Graphs of quantum groups and K-amenability
Building on a construction of J-P. Serre, we associate to any graph of C*-algebras a maximal and a reduced fundamental C*-algebra and use this theory to construct the fundamental 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 fundamental quantum group. To illustrate the properties of this construction, we prove that if all the vertex qantum groups are amenable, then the fundamental quantum group is K-amenable. This generalizes previous results of P. Julg, A. Valette, R. Vergnioux and the first author.
Key words and phrases:Quantum groups, quantum Bass-Serre tree, K-amenability
2010 Mathematics Subject Classification:46L09, 46L65
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 been a very active field of research, in particular in relation with algebraic and geometric problems, culminating in the celebrated Baum-Connes conjecture (see for example [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-theory of any reduced crossed-product by the group is equal to the K-theory of the corresponding 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. Skandalis in [Ska88] and further studied by E. Germain who proved in [Ger96] that a free product of amenable C*-algebras is K-nuclear. However, we will concentrate in the present paper on the 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 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 to include the additional data of the groups and 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 extensions of amenable discrete quantum groups in [Fim13]. The use of quantum Bass-Serre trees 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 result of [JV84]. Note that this gives a large class of K-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 and a reduced fundamental C*-algebra and give some structure results. This section is rather long 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 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.
2.1. 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 and are supposed to be linear in the second variable. For two Hilbert spaces and , will denote the set of bounded linear maps from to and . For a C*-algebra and Hilbert -modules a nd , we denote by the set of bounded adjointable -linear operators from to and .
We will also use the following terminology : if is a Hilbert -module and is a state, the GNS construction of is the triple , where is the Hilbert space obtained by separation and completion of with respect to the scalar product , is the canonical linear map with dense range and is the induced unital -homomorphism. Note that and are faithful as soon as is. Observe also that if is another Hilbert -module and if denotes the GNS construction of , then we also have an obvious induced linear map which respects the adjoint and the composition (if we take a third Hilbert -module).
If is a graph in the sense of [Ser77, Def 2.1], its vertex set will be denoted and its edge set will be denoted . For we denote by and respectively the source and range of and by the inverse edge of . An orientation of is a partition such that if and only if .
Finally, we will always denote by the identity map.
2.2. Compact quantum groups
A compact quantum group is a pair where is a unital C*-algebra and is a unital -homomorphism such that
and the linear span of as well as the linear span of are dense in .
Theorem 2.2 (Woronowicz).
Let be a compact quantum group. There is a unique state , called the Haar state of , such that for every ,
The Haar state need not be faithful. Let be the C*-algebra obtained by the GNS construction of the Haar state. is called the reduced C*-algebra of the compact quantum group . By the invariance properties of the Haar state, the coproduct induces a coproduct on which turns it into a compact quantum group called the reduced form of . The Haar state on the reduced form of is faithful by construction.
Let be a compact quantum group. A representation of of dimension is a matrix such that for all ,
A representation is called unitary if is a unitary. An intertwiner between two representations and of dimension respectively and is a linear map such that . If there exists a unitary intertwiner between and , they are said to be unitarily equivalent. A representation is said to be irreducible if its only self-intertwiners are the scalar multiples of the identity. The tensor product of the two representations and is the representation
Theorem 2.4 (Woronowicz).
Every unitary representation of a compact quantum group is unitarily equivalent to a direct sum of irreducible unitary representations.
Let be the set of equivalence classes of irreducible unitary representations of and, for , denote by a representative of . The linear span of the elements for forms a Hopf--algebra which is dense in . Its enveloping C*-algebra is denoted and it has a natural quantum group structure called the maximal form of . By universality, there is a surjective -homomorphism
which intertwines the coproducts.
The C*-algebras and should be thought of as the reduced and maximal C*-algebras of the dual discrete quantum group . This point of view justifies the terminology "discrete quantum groups" used in the paper.
admits a one-dimensional representation , called the trivial representation (or the counit) and defined by for all and every . The counit is the unique unital -homomorphism such that .
A compact quantum group is said to be co-amenable if is an isomorphism. We will equivalently say that is amenable.
Like in the classical case, co-amenability has several equivalent characterizations, we only give the one which will be needed in the sequel (see [BMT01, Thm 3.6] for a proof).
A compact quantum group is co-amenable if and only if the trivial representation factors through .
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 be a compact quantum group. The following are equivalent
There exists such that in .
The element in invertible in .
In any of those two equivalent situations, we will say that is K-amenable.
3. Graphs of C*-algebras
In this section we give the general construction of a maximal and a reduced fundamental C*-algebra associated to a graph of C*-algebras.
A graph of C*-algebras is a tuple
is a connected graph.
For every and every , and are unital C*-algebras.
For every , .
For every , is a unital faithful -homomorphism.
For every , we set , and .
The notation will always be simplified in .
3.1. The maximal fundamental C*-algebra
Like in the case of free products, the definition of the maximal fundamental C*-algebra is quite obvious and simple. However, it requires the choice of a maximal subtree of the graph (which is implicit in the case of free products since the graph is already a tree, see Example 3.4).
Let be a graph of C*-algebras and let be a maximal subtree of . The maximal fundamental C*-algebra with respect to is the universal C*-algebra generated by the C*-algebras for and by unitaries for such that
For every , .
For every and every , .
For every , .
This C*-algebra will be denoted .
It is not obvious that this C*-algebra is not (i.e. that the relations admit a non-trivial representation). With natural additional assumptions, the non-triviality will be proved by the construction of the reduced fundamental C*-algebra and it will be clear that the inclusions of in the maximal fundamental C*-algebra are faithful.
Let and be two C*-algebras and let be a C*-algebra together with injective -homomorphisms for . Let be the graph with two vertices and and two edges and , where and . This graph is obviously a tree. Setting , , and yields a graph of C*-algebras whose maximal fundamental C*-algebra with respect to is the maximal free product of and amalgamated over .
Let be a C*-algebra, a C*-subalgebra of and an injective -homomorphism. Let be a graph with one vertex and two edges and , where is a loop from to . Obviously, the only maximal subtree of is the graph with one vertex and no edge. Setting , , and yields a graph of C*-algebras whose maximal fundamental C*-algebra with respect to is the maximal HNN extension as defined in [Ued05, Rmk 7.3].
By construction, the maximal fundamental C*-algebra of satisfies the following universal property.
Let be a graph of C*-algebras, let be a maximal subtree of and let be a Hilbert space. Assume that for every , we have a representation of on and that for every , we have a unitary such that , for all and for every ,
Then, there is a unique representation of on such that for every and every ,
Let . Define to be the linear span of and elements of the form where is a path in from to , and for . Observe that is a dense -subalgebra of . Indeed, it suffices to show that it contains for every and for every . Let and . Let be the unique geodesic path in from to . Since , we have for every . Hence, . Now, let and let (resp. ) be the geodesic path in from to (resp. ). Then, .
We will need in the sequel the following slightly more general version of the universal property.
Let be a graph of C*-algebras, let be a maximal subtree of and let . Assume that for every , there is a Hilbert space together with a representation of and that, for every , there is a unitary such that and, for every ,
Then, there exists a unique representation of on such that and, for every path from to in and every ,
The proof amounts to a suitable application of Proposition 3.6. Let and let be the unique geodesic path in from to . Set and observe that . For every and every , we can define a representation of on and a unitary by . It is easily checked that these satisfy the hyptohesis of Proposition 3.6, yielding the result. ∎
3.2. The reduced fundamental C*-algebra
We now turn to the construction of the reduced fundamental C*-algebra, which is more involved. The basic idea is to build a concrete representation of the C*-algebras forming the graph together with unitaries satisfying the required relations. To be able to carry out this construction, we will need an extra assumption. From now on, we assume that for every , there exists a conditional expectation and we set .
3.2.1. Path Hilbert modules
For every let be the GNS construction associated to the completely positive map . This means that is the right Hilbert -module obtained by separation and completion of with respect to the -valued inner product
The right action of an element is given by right multiplication by and the representation
is induced by the left multiplication. Finally, is the standard linear map with dense range. Let denote the image of in . The triple will be denoted . Although it is not necessary, we will assume, for convenience and simplicity of notations, that for every , the conditional expectations are GNS-faithful (i.e. that the representations are faithful). This allows us to identify with its image in . We will also use, for every , the notation for . One should however keep in mind that may be zero for some non-zero . Let us also notice that the submodule of is orthogonally complemented. In fact, its orthogonal complement is the closure of . We thus have an orthogonal decomposition
with . Similarly, the orthogonal complement of in will be denoted .
We now turn to the construction of the Hilbert C*-module which will carry our faithful representation of the fundamental C*-algebra. Let and let be a path in . We define Hilbert C*-modules , and for by
If , then
If , then
For , is a right Hilbert -module and will be seen as a right Hilbert -module. We can put compatible left module structures on these Hilbert C*-modules in order to make tensor products. In fact, for , the map
yields a suitable action of on and left multiplication by for induces a representation
We can now define a right Hilbert -module
endowed with a faithful left action of which is induced by its action on by left multiplication. This will be called a path Hilbert module. Let us describe more precisely the inner product.
Let and let be a path in . Let and be two elements in . Set and, for , set
The proof is by induction on . For , we have where , and , where and . By definition of we have
Assume that the formula holds for a given . Let be a path and fix . Write
where is the Hilbert -module . We have,
By definition of the inner product, we get, with ,
This concludes the proof using the induction hypothesis. ∎
For any two vertices , we define a right Hilbert -module
where the sum runs over all paths in connecting with . By convention, when , the sum also runs over the empty path, where with its canonical Hilbert -bimodule structure. We equip this Hilbert C*-module with the faithful left action of which is given by the sum of its left actions on every .
3.2.2. The C*-algebra
For every and , we can define an operator
which "adds the edge on the left". To construct this operator, let be a path in from to and let .
If and is the empty path, then .
If , , with and , then
If , .
If , , with and , then
If , .
One easily checks that the operators commute with the right actions of on and and extend to unitary operators (still denoted ) in such that . Moreover, for every and every , the definition implies that
as operators in . Let be a path in and let , we set
We are now ready to define the reduced fundamental C*-algebra.
Let be a graph of C*-algebras and let . The reduced fundamental C*-algebra rooted in with base is the C*-algebra
If the root is equal to the base , we will simply call it the reduced fundamental C*-algebra in . We will use the shorthand notation (and when ) to denote the reduced fundamental C*-algebra in the sequel.
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 will be needed later on. This also gives a more tractable object when it turns to making products or computing norms.
Because the graph is connected, the previous construction does not really depend on . In fact, let be three vertices of and let be a maximal subtree in . If denotes the unique geodesic path in from to , then we have an isomorphism
which is given by
Note, however, that there is no truly canonical way to identify these C*-algebras.
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 and consider the C*-algebra . Let be a maximal subtree in . As before, given a vertex , we denote by the unique geodesic path in from to . For every , we define a unitary operator by
For every , we define a unital faithful -homomorphism by
Observe that the following relations hold:
for every ,
for every ,
for every , .
The first and the last relations are clear from the definitions. To check the second one, observe that if , then the path is a cycle in . This means that either or . In both cases we get .
Thus, we can apply the universal property of Proposition 3.6 to get a surjective -homomorphism
3.2.4. Reduced operators
Like in the case of groups, we have a notion of reduced element.
Let be a graph of C*-algebras and let . Let be of the form , where is a path in from to , and, for , . The operator is said to be reduced (from to ) if for all such that , we have .
Let be a path from to . Observe that any reduced operator of the form is in and that the linear span of and the reduced operators from to is a dense -subalgebra of . Indeed, we can write where , and, for ,