e e e
notations, that for every e ∈ E(G), the conditional expectations Es are GNS-faithful (i.e. that
e
the representations πs are faithful). This allows us to identify A with its image in L (Hs).
e s(e) Be e
We will also use, for every a ∈ A , the notation a for ηs(a) ∈ Hs. One should however keep
s(e) e e
in mind that a may be zero for some non-zero a. Let us also notice that the submodule ξs.B
e e
of Hs is orthogonally complemented. In fact, its orthogonal complement is the closure (Hs)◦ of
e b e
{a|a ∈A ,Es(a) = 0}. We thus have an orthogonal decomposition
s(e) be
Hs = (ξs.B )⊕(Hs)◦
e e e e
b
with (Hs)◦.Bs = (Hs)◦. Similarly, the orthogonal complement of ξr.B in Hr will be denoted
e e e e e e
(Hr)◦ .
e
We now turn to the construction of the Hilbert C*-module which will carry our faithful repre-
sentation of the fundamental C*-algebra. Let n > 1 and let w = (e ,...,e ) be a path in G. We
1 n
define Hilbert C*-modules K , K and K for 1 6 i6 n−1 by
0 n i
• K = Hs
0 e1
• If e 6= e , then K = Hs
i+1 i i ei+1
• If e = e , then K = (Hs )◦
i+1 i i ei+1
• K = A
n r(en)
For 0 6 i 6 n−1, K is a right Hilbert B -module and K will be seen as a right Hilbert
i ei+1 n
A -module. We can put compatible left module structures on these Hilbert C*-modules in
r(en)
order to make tensor products. In fact, for 1 6 i6 n−1, the map
ρ = πs ◦r :B → L (K )
i ei+1 ei ei Bei+1 i
yields a suitable action of B on K and left multiplication by r (b) for b ∈ B induces a
ei i en en
representation
ρ :B → L (K ).
n en Ar(en) n
We can now define a right Hilbert A -module
r(en)
H = K ⊗...⊗K
w 0 n
ρ1 ρn
endowed with a faithful left action of A which is induced by its action on K by left mul-
s(e1) 0
tiplication. This will be called a path Hilbert module. Let us describe more precisely the inner
product.
8 PIERREFIMAANDAMAURYFRESLON
Lemma 3.9. Let n > 1 and let w = (e ,...,e ) be a path in G. Let a = a ⊗··· ⊗ a and
1 n 0 n
b = b ⊗···⊗b be two elements in H . Set x = a∗b and, for 1 6 k 6 n, set
0 n w 0 0 0
b xk = a∗k(rek ◦s−ek1◦Esek(xk−1))bk. b
Then, ha,biHw = xn ∈ Ar(en).
Proof. The proof is by induction on n. For n = 1, we have w = (e) where e ∈E(G), a = a ⊗a
0 1
e
and b = b ⊗b , where a ,b ∈A and a ,b ∈A . By definition of H we have
0 1 0 0 s(e) 1 1 r(e) w
e b
b ha,biHw = ha1,ρ1(s−e1◦Ese(a∗0b0))b1iAr(e) = a∗1(re◦s−e1◦Ese(a∗0b0))b1 = x1.
Assume that the formula holds for a given n ≥ 1. Let w = (e ,...e ) be a path and fix
1 n+1
a = a ⊗···⊗a ,b = b ⊗···⊗b ∈H . Write
0 n+1 0 n+1 w
′
H = K ⊗... ⊗ K = H ⊗ A ,
b b w 0ρ1 ρn+1 n+1 w ρn+1 r(en+1)
where H′ is the Hilbert B -module H′ = K ⊗...⊗K . We have,
w en+1 w 0ρ1 ρn n
∗
ha,biHw = an+1ren+1 ha0⊗...⊗an,b0 ⊗...⊗bniH′w bn+1.
By definition of the inner product, we (cid:16)get, with w′ = (e ,...,e ), (cid:17)
b b1 n b
ha0⊗...⊗an,b0⊗...⊗bniH′w = s−en1+1 ◦Esen+1 ha0⊗...⊗an,b0⊗...⊗bniHw′ .
This concludes the proof using the induction hypothes(cid:16)is. (cid:17) (cid:3)
b b b b b
For any two vertices p ,q ∈ V(G), we define a right Hilbert A -module
0 q
H = H
p0,q w
w
M
where the sum runs over all paths w in G connecting p with q. By convention, when q = p ,
0 0
the sum also runs over the empty path, where H = A with its canonical Hilbert (A ,A )-
∅ p0 p0 p0
bimodule structure. We equip this Hilbert C*-module with the faithful left action of A which
p0
is given by the sum of its left actions on every H .
w
3.2.2. The C*-algebra. For every e∈ E(G) and p ∈ V(G), we can define an operator
up :H → H
e r(e),p s(e),p
which "adds the edge e on the left". To construct this operator, let w be a path in G from r(e)
to p and let ξ ∈H .
w
• If p = r(e) and w is the empty path, then up(ξ) = ξs⊗ξ ∈ H .
e e (e)
• If n = 1, w = (e ), ξ = a⊗ξ′ with a ∈ A and ξ′ ∈ A , then
1 s(e1) p
– If e 6= e, up(ξ) = ξs⊗ξ ∈ H .
1 e e (e,e1)
– If e1 = e, upe(ξ) = b r ◦ξess−⊗1ξ(a)ξ′ ∈∈HA(e,e1) iiff aa∈∈(HBess1).◦,
• If n > 2, w = (e1,...,e(cid:26)n), ξe1= ae⊗1 ξ′ with a ∈ Ap s(e1) anbd ξ′ ∈ Ke11⊗ρ2...ρ⊗nKn, then
– If e 6= e, up(ξ) = ξs⊗ξ ∈ H .
1 e e b (e,e1,...,en)
GRAPHS OF QUANTUM GROUPS AND K-AMENABILITY 9
ξs⊗ξ ∈ H if a ∈ (Hr)◦,
– If e = e, up(ξ) = e (e,e1,...,en) e
1 e r ◦s−1(a)ξ′ ∈ H if a ∈ Bs .
(cid:26) e1 e1 (e2,...,en) e1
p
One easily checks that the operators u commute with the right abctions of A on H and
e p r(e),p
H andextendtounitaryoperators(stilldenotedup)inL (H ,H )suchthat(up)∗ =
s(e),p e Ap r(e),p s(e),p e
p
u . Moreover, for every e ∈ E(G) and every b ∈ B , the definition implies that
e e
ups (b)up = r (b)
e e e e
as operators in L (H ). Let w = (e ,...,e ) be a path in G and let p ∈V(G), we set
Ap r(e),p 1 n
up =up ...up ∈ L (H ,H ).
w e1 en Ap r(en),p s(e1),p
We are now ready to define the reduced fundamental C*-algebra.
Definition 3.10. Let (G,(A ) ,(B ) ) be a graph of C*-algebras and let p,p ∈ V(G). The
q q e e 0
reduced fundamental C*-algebra rooted in p with base p is the C*-algebra
0
πp(G,(A ) ,(B ) ,p )= h(up)∗A up|q ∈ V(G),w,z paths from q to p i⊂ L (H ).
1 q q e e 0 z q w 0 Ap p0,p
If the root p is equal to the base p, we will simply call it the reduced fundamental C*-algebra in
0
p . We will use the shorthand notation P (p ) (and P(p ) when p = p ) to denote the reduced
0 p 0 0 0
fundamental C*-algebra in the sequel.
Remark 3.11. The above definition may seem unsatisfying because of the two arbitrary vertices
involved. However, this will give many natural representations of the reduced C*-algebra which
willbeneeded later on. This alsogives amoretractable objectwhenitturns tomaking products
or computing norms.
Remark 3.12. BecausethegraphG isconnected, thepreviousconstructiondoesnotreallydepend
on p . In fact, let p,p ,q be three vertices of G and let T be a maximal subtree in G. If g
0 0 0 q0p0
denotes the unique geodesic path in T from q to p , then we have an isomorphism
0 0
p
Φ : P (p ) → P (q )
T,p0,q0 p 0 p 0
which is given by
x 7→ up x(up )∗.
gq0p0 gq0p0
Note, however, that there is no truly canonical way to identify these C*-algebras.
Example 3.13. Carrying out this construction with the graphs of Examples 3.4 and 3.5, one
recovers respectively the reduced amalgamated free product construction of [Voi85] and the
reduced HNN extension construction of [Ued05, Sec 7.4].
3.2.3. The quotient map. We now investigate the link beteween the reduced fundamental C*-
algebra and the maximal one. From now on, we fix two vertices p ,p ∈ V(G) and consider the
0
C*-algebra P (p ). Let T be a maximal subtree in G. As before, given a vertex q ∈ V(G), we
p 0
denote by g the unique geodesic path in T from q to p. For every e ∈E(G), we define a unitary
qp
p
operator w ∈ P (p ) by
e p 0
wp = (up )∗up .
e gs(e)p (e,gr(e)p)
p
For every q ∈ V(G), we define a unital faithful ∗-homomorphism π :A → P (p ) by
q,p0 q p 0
πp (a) = (up )∗aup
q,p0 gqp0 gqp0
Observe that the following relations hold: