A cohomological description of property (T) for quantum groups

Abstract

We prove a Delorme-Guichardet type theorem for discrete quantum groups
expressing property (T) of the quantum group in question in terms of its first
cohomology groups. As an application, we show that the first L^2-Betti number
of a discrete property (T) quantum group vanishes.

Full text

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arXiv:1003.5181v1 [math.OA] 26 Mar 2010
A DELORME-GUICHARDET THEOREM FOR QUANTUM
GROUPS
DAVID KYED
Abstract. We prove a Delorme-Guichardet Theorem for discrete quantum
groups, expressing property (T) of a quantum group in terms of vanishing of
its first cohomology groups. As an application, we show that the first L2-Betti
number of a discrete property (T) quantum group vanishes.
0. Introduction
The notion of property (T) was introduced by Kazhdan in his influential paper
[Kaˇ z67] and has since then played a prominent role in a variety of mathematical
disciplines, including topology, ergodic theory and operator algebras. Recall, that
a discrete group Γ has property (T) if any unitary representation of Γ with almost
invariant vectors has a non-zero invariant vector. For details on these definitions
and proofs of the classical theorems mentioned below we refer the reader to book
[BdlHV08] by Bekka, de la Harpe and Valette. One reason why property (T) is
an important notion is that it allows many different descriptions. Firstly, it can
be described using the functions on Γ by means of the following theorem.
Theorem (de la Harpe-Valette). The group Γ has property (T) if and only if any
sequence of positive definite functions ϕn: Γ → C with ϕn(e) = 1 and converging
pointwise to 1, converges uniformly to the constant function 1.
Property (T) can also be described in terms of the first cohomology of Γ which,
among other things, provides a link between property (T) and Serre’s property
(FA). The precise cohomological description is given by the celebrated Delorme-
Guichardet Theorem.
Theorem (Delorme-Guichardet). The group Γ has property (T) if and only if the
first group cohomology H1(Γ,H) vanishes for all Hilbert spaces H with a unitary
Γ-action.
Recently Fima introduced the notion of property (T) for the class of discrete
quantum groups in the paper [Fim08] in which he also proved that several classical
facts (see Theorem 1.7) can be transfered (mutatis-mutandi) from the theory of
discrete groups to the more general setting of discrete quantum groups. The
2010 Mathematics Subject Classification. 16T05, 46L52.
Key words and phrases. Property (T), Quantum groups, L2-Betti numbers.
Research supported by The Danish Council for Independent Research | Natural Sciences.
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2DAVID KYED
aim of the present paper is to show how the two classical theorems stated above
can be generalized to the quantum group context. IfˆG is a discrete quantum
group and G is its compact dual, we denote by (Pol(G),∆,S,ε) the associated
Hopf ∗-algebra of matrix coefficients and by C(Gu) the universal C∗-completion
of Pol(G). These objects will be introduced and discussed in greater detail in
Section 1 where we also elaborate on Fima’s definition of property (T) and the
results obtained in [Fim08]. The first main result (Theorem 3.1) of the present
paper is a quantum group analogue of the result of de la Harpe and Valette and
its precise statement is as follows.
Theorem. The discrete quantum groupˆG has property (T) if and only if any
net of states ϕi: C(Gu) → C converging pointwise to the counit ε converges in
the uniform norm.
Secondly, we prove in Theorem 5.1 the following quantum group version of the
Delorme-Guichardet Theorem.
Theorem. The discrete quantum groupˆG has property (T) if and only if the
following holds: for every ∗-representation π: Pol(G) → B(H) on a Hilbert space
H the first Hochschild cohomology of Pol(G) with values in the bimoduleπHε
vanishes.
The relevant definitions concerning the first Hochschild cohomology are given
in Section 4. Along the way, we obtain in Theorem 5.1 a characterization of
property (T) in terms of conditionally negative functionals ψ: Pol(G) → C that
parallels the classical description stating that a discrete group Γ has property (T)
if and only if any conditionally negative function ψ: Γ → C is bounded. Finally,
using the quantum group version of the Delorme-Guichardet Theorem, we obtain
in Corollary 6.1 the following extension of a well known result for groups.
Corollary. IfˆG has property (T) then its first L2-Betti number vanishes.
The paper is organized as follows.
Structure. The first section provides the reader with the necessary background
concerning the theory of compact quantum groups and their discrete duals and
discusses the definition of property (T) for discrete quantum groups. In Section 2
we show how property (T) can be described in terms of the dual compact quantum
group and use this description to give a spectral interpretation of property (T).
Section 3 is devoted to the proof of the characterization of property (T) in terms
of convergence of states on the associated universal C∗-algebra, and in Section 4
the proof of the quantum Delorme-Guichardet Theorem is given. In the sixth and
final section we show how the results obtained can be used to derive information
about the L2-invariants of the quantum group in question.

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A DELORME-GUICHARDET THEOREM FOR QUANTUM GROUPS3
Notation. Throughout the paper, the symbol ⊙ will be used to denote algebraic
tensor products while the symbol ¯ ⊗ will be used to denote tensor products in
the category of Hilbert spaces or in the category of von Neumann algebras. All
tensor products between C∗-algebras are assumed minimal/spatial and these will
be denoted by the symbol ⊗. All Hilbert spaces are assumed to be complex
and their inner products are assumed linear in the first variable. Furthermore,
∗-representations of unital algebras on Hilbert spaces will always be assumed to
be unit-preserving.
Acknowledgements. The author wishes to thank Pierre Fima, Ryszard Nest, Jesse
Peterson and Roland Vergnioux for discussions revolving around the notion of
property (T). The work presented in the paper was initiated during the authors
stay at the Hausdorff Research Institute for Mathematics and it is a pleasure to
thank the organizers, Wolfgang L¨ uck and Nicolas Monod, as well as the partici-
pants of the trimester program on Rigidity.
1. Preliminaries on quantum groups
We choose here the approach to compact quantum groups developed by Woro-
nowicz in [Wor87a], [Wor87b] and [Wor98]. Thus, a compact quantum group G
consists of a (not necessarily commutative) separable, unital C∗-algebra C(G)
together with a unital, coassociative ∗-homomorphism ∆: C(G) → C(G)⊗C(G)
satisfying a certain density condition. The map ∆ is referred to as the comul-
tiplication. Such a quantum group possesses a unique Haar state; i.e. a state
h: C(G) → C such that
(id⊗h)∆a = h(a)1 = (h ⊗ id)∆(a) for every a ∈ C(G).
The GNS-construction applied to the Haar state yields a separable Hilbert space
L2(G) together with a representation λ: C(G) → B(L2(G)) and a linear map
Λ: C(G) → L2(G) with dense image. In general, the Haar state need not be
faithful and hence λ might have a kernel. We denote by C(Gred) the image
λ(C(G)). This C∗-algebra inherits a quantum group structure from G and the
comultiplication ∆redon C(Gred) is implemented by the so-called multiplicative
unitary W ∈ B(L2(G)¯ ⊗L2(G)) given by
W∗(Λ(a) ⊗ Λ(b)) = Λ ⊗ Λ(∆(b)(a ⊗ 1)).
The statement that W implements ∆redmeans that ∆red(λ(a)) = W∗(1⊗λ(a))W
for every a ∈ C(G). One notes that the right hand side of this formula makes
sense for any a ∈ B(L2(G)) and it turns out that the enveloping von Neumann
algebra L∞(G) = C(Gred)′′is turned into a compact von Neumann algebraic
quantum group (see [KV03]) when endowed with this map as comultiplication.
A unitary corepresentation of G on a Hilbert space H is of a unitary element
u ∈ M(K(H) ⊗ C(G)) such that (id⊗∆)u = u(12)u(13). Here K(H) denotes

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4 DAVID KYED
the compact operators on H, M(K(H) ⊗ C(G)) is the multiplier algebra of
K(H) ⊗ C(G) and the subscripts (12) and (13) is the standard leg-numbering
convention. Representation theoretic notions from the theory of compact groups,
such as direct sums, tensor products, intertwiners and irreducibility, have natu-
ral counterparts in the corepresentation theory for compact quantum groups. In
particular the following important theorem holds true.
Theorem 1.1 (Woronowicz). Any irreducible unitary corepresentation of G is
finite dimensional and an arbitrary unitary corepresentation decomposes as a
direct sum irreducible ones.
We denote by Irred(G) the set of equivalence classes of irreducible, unitary
corepresentations of G. The separability assumption on C(G) together with the
quantum Peter-Weyl theorem [Wor98] ensures that Irred(G) is a countable set.
We label its elements by an auxiliary countable set I and choose for each α ∈ I
a Hilbert space Hαand a concrete representative uα∈ B(Hα) ⊗ C(G). Abusing
notation slightly, we shall often identify the index α with the corresponding class
of uα. Fix an orthonormal basis {e1,...,enα} for Hαand consider the corre-
sponding functionals ωij: B(Hα) → C given by ωij(T) = ?Tei|ej?. The matrix
coefficients of uα, relative to the chosen basis, are then defined as
uα
ij= (ωij⊗ id)uα∈ C(G).
It turns out that these matrix coefficients are linearly independent and that their
linear span constitutes a dense ∗-subalgebra Pol(G) of C(G). Furthermore, the
comultiplication descends to a comultiplication ∆: Pol(G) → Pol(G) ⊙ Pol(G)
and with this comultiplication Pol(G) becomes a Hopf ∗-algebra; i.e. there exists
an antipode S: Pol(G) → Pol(G) as well as a counit ε: Pol(G) → C satisfying
the standard Hopf ∗-algebra relations [KS97]. The fact that Pol(G) is spanned
by matrix coefficients arising from finite dimensional, unitary corepresentations
of G also ensures that the relation
?a?u= sup{?π(x)? | π: Pol(G) → B(H) a ∗-representation}
defines a C∗-norm ? · ?uon Pol(G). The C∗-completion of Pol(G) with respect
to this norm is called the universal C∗-algebra associated with G and is denoted
C(Gu). By definition of ?·?u, the comultiplication extends to a comultiplication
∆u: C(Gu) → C(Gu) ⊗ C(Gu) turning C(Gu) into a compact quantum group.
Note that the ∗-representations of C(Gu) are in one-to-one correspondence with
the ∗-representations of Pol(G) via restriction/extension.
Example 1.2. The fundamental example, on which the general definition is mod-
eled, is obtained by considering a compact, second countable, Hausdorff topologi-
cal group G and its commutative C∗-algebra C(G) of continuous, complex valued
functions. In this case the comultiplication is the Gelfand dual of the multipli-
cation map G × G → G and the Haar state is given by integration against the
unique Haar probability measure µ on G. The GNS-space therefore identifies

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A DELORME-GUICHARDET THEOREM FOR QUANTUM GROUPS5
with L2(G,µ) and the representation λ with the action of C(G) on L2(G,µ)
by pointwise multiplication. Similarly, the von Neumann algebra identifies with
L∞(G,µ) and the Hopf ∗-algebra Pol(G) is the subalgebra of C(G) generated by
matrix coefficients arising from irreducible, unitary representations of G. The
antipode in Pol(G) is the Gelfand dual of the inversion map and the counit is
given by evaluation at the identity element in G.
In the previous example there is no real difference between the reduced and
universal version of the compact quantum group. The next example, however,
will illustrate this difference more clearly.
Example 1.3. Consider a countable, discrete group Γ. Denote by C∗
reduced group C∗-algebra acting on ℓ2(Γ) via the left regular representation and
define a comultiplication on group elements by ∆redγ = γ⊗γ. This turns C∗
into a compact quantum group whose Haar state is given by the natural trace
on C∗
respectively, with ℓ2(Γ) and the left regular representation, and the enveloping
von Neumann algebra is therefore nothing but the group von Neumann algebra
L(Γ). Each element in Γ is a one-dimensional corepresentation for this quantum
group and the Hopf ∗-algebra therefore identifies with the complex group algebra
CΓ. Thus, the universal C∗-algebra is, by definition, equal to the maximal group
C∗-algebra C∗
Remark 1.4. The three C∗-algebras C(G),C(Gred) and C(Gu), together with
their comultiplications, can be thought of as ”different pictures of the same quan-
tum group”, each having its advantages and disadvantages. For instance, whereas
the the Haar state is always faithful on C(Gred) this is in general not the case
on C(Gu) and, conversely, the counit is always well defined on all of C(Gu) but
not necessarily on C(Gred). The latter difference is the fundamental observa-
tion leading to the notion of (co)-amenability for quantum groups as studied in
[BMT01].
Any compact quantum group G has a dual quantum groupˆG of so-called
discrete type. As in the compact case,ˆG comes with both a C∗-algebra c0(ˆG)
and a von Neumann algebra ℓ∞(ˆG) defined, respectively, as
red(Γ) its
red(Γ)
red(Γ). Hence the GNS-space and GNS-representation can be identified,
u(Γ).
c0(ˆG) =
c0
?
α∈I
B(Hα) and ℓ∞(ˆG) =
ℓ∞
?
α∈I
B(Hα).
In the discrete picture we will primarily be working with the von Neumann algebra
ℓ∞(ˆG). This algebra is endowed with a natural comultiplicationˆ∆: ℓ∞(ˆG) →
ℓ∞(ˆG)¯ ⊗ℓ∞(ˆG) arising from the quantum group structure on G. Since c0(ˆG) is a
direct sum of finite dimensional C∗-algebras we have isomorphisms
ℓ∞(ˆG)¯ ⊗B(H) ≃ M(c0(ˆG) ⊗ B(H)) ≃
ℓ∞
?
α∈I
B(Hα) ⊗ B(H)