aa r X i v : . [ m a t h . OA ] O c t THE PLANCHEREL FORMULA FOR COUNTABLEGROUPS
BACHIR BEKKA
Abstract.
We discuss a Plancherel formula for countable groups,which provides a canonical decomposition of the regular represen-tation of such a group Γ into a direct integral of factor represen-tations. Our main result gives a precise description of this decom-position in terms of the Plancherel formula of the FC-center Γ fc ofΓ (that is, the normal sugbroup of Γ consisting of elements witha finite conjugacy class); this description involves the action of anappropriate totally disconnected compact group of automorphismsof Γ fc . As an application, we determine the Plancherel formula forlinear groups. In an appendix, we use the Plancherel formula toprovide a unified proof for Thoma’s and Kaniuth’s theorems whichrespectively characterize countable groups which are of type I andthose whose regular representation is of type II. Introduction
Given a second countable locally compact group G, a fundamentalobject to study is its unitary dual space b G , that is, the set irreducibleunitary representations of G up to unitary equivalence. The space b G carries a natural Borel structure, called the Mackey Borel structure(see [Mac57, §
6] or [Dix77, § b G is consideredas being possible only if b G is a standard Borel space; according toGlimm’s celebrated theorem ([Gli61]), this is the case if and only if G is of type I in the following sense.Recall that a von Neumann algebra is a self-adjoint subalgebra of L ( H ) which is closed for the weak operator topology of L ( H ), where H is a Hilbert space. A von Neumann algebra is a factor if its centeronly consists of the scalar operators.Let π be a unitary representation of G in a Hilbert space H (aswe will only consider representations which are unitary, we will oftendrop the adjective “unitary”). The von Neumann subalgebra of L ( H ) Date : October 27, 2020.2000
Mathematics Subject Classification. generated by π ( G ) coincides with the bicommutant π ( G ) ′′ of π ( G ) in L ( H ); we say that π is a factor representation if π ( G ) ′′ is a factor. Definition 1.
The group G is of type I if, for every factor representa-tion π of G , the factor π ( G ) ′′ is of type I, that is, π ( G ) ′′ is isomorphicto the von Neumann algebra L ( K ) for some Hilbert space K ; equiva-lently, the Hilbert space H of π can written as tensor product K ⊗ K ′ of Hilbert spaces in such a way that π is equivalent to σ ⊗ I K ′ for anirreducible representation σ of G on K . Important classes of groups are known to be of type I, such as semi-simple or nilpotent Lie groups. A major problem in harmonic analysisis to decompose the left regular representation λ G on L ( G, µ G ) fora Haar measure µ G as a direct integral of irreducible representations.When G is of type I and unimodular, this is the content of the classicalPlancherel theorem: there exist a unique measure µ on b G and a uni-tary isomorphism between L ( G, µ G ) and the direct integral of Hilbertspaces R ⊕ b G ( H π ⊗ H π ) dµ ( π ) which transforms λ G into R ⊕ b G ( π ⊗ I H π ) dµ ( x ),where H π is the conjugate of the Hilbert space H π of π ; in particular,we have a Plancherel formula k f k = Z b G Tr( π ( f ) ∗ π ( f )) dµ ( π ) for all f ∈ L ( G, µ G ) ∩ L ( G, µ G ) , where k f k is the L -norm of f, π ( f ) is the value at f of the naturalextension of π to a representation of L ( G, µ G ), and Tr denotes thestandard trace on L ( H π ); for all this, see [Dix77, 18.8.1].When G is not type I, λ G usually admits several integral decompo-sitions into irreducible representations and it is not possible to singleout a canonical one among them. However, when G is unimodular, λ G does admit a canonical integral decomposition into factor representa-tions representations; this the content of a Plancherel theorem whichwe will discuss in the case of a discrete group (see Theorem A).Let Γ be a countable group. As discussed below (see Theorem E), Γis usually not of type I. In order to state the Plancherel theorem for Γ , we need to replace the dual space b Γ by the consideration of Thoma’sdual space Ch(Γ) which we now introduce.Recall that a function t : Γ → C is of positive type if the complex-valued matrix ( t ( γ − j γ i )) ≤ i,j ≤ n is positive semi-definite for any γ , . . . , γ n in ΓA function of positive type t on Γ which is constant on conjugacyclasses and normalized (that is, t ( e ) = 1) will be called a trace on G .The set of traces on Γ will be denoted by Tr(Γ). LANCHEREL FORMULA FOR COUNTABLE GROUPS 3
Let t ∈ Tr(Γ) and ( π t , H t , ξ t ) be the associated GNS triple (see[BHV08, C.4]). Then τ t : π (Γ) ′′ → C , defined by τ t ( T ) = h T ξ t | ξ t i is a trace on the von Neumann algebra π t (Γ) ′′ , that is, τ t ( T ∗ T ) ≥ τ t ( T S ) = τ t ( ST ) for all T, S ∈ π t (Γ) ′′ ; moreover, τ t is faithful inthe sense that τ t ( T ∗ T ) > T ∈ π t (Γ) ′′ , T = 0 . Observe that τ t ( π ( f )) = t ( f ) for f ∈ C [Γ] , where t denotes the linear extension of t to the group algebra C [Γ] . The set Tr(Γ) is a convex subset of the unit ball of ℓ ∞ (Γ) which iscompact in the topology of pointwise convergence. An extreme pointof Tr(Γ) is called a character of Γ; we will refer to Ch(Γ) as Thoma’sdual space .Since Γ is countable, Tr(Γ) is a compact metrizable space and Ch(Γ)is easily seen to be a G δ subset of Tr(Γ). So, in contrast to b Γ, Thoma’sdual space Ch(Γ) is always a standard Borel space.An important fact is that Tr(Γ) is a simplex (see [Tho64, Satz 1] or[Sak71, 3.1.18]); by Choquet theory, this implies that every τ ∈ Tr(Γ)can be represented as integral τ = R Ch(Γ) tdµ ( t ) for a unique probabilitymeasure µ on Ch(Γ) . As we now explain, the set of characters of Γ parametrizes the factorrepresentations of finite type of Γ , up to quasi-equivalence; for moredetails, see [Dix77, § § π and π of Γ are quasi-equivalentif there exists an isomorphism Φ : π (Γ) ′′ → π (Γ) ′′ of von Neumannalgebras such that Φ( π ( γ )) = π ( γ ) for every γ ∈ Γ . Let t ∈ Ch(Γ) and π t the associated GNS representation. Then π t (Γ) ′′ is a factor of finite type. Conversely, let π be a representationof Γ such that π (Γ) ′′ is a factor of finite type and let τ be the uniquenormalized trace on π (Γ) ′′ . Then t := τ ◦ π belongs to Ch(Γ) and onlydepends on the quasi-equivalence class [ π ] of π. The map t → [ π t ] is a bijection between Ch(Γ) and the set of quasi-equivalence classes of factor representations of finite type of Γ . The following result is a version for countable groups of a Planchereltheorem due to Mautner [Mau50] and Segal [Seg50] which holds moregenerally for any unimodular second countable locally compact group;its proof is easier in our setting and will be given in Section 3 for theconvenience of the reader.
Theorem A. ( Plancherel theorem for countable groups ) Let Γ be a countable group. There exists a probability measure µ on Ch(Γ) ,a measurable field of representations t ( π t , H t ) of Γ on the standardBorel space Ch(Γ) , and an isomorphism of Hilbert spaces between ℓ (Γ) BACHIR BEKKA and R ⊕ Ch(Γ) H t dµ ( t ) which transforms λ Γ into R ⊕ Ch(Γ) π t dµ ( t ) and has thefollowing properties: (i) π t is quasi-equivalent to the GNS representation associated to t ; in particular, the π t ’s are mutually disjoint factor represen-tations of finite type, for µ -almost every t ∈ Ch(Γ) ; (ii) the von Neumann algebra L (Γ) := λ Γ (Γ) ′′ is mapped onto thedirect integral R ⊕ Ch(Γ) π t (Γ) ′′ dν ( t ) of factors; (iii) for every f ∈ C [Γ] , the following Plancherel formula holds: k f k = Z Ch(Γ) τ t ( π t ( f ) ∗ π t ( f )) dµ ( t ) = Z Ch(Γ) t ( f ∗ ∗ f ) dµ ( t ) . The measure µ is the unique probability measure on Ch(Γ) such thatthe Plancherel formula above holds.
The probability measure µ on Ch(Γ) from Theorem A is called the Plancherel measure of Γ.
Remark 2.
The Plancherel measure gives rise to what seems to bean interesting dynamical system on Ch(Γ) involving the group Aut(Γ)of automorphisms of Γ . We will equip Aut(Γ) with the topology ofpointwise convergence on Γ , for which it is a totally disconnectedtopological group. The natural action of Aut(Γ) on Ch(Γ), given by t g ( γ ) = t ( g − ( γ )) for g ∈ Aut(Γ) and t ∈ Ch(Γ) , is clearly continuous.Since the induced action of Aut(Γ) on ℓ (Γ) is isometric, the followingfact is an immediate consequence of the uniqueness of the Plancherelmeasure µ of Γ :the action of Aut(Γ) on Ch(Γ) preserves µ .For example, when Γ = Z d , Thoma’s dual Ch(Γ) is the torus T d ,the Plancherel measure µ is the normalized Lebesgue measure on T d which is indeed preserved by the group Aut( Z d ) = GL d ( Z ). Dynamicalsystems of the form (Λ , Ch(Γ) , µ ) for a subgroup Λ of Aut(Γ) may beviewed as generalizations of this example.We denote by Γ fc the FC-centre of Γ , that is, the normal subgroup ofelements in Γ with a finite conjugacy class. It turns out (see Remark 6)that t = 0 on Γ \ Γ fc for µ -almost every t ∈ Ch(Γ). In particular, when Γis ICC, that is, when Γ fc = { e } , the regular representation λ Γ is factorial(see also Corollary 5) so that the Plancherel formula is vacuous in thiscase.In fact, as we now see, the Plancherel measure of Γ is entirely de-termined by the Plancherel measure of Γ fc . Roughly speaking, we willsee that the Plancherel measure of Γ is the image of the Plancherel
LANCHEREL FORMULA FOR COUNTABLE GROUPS 5 measure of Γ fc under the quotient map Ch(Γ fc ) → Ch(Γ fc ) /K Γ , for acompact group K Γ which we now define.Let K Γ be the closure in Aut(Γ fc ) of the subgroup Ad(Γ) | Γ fc givenby conjugation with elements from Γ . Since every Γ-conjugation classin Γ fc is finite, K Γ is a compact group. By a general fact about actionsof compact groups on Borel spaces (see Corollary 2.1.21 and Appendixin [Zim84]), the quotient space Ch(Γ fc ) /K Γ is a standard Borel space.Given a function t : H → C of positive type on a subgroup H of agroup Γ , we denote by e t the extension of t to Γ given by e t = 0 outside H. Observe that e t is of positive type on Γ (see for instance [BH, 1.F.10]).Here is our main result. Theorem B. ( Plancherel measure: reduction to the FC-center )Let Γ be a countable group. Let ν be the Plancherel measure of Γ fc and λ Γ fc = R ⊕ Ch(Γ fc ) π t dν ( t ) the integral decomposition of the regular repre-sentation of Γ fc as in Theorem A. Let ˙ ν be the image of ν under thequotient map Ch(Γ fc ) → Ch(Γ fc ) /K Γ . (i) For every K Γ -orbit O in Ch(Γ fc ) , let m O be the unique nor-malized K Γ -invariant probability measure on O and let π O := R ⊕O π t m O ( t ) . Then the induced representation f π O := Ind ΓΓ fc π O is factorial for ˙ ν -almost every O and we have a direct integraldecomposition of the von Neumann algebra L (Γ) into factors L (Γ) = Z ⊕ Ch(Γ fc ) /K Γ f π O (Γ) ′′ d ˙ ν ( O ) . (ii) The Plancherel measure of Γ is the image Φ ∗ ( ν ) of ν under themap Φ : Ch(Γ fc ) → Tr(Γ) , t Z K Γ e t g dm ( g ) , where m is the normalized Haar measure on K Γ . It is worth mentioning that the support of µ was determined in[Tho67]. For an expression of the map Φ as in Theorem B.ii with-out reference to the group K Γ , see Remark 7.As we next see, the Plancherel measure on Γ fc can be explicitly de-scribed in the case of a linear group Γ . We first need to discuss thePlancherel formula for a so-called central group , that is, a centralextension of a finite group.Let Λ be a central group. Then Λ is of type I (see Theorem E). Infact, b Λ can be described as follows; let r : b Λ → [ Z (Λ) be the restric-tion map, where Z (Λ) is the center of Λ . Then, for χ ∈ [ Z (Λ) , every BACHIR BEKKA π ∈ r − ( χ ) is equivalent to a subrepresentation of the finite dimen-sional representation Ind Λ Z (Λ) χ , by a generalized Frobenius reciprocitytheorem (see [Mac52, Theorem 8.2]); in particular, r − ( χ ) is finite.The Plancherel measure ν on Ch(Λ) is, in principle, easy to deter-mine: we identify every π ∈ b Λ with its normalized character given by x π Tr π ( x ); for every Borel subset A of Ch(Λ), we have ν ( A ) = Z \ Z (Λ) A ∩ r − ( χ )) P π ∈ r − ( χ ) (dim π ) dχ, where dχ is the normalized Haar measure on the abelian compact group [ Z (Λ). Corollary C. ( The Plancherel measure for linear groups ) Let Γ be a countable linear group. (i) Γ fc is a central group; (ii) K Γ coincides with Ad(Γ) | Γ fc and is a finite group; (iii) the Plancherel measure of Γ is the image of the Plancherel mea-sure of Γ fc under the map Φ : Ch(Γ fc ) → Tr(Γ) given by Φ( t ) = 1 | Γ fc X s ∈ Ad(Γ) | Γfc t s . When the Zariski closure of the linear group Γ is connected, thePlancherel measure of Γ has a particularly simple form.
Corollary D. ( The Plancherel measure for linear groups-bis )Let G be a connected linear algebraic group over a field k and let Γ bea countable Zariski dense subgroup of G . The Plancherel measure of Γ is the image of the normalized Haar measure dχ on [ Z (Γ) under themap [ Z (Γ) → Tr(Γ) , χ e χ and the Plancherel formula is given for every f ∈ C [Γ] by k f k = Z [ Z (Γ) F (( f ∗ ∗ f ) | Z (Γ) )( χ ) dχ, where F is the Fourier transform on the abelian group Z (Γ) . The previous conclusion holds in the following two cases: (i) k is a countable field of characteristic and Γ = G ( k ) is thegroup of k -rational points in G ; (ii) k is a local field (that is, a non discrete locally compact field), G has no proper k -subgroup H such that ( G / H )( k ) is compact,and Γ is a lattice in G ( k ) . LANCHEREL FORMULA FOR COUNTABLE GROUPS 7
Corollary D generalizes the Plancherel theorem obtained in [CPJ94,Theorem 4] for Γ = G ( Q ) and in [PJ95, Theorem 3.6] for Γ = G ( Z ), inthe case where G is a unipotent linear algebraic group over Q ; indeed, G is connected (since the exponential map identifies G with its Liealgebra, as affine varieties) and these two results follow from (i) and(ii) respectively.In an appendix to this article, we use Theorem A to give a unifiedproof of Thoma’s and Kaniuth’s results ([Tho64],[Tho68], [Kan69]) asstated in the following theorem. For a group Γ , we denote by [Γ , Γ] itscommutator subgroup. Recall that Γ is said to be virtually abelian ifit contains an abelian subgroup of finite index.The regular representation λ Γ is of type I (or type II) if the von Neu-mann algebra L (Γ) is of type I (or type II); equivalently (see Corollaire2 in [Dix69, Chap. II, §
3, 5]), if π t (Γ) ′′ is a finite dimensional factor(or a factor of type II) for µ -almost every t ∈ Ch(Γ) in the Planchereldecomposition λ Γ = R ⊕ Ch(Γ) π t dµ ( t ) from Theorem A. Theorem E ( Thoma , Kaniuth ) . Let Γ be a countable group. Thefollowing properties are equivalent: (i) Γ is type I; (ii) Γ is virtually abelian; (iii) the regular representation λ Γ is of type I; (iv) every irreducible unitary representation of Γ is finite dimen-sional; (iv’) there exists an integer n ≥ such that every irreducible unitaryrepresentation of Γ has dimension ≤ n. Moreover, the following properties are equivalent: (v) λ Γ is of type II; (vi) either [Γ : Γ fc ] = ∞ or [Γ : Γ fc ] < ∞ and [Γ , Γ] is infinite. Our proof of Theorem E is not completely new as it uses severalcrucial ideas from [Tho64] and especially from [Kan69] (compare withthe remarks on p.336 after Lemma in [Kan69]); however, we felt itcould be useful to have a short and common treatment of both resultsin the literature. Observe that the equivalence between (i) and (iii)above does not carry over to non discrete groups (see [Mac61]).
Remark 3.
Theorem E holds also for non countable discrete groups.Write such a group Γ as Γ = ∪ j H j for a directed net of countablesubgroups H j . If L (Γ) ′′ is not of type II (or is of type I), then L ( H j ) isnot of type II (or is of type I) for j large enough. This is the crucialtool for the extension of the proof of Theorem E to Γ; for more details,proofs of Satz 1 and Satz 2 in [Kan69]. BACHIR BEKKA
This paper is organized as follows. In Section 2, we recall the well-known description of the center of the von Neumann algebra of a dis-crete group. Sections 3 and 4 contains the proofs of Theorems A andB. In Section 5, we prove Corollaries C and D. Section 6 is devoted tothe explicit computation of the Plancherel formula for a few examplesof countable groups. Appendix A contains the proof of Theorem E.
Acknowledgement.
We are grateful to Pierre-Emmanuel Capraceand Karl-Hermann Neeb for useful comments on the first version ofthis paper.2.
On the center of the group von Neumann algebra
Let Γ be a countable group. We will often use the following well-known description of the center Z = λ (Γ) ′′ ∩ λ (Γ) ′ of L (Γ) = λ Γ (Γ) ′′ .Observe that λ Γ ( H ) ′′ is a von Neumann subalgebra of L (Γ), for everysubgroup H of Γ . For h ∈ Γ fc , we set T [ h ] := λ Γ ( [ h ] ) = X x ∈ [ h ] λ Γ ( x ) ∈ λ Γ (Γ fc ) ′′ , where [ h ] denotes the Γ-conjugacy class of h. Lemma 4.
The center Z of L (Γ) = λ Γ (Γ) ′′ coincides with the closureof the linear span of { T [ h ] | h ∈ Γ fc } , for the strong operator topology;in particular, Z is contained in λ Γ (Γ fc ) ′′ . Proof.
It is clear that T [ h ] ∈ Z for every h ∈ Γ fc . Observe also that thelinear span of { T [ h ] | h ∈ Γ fc } is a unital selfadjoint algebra; indeed, T [ h − ] = T ∗ [ h ] for every h ∈ Γ fc and { [ h ] | h ∈ Γ fc } is a vector spacebasis of the algebra C [Γ fc ] Γ of Γ-invariant functions in C [Γ fc ].Let T ∈ Z . We have to show that T ∈ { T [ h ] | h ∈ Γ fc } ′′ . For every γ ∈ Γ , we have λ Γ ( γ ) ρ Γ ( γ )( T δ e ) = ( λ Γ ( γ ) ρ Γ ( γ ) T ) δ e = ( T λ Γ ( γ ) ρ Γ ( γ )) δ e = T δ e . and this shows that f := T δ e , which is a function in ℓ (Γ) , is invariantunder conjugation by γ. The support of f is therefore contained in Γ fc . Write f = P [ h ] ∈C c [ h ] [ h ] for a sequence ( c [ h ] ) [ h ] ∈C of complex numberswith P [ h ] ∈C h ] | c [ h ] | < ∞ , where C is a set of representatives for theΓ-conjugacy classes in Γ fc . Let ρ Γ be the right regular representationof Γ. Since T ∈ λ Γ (Γ) ′′ and ρ Γ (Γ) ⊂ λ Γ (Γ) ′ , we have, for every x ∈ Γ, T ( δ x ) = T ρ Γ ( x )( δ e ) = ρ Γ ( x )( f ) = ρ Γ ( x ) X [ h ] ∈C c [ h ] [ h ] = X [ h ] ∈C c [ h ] T [ h ] ( δ x ) , LANCHEREL FORMULA FOR COUNTABLE GROUPS 9 where the last sum is convergent in ℓ (Γ) . We also have T ∗ ( δ x ) = P [ h ] ∈C c [ h ] T [ h − ] ( δ x ) . Let S ∈ { T [ h ] | h ∈ Γ fc } ′ . For every x, y ∈ Γ , we have h ST ( δ x ) | δ y i = h S X [ h ] ∈C c [ h ] T [ h ] ( δ x ) | δ y i = h X [ h ] ∈C c [ h ] ST [ h ] ( δ x ) | δ y i = X [ h ] ∈C c [ h ] h T [ h ] S ( δ x ) | δ y i = h S ( δ x ) | X [ h ] ∈C c [ h ] T [ h − ] ( δ y ) i = h S ( δ x ) | T ∗ ( δ y ) i = h T S ( δ x ) | δ y ) i and it follows that ST = T S. (cid:3)
The following well-known corollary shows that the Plancherel mea-sure is the Dirac measure at δ e in the case where Γ is ICC group, thatis, when Γ fc = { e } . Corollary 5.
Assume that Γ is ICC. Then L (Γ) = λ Γ (Γ) ′′ is a factor. Proof of Theorem A
Consider a direct integral decomposition R ⊕ X π x dµ ( x ) of λ Γ associ-ated to the centre Z of L (Γ) = λ Γ (Γ) ′′ (see [8.3.2][Dix77]); so, X isa standard Borel space equipped with a probability measure µ and( π x , H x ) x ∈ X is measurable field of representations of Γ over X , suchthat there exists an isomorphism of Hilbert spaces U : ℓ (Γ) → Z ⊕ X H x dµ ( x )which transforms λ Γ into R ⊕ X π x dµ ( x ) and for which U Z U − is thealgebra of diagonal operators on R ⊕ X H x dµ ( x ). (Recall that a diagonaloperator on R ⊕ X H x dµ ( x ) is an operator of the form R ⊕ X ϕ ( x ) I H x dµ ( x )for an essentially bounded measurable function ϕ : X → C .)Then, upon disregarding a subset of X of µ -measure 0, the followingholds (see [8.4.1][Dix77]):(1) π x is a factor representation for every x ∈ X ;(2) π x and π y are disjoint for every x, y ∈ X with x = y ;(3) we have U λ Γ (Γ) ′′ U − = R ⊕ X π x (Γ) ′′ dµ ( x ) . Let ρ Γ be the right regular representation of Γ. Let γ ∈ Γ . Then
U ρ Γ ( γ ) U − commutes with every diagonalisable operator on R ⊕ X H x dµ ( x ),since ρ Γ ( γ ) ∈ L (Γ) ′ . It follows (see [Dix69, Chap. II, §
2, No 5,Th´eor`eme 1]) that
U ρ Γ ( γ ) U − is a decomposable operator, that is,there exists a measurable field of unitary operators x σ x ( γ ) such that U ρ Γ ( γ ) U − = R ⊕ X σ x ( γ ) dµ ( x ) . So, we have a measurable field x σ x of representations of Γ in R ⊕ X H x dµ ( x ) such that U ρ Γ ( γ ) U − = Z ⊕ X σ x ( γ ) dµ ( x ) for all γ ∈ Γ . Let ( ξ x ) x ∈ X ∈ R ⊕ X H x dµ ( x ) be the image of δ e ∈ ℓ (Γ) under U. Weclaim that ξ x is a cyclic vector for π x and σ x , for µ -almost every x ∈ X .Indeed, since δ e ∈ ℓ (Γ) is a cyclic vector for both λ Γ and ρ Γ , { ( π x ( γ ) ξ x ) x ∈ X | γ ∈ Γ } and { ( σ x ( γ ) ξ x ) x ∈ X | γ ∈ Γ } are countable total subsets of R ⊕ X H x dµ ( x ) and the claim follows froma general fact about direct integral of Hilbert spaces (see Proposition8 in Chap. II, § λ Γ ( γ ) δ e = ρ Γ ( γ − ) δ e for every γ ∈ Γ and since Γ is countable,upon neglecting a subset of X of µ -measure 0, we can assume that(4) π x ( γ ) ξ x = σ x ( γ − ) ξ x ;(5) π x ( γ ) σ x ( γ ′ ) = σ x ( γ ′ ) π x ( γ );(6) ξ x is a cyclic vector for both π x and σ x ,for all x ∈ X and all γ, γ ′ ∈ Γ.Let x ∈ X and let ϕ x be the function of positive type on Γ definedby ϕ x ( γ ) = h π x ( γ ) ξ x | ξ x i for every γ ∈ Γ . We claim that ϕ x ∈ Ch(Γ) . Indeed, using (4) and (5), we have, forevery γ , γ ∈ Γ, ϕ x ( γ γ γ − ) = h π x ( γ γ γ − ) ξ x | ξ x i = h π x ( γ γ ) σ x ( γ ) ξ x | ξ x i = h σ x ( γ ) π x ( γ γ ) ξ x | ξ x i = h π x ( γ ) ξ x | π x ( γ − ) σ x ( γ − ) ξ x i = h π x ( γ ) ξ x | ξ x i = ϕ x ( γ ) . So, ϕ x is conjugation invariant and hence ϕ x ∈ Tr(Γ) . Moreover, ϕ x isan extreme point in Tr(Γ) , since π x is factorial and ξ x is a cyclic vectorfor π x . Finally, since U : ℓ (Γ) → R ⊕ X H x dµ ( x ) is an isometry, we have forevery f ∈ C [Γ] , k f k = f ∗ ∗ f ( e ) = h λ Γ ( f ∗ ∗ f ) δ e | δ e i = k λ Γ ( f ) δ e k = k U ( λ Γ ( f ) δ e ) k = Z X k π x ( f ) ξ x k dµ ( x ) = Z X ϕ x ( f ∗ ∗ f ) dµ ( x ) . The measurable map Φ : X → Ch(Γ) given by Φ( x ) = ϕ x is injective,since π x and π y are disjoint by (2) and hence ϕ x = ϕ y for x, y ∈ X with LANCHEREL FORMULA FOR COUNTABLE GROUPS 11 x = y . It follows that Φ( X ) is a Borel subset of Ch(Γ) and that Φ is aBorel isomorphism between X and Φ( X ) (see [Mac57, Theorem 3.2]).Pushing forward µ to Ch(Γ) by Φ, we can therefore assume withoutloss of generality that X = Ch(Γ) and that µ is a probability measureon Ch(Γ). With this identification, it is clear that Items (i), (ii) and(iii) of Theorem A are satisfied and that the Plancherel formula holds.It remains to show the uniqueness of µ. Let ν any probability measureon Ch(Γ) such that the Plancherel formula. By polarization, we havethen δ e = R Ch(Γ) tdν ( t ), which is an integral decomposition of δ e ∈ Tr(Γ)over extreme points of the convex set Tr(Γ) . The uniqueness of such adecomposition implies that ν = µ. Remark 6. (i) For µ -almost every t ∈ Ch(Γ) , we have t = 0 on Γ \ Γ fc . Indeed, let γ / ∈ Γ fc . Then h λ Γ ( γ ) λ Γ ( h ) δ e | δ e i = 0 for every h ∈ Γ fc andhence( ∗ ) h λ Γ ( γ ) T δ e | δ e i = 0 for all T ∈ λ Γ (Γ fc ) ′′ . With the notation as in the proof above, let E be a Borel subset of X. Then T E := U − P E U is a projection in Z , where P E is the diagonaloperator R ⊕ X E ( x ) I H x dµ ( x ). It follows from Lemma 4 and ( ∗ ) that Z E ϕ x ( γ ) dµ ( x ) = h T E λ Γ ( γ ) δ e | δ e i = h λ Γ ( γ ) T E δ e | δ e i = 0 . Since this holds for every Borel subset E of X, this implies that ϕ x ( γ ) =0 for µ -almost every x ∈ X .As Γ is countable, for µ -almost every x ∈ X , we have ϕ x ( γ ) = 0 forevery γ / ∈ Γ fc .(ii) Let λ Γ = R ⊕ Ch(Γ) π t dµ ( t ), ρ Γ = R ⊕ Ch(Γ) σ t dµ ( t ), and δ e = ( ξ t ) t ∈ Ch(Γ) bethe decompositions as above. For µ -almost every t ∈ Ch(Γ) , the linearmap π t (Γ) ′′ → H t , T T ξ t is injective. Indeed, this follows from the fact that ξ t is cyclic for σ t and that σ t (Γ) ⊂ π t (Γ) ′ . Proof of Theorem B
Set N := Γ fc and X := Ch( N ) . Consider the direct integral decom-position R ⊕ X π t dν ( t ) of λ N into factor representations ( π t , K t ) of N withcorresponding traces t ∈ X, as in Theorem A.Let K Γ be the compact group which is the closure in Aut( N ) ofAd(Γ) | N . Since the quotient space X/K Γ is a standard Borel space,there exists a Borel section s : X/K Γ → X for the projection map X → X/K Γ . Set Ω := s ( X/K Γ ).Then Ω is a Borel transversal for X/K Γ . The Plancherel measure ν can accordingly be decomposed overΩ : we have ν ( f ) = Z Ω Z O ω f ( t ) dm ω ( t ) d ˙ ν ( ω )for every bounded measurable function f on Ch(Γ fc ) , where m ω be theunique normalized K Γ -invariant probability measure on the K Γ -orbit O ω of ω and ˙ ν is the image of ν under s. Let ω ∈ Ω and set π O ω := Z ⊕O ω π t dm ω ( t ) , which is a unitary representation of N on the Hilbert space K ω := Z ⊕O ω K t dm ω ( t ) . For g ∈ Aut( N ) , let π g O ω be the conjugate representation of N on K ω given by π g O ω ( h ) = π O ω ( g ( h )) for h ∈ N. Step 1
There exists a unitary representation U ω : g U ω,g of K Γ on K ω such that U ω,g π O ω ( h ) U − ω,g = π O ω ( g ( h )) for all g ∈ K Γ , h ∈ N ;in particular, π g O ω is equivalent to π O ω for every g ∈ K Γ . Indeed, observe that the representations π t for t ∈ O ω are conjugateto each other (up to equivalence) and may therefore be considered asdefined on the same Hilbert space.Let g ∈ K Γ . Then π g O ω is equivalent to R ⊕O ω π tg dm ω ( t ) . Define a linearoperator U ω,g : K ω → K ω by U ω,g (( ξ t ) t ∈O ω ) = ( ξ t g ) t ∈O ω for all ( ξ t ) t ∈O ω ∈ K ω . Then U ω,g is an isometry, by K Γ -invariance of the measure m ω . It isreadily checked that U ω intertwines π O ω and π g O ω and that U ω is ahomomorphism. To show that U ω is a representation of K Γ , it remainsto prove that g U ω,g ξ is continuous for every ξ ∈ K ω . For this, observe that K ω can be identified with the Hilbert space L ( O ω , m ω ) ⊗ K , where K is the common Hilbert space of the π t ’s for t ∈ O ω ; under this identification, U ω corresponds to κ ⊗ I K , where κ is the Koopman representation of K Γ on L ( O ω , m ω ) associated to theaction K Γ y O ω (for the fact that κ is indeed a representation of K Γ , see [BHV08, A.6]) and the claim follows. LANCHEREL FORMULA FOR COUNTABLE GROUPS 13
Next, let g π O ω := Ind Γ N π O ω be the representation of Γ induced by π O ω . We recall how g π O ω can be realized on ℓ ( R, K ω ) = ℓ ( R ) ⊗ K ω , where R ⊂ Γ is a set of representatives for the cosets of N with e ∈ R. Forevery γ ∈ Γ and r ∈ R , let c ( r, γ ) ∈ N and r · γ ∈ R be such that rγ = c ( r, γ ) r · γ. Then g π O ω is given on ℓ ( R, K ω ) by( g π O ω ( γ ) F )( r ) = π O ω ( c ( r, γ ))( F ( r · γ )) for all F ∈ ℓ ( R, K ω ) . Step 2
We claim that there exists a unitary map f U ω : ℓ ( R, K ω ) → ℓ ( R, K ω )which intertwines the representation I ℓ ( R ) ⊗ π O ω and the restriction g π O ω | N of g π O ω to N ; moreover, ω → f U ω is a measurable field of unitaryoperators on Ω . Indeed, we have an orthogonal decomposition ℓ ( R, K ω ) = ⊕ r ∈ R ( δ r ⊗ K ω )into g π O ω ( N )-invariant; moreover, the action of N on every copy δ r ⊗K ω is given by π r O ω . For every r ∈ R, the unitary operator U ω,r : K ω → K ω from Step 1 intertwines π O ω and π r O ω . In view of the explicit formulaof U ω,r , the field ω → U ω,r is measurable on Ω . Define a unitary operator f U ω : ℓ ( R, K ω ) → ℓ ( R, K ω ) by f U ω ( δ r ⊗ ξ ) = δ r ⊗ U ω,r ( ξ ) for all ξ ∈ K ω . Then f U ω intertwines I ℓ ( R ) ⊗ π O ω and g π O ω | N ; moreover, ω → f U ω is ameasurable field on Ω . Observe that the representation λ N is equivalent to R ⊕ Ω π O ω d ˙ ν ( ω ) . Since λ Γ is equivalent to Ind Γ N λ N , it follows that λ Γ is equivalent to R ⊕ Ω g π O ω d ˙ ν ( ω ) . In the sequel, we will identify the representations λ N on ℓ ( N ) and λ Γ on ℓ (Γ) with respectively the representations Z ⊕ Ω π O ω d ˙ ν ( ω ) on K := Z ⊕ Ω K ω d ˙ ν ( ω )and Z ⊕ Ω g π O ω d ˙ ν ( ω ) on H := Z ⊕ Ω ℓ ( R, K ω ) d ˙ ν ( ω ) . Step 3
The representations g π O ω are factorial and are mutually dis-joint, outside a subset of Ω of ˙ ν -measure 0.To show this, it suffices to prove (see [Dix77, 8.4.1]) that the algebra D of diagonal operators in L ( H ) coincides with the center Z of λ Γ (Γ) ′′ .Let us first prove that D ⊂ Z . For this, we only have to prove that
D ⊂ λ Γ (Γ) ′′ , since it is clear that D ⊂ λ Γ (Γ) ′ . By Step 2, for every ω ∈ Ω , there exists a measurable field ω → f U ω ofunitary operators f U ω : ℓ ( R, K ω ) → ℓ ( R, K ω ) intertwining I ℓ ( R ) ⊗ π O ω and π O ω | N . So, e U := Z ⊕ Ω f U ω d ˙ ν ( ω )is a unitary operator on H which intertwines I ℓ ( R ) ⊗ λ N and λ Γ | N ; itis obvious that e U commutes with the diagonal operators on H . Let ϕ : Ω → C be a measurable essential bounded function on Ω.By Theorem A.ii, the corresponding diagonal operator T = Z ⊕ Ω ϕ ( ω ) I K ω d ˙ ν ( ω )on K belongs to λ N ( N ) ′′ . For the corresponding diagonal operator e T = Z ⊕ Ω ϕ ( ω ) I ℓ ( R, K ω ) ˙ ν ( ω )on H , we have e T = I ℓ ( R ) ⊗ T . So, e T belongs to ( I ℓ ( R ) ⊗ λ N )( N ) ′′ . Since, e U commutes with e T and intertwines I ℓ ( R ) ⊗ λ N and λ Γ | N , itfollows that e T = e U ( I ℓ ( R ) ⊗ T ) e U − ∈ λ Γ ( N ) ′′ ⊂ λ Γ (Γ) ′′ . So, we have shown that
D ⊂ Z . Observe that this implies (seeTh´eor`eme 1 in Chap. II, § L (Γ) is the direct in-tegral R ⊕ Ω g π O ω (Γ) ′′ d ˙ ν ( ω ) and that Z is the direct integral R ⊕ Ω Z ω d ˙ ν ( ω ) , where Z ω is the center of g π O ω (Γ) ′′ . Let e T ∈ Z . Then e T ∈ λ Γ ( N ) ′′ , by Lemma 4. So, e T = R ⊕ Ω T ω d ˙ ν ( ω ) , where T ω belongs to the center of g π O ω ( N ) ′′ , for ˙ ν -almost every ω. Since g π O ω | N is equivalent to I ℓ ( R ) ⊗ π O ω and since π O ω and hence I ℓ ( R ) ⊗ π O ω is a factor representation, it follows that T ω is a scalar operator, for˙ ν -almost every ω. So, e T ∈ D . As a result, we have a decomposition λ Γ = Z ⊕ Ω g π O ω d ˙ ν ( ω ) LANCHEREL FORMULA FOR COUNTABLE GROUPS 15 of λ Γ as a direct integral of pairwise disjoint factor representations. Bythe argument of the proof of Theorem A, it follows that, for ˙ ν -almostevery ω ∈ X, there exists a cyclic unit vector ξ ω ∈ ℓ ( R, K ω ) for g π O ω so that ϕ ω := h g π O ω ( · ) ξ ω | ξ ω i belongs to Ch(Γ). In particular, g M ω := g π O ω (Γ) ′′ is a factor of type II and its normalized trace is the extension of ϕ ω to g M ω , which we again denote by ϕ ω . Our next goal is to determine ϕ ω in terms of the character ω ∈ Ch( N ) . Fix ω ∈ Ω such that g M ω is a factor. We identity the Hilbert space K ω of π O ω with the subspace δ e ⊗ K ω and so π O ω with a subrepresentationof the restriction of g π O ω to N. Let η ω ∈ K ω be a cyclic vector for π O ω such that ω = h π O ω ( · ) η ω | η ω i . For g ∈ K Γ , define a normal state ψ ω,g on g M ω by the formula ψ ω,g ( e T ) = h e T U − g,ω η ω | U − g,ω η ω i for all e T ∈ g M ω , where U g,ω is the unitary operator on K ω from Step 1.Consider the linear functional ψ ω : g M ω → C given by ψ ω ( e T ) = Z K Γ ψ ω,g ( e T ) dm ( g ) for all e T ∈ g M ω , where m is the normalized Haar measure on K Γ . Step 4
We claim that ψ ω is a normal state on g M ω Indeed, it is clear that ψ ω is a state on g M ω . Let ( e T n ) n be an increasingsequence of positive operators in g M ω with e T = sup n e T n ∈ g M ω .For every g ∈ K Γ , the sequence ( ψ ω,g ( e T n )) n is increasing and its limitis ψ ω,g ( e T ). It follows from the monotone convergence theorem thatlim n ψ ω ( T n ) = lim n Z K Γ ψ ω,g ( T n ) dm ( g ) = Z K Γ ψ ω,g ( T ) dm ( g ) = ψ ω ( T ) . So, ψ ω is normal, as g M ω acts on a separable Hilbert space. Step 5
We claim that, writing γ instead of g π O ω ( γ ) for γ ∈ Γ, wehave ψ ω ( γ ) = (R K Γ ω g ( γ ) dm ( g ) if γ ∈ N γ / ∈ N. Moreover, ψ ω coincides with the trace ϕ ω on g M ω from above.Indeed, Let γ ∈ N. Since U ω,g π O ω ( γ ) U − ω,g = π O ω ( g ( γ )) , we have ψ ω ( γ ) = Z K Γ h e π O ω ( g ( γ )) η ω | η ω i dm ( g ) = Z K Γ h π O ω ( g ( γ )) η ω | η ω i dm ( g )= Z K Γ ω g ( γ ) dm ( g ) . Let γ ∈ Γ \ N. Then, by the usual properties of an induced represen-tation, e π O ω ( γ )( K ω ) is orthogonal to K ω . It follows that ψ ω,g ( γ ) = h e π O ω ( γ ) U − g,ω η ω | U − g,ω η ω i = 0and hence ψ ω ( γ ) = 0 . In particular, this shows that ψ ω is a Γ-invariant state on g M ω ; since ψ ω is normal (Step 4), it follows that ψ ω is a trace on g M ω . As g M ω is afactor (see Step 3), the fact that f ψ ω = ϕ ω follows from the uniquenessof normal traces on factors (see Corollaire p. 92 and Corollaire 2 p.83in [Dix69, Chap. I, § Step 6
The Plancherel measure µ on Γ is the image of ν under themap Φ as in the statement of Theorem B.ii.Indeed, for f ∈ C [Γ], we have by Step 5 k f k = Z Ω k g π O ω ( f ) k d ˙ ν ( ω ) = Z Ω ψ ω ( f ∗ ∗ f ) d ˙ ν ( ω )= Z Ω Z K Γ ω g (( f ∗ ∗ f ) | N ) dm ( g ) d ˙ ν ( ω )= Z Ω Z O ω (cid:18)Z K Γ ω g (( f ∗ ∗ f ) | N ) dm ( g ) (cid:19) dm ω ( t ) d ˙ ν ( ω )= Z Ch(Γ fc ) Z K Γ t g (( f ∗ ∗ f ) | N ) dm ( g ) dν ( t )= Z Ch(Γ) Φ( t )( f ∗ ∗ f ) dν ( t )and the claim follows. Remark 7.
The map Φ in Theorem B.ii can be described withoutreference to the group K Γ as follows. For t ∈ Ch(Γ fc ) and γ ∈ Γ , we LANCHEREL FORMULA FOR COUNTABLE GROUPS 17 have Φ( t )( γ ) = γ ] P x ∈ [ γ ] t ( x ) if γ ∈ Γ fc γ / ∈ Γ fc , where [ γ ] denotes the Γ-conjugacy class of γ. Indeed, it suffices toconsider the case where γ ∈ Γ fc . The stabilizer K of γ in K Γ is anopen and hence cofinite subgroup of K Γ ; in particular, the K Γ -orbitof γ coincides with the Ad(Γ)-orbit of γ and so { Ad( x ) | x ∈ [ γ ] } is a system of representatives for K Γ /K . Let m be the normalizedHaar measure on K . The normalized Haar measure m on K Γ is thengiven by m ( f ) = 1[ γ ] P x ∈ [ γ ] R K f (Ad( x ) g ) dm ( g ) for every continuousfunction f on K Γ . It follows thatΦ( t )( γ ) = Z K Γ t ( g ( γ )) dm ( g ) = 1[ γ ] X x ∈ [ γ ] t ( x ) . Proofs of Corollary C and Corollary D
Let Γ be a countable linear group. So, Γ is a subgroup of GL n ( k )for a field k , which may be assumed to be algebraically closed. Let G be the closure of Γ in the Zariski topology of GL n ( k ) and let G bethe irreducible component of G . As is well-known, G has finite indexin G and hence Γ := G ∩ Γ is a normal subgroup of finite index in ΓLet γ ∈ Γ fc . On the one hand, the centralizer Γ γ of γ in Γ is asubgroup of finite index of Γ; therefore, the irreducible component ofthe Zariski closure of Γ γ coincides with G . On the other hand, thecentralizer G γ of γ in G is clearly a Zariski-closed subgroup of G . Itfollows that the irreducible component of G γ contains (in fact coincideswith) G and hence Γ = G ∩ Γ ⊂ G γ ∩ Γ = Γ γ . As a consequence, we see that Γ acts trivially on Γ fc and henceAd(Γ) | Γ fc is a finite group. In particular, Γ ∩ Γ fc is contained in thecenter Z (Γ fc ) of Γ fc ; so Z (Γ fc ) has finite index in Γ fc which is thereforea central group. This proves Items (i) and (ii) of Corollary C. Item (iii)follows from Theorem B.ii.Assume now that G is connected, that is G = G . Then Γ = Γ actstrivially on Γ fc and so Γ fc coincides with the center Z (Γ) of Γ . Thisproves the first part of Corollary D.It remains to prove that the assumption G = G is satisfied in Cases(i) and (ii) of Corollary D: (i) Let G be a connected linear algebraic group over a countablefield k of characteristic 0. Then Γ = G ( k ) is Zariski dense in G , by [Ros57, Corollary p.44]).(ii) Let G be a connected linear algebraic group G over a localfield k . Assume that G has no proper k -subgroup H such that( G / H )( k ) is compact. Then every lattice Γ in G ( k ) is Zariskidense in G , by [Sha99, Corollary 1.2].6. The Plancherel Formula for some countable groups
Restricted direct product of finite groups.
Let ( G n ) n ≥ be asequence of finite groups. Let Γ = Q ′ n ≥ G n be the restricted directproduct of the G n ’s, that is, Γ consists of the sequences ( g n ) n ≥ with g n ∈ G n for all n and g n = e for at most finitely many n. It is clearthat Γ is an FC-group.Set X n := Ch( G n ) for n ≥ X = Q n ≥ X n be the cartesianproduct equipped with the product topology, where each X n carriesthe discrete topology. Define a map Φ : X → Tr(Γ) byΦ(( t n ) n ≥ )(( g n ) n ≥ ) = Y n ≥ t n ( g n ) for all ( t n ) n ≥ ∈ X, ( g n ) n ≥ ∈ Γ(observe that this product is well-defined, since g n = e and hence t n ( g n ) = 1 for almost every n ≥ X ) = Ch(Γ) andΦ : X → Ch(Γ) is a homeomorphism (see [Mau51, Lemma 7.1])For every n ≥ , let ν n be the measure on X n given by ν n ( { t } ) = d t G n for all t ∈ X n , where d t is the dimension of the irreducible representation of G n with t as character; observe that ν n is a probability measure, since P t ∈ X n d t = G n .Let ν = ⊗ n ≥ ν be the product measure on the Borel subsets of X. The Plancherel measure on Γ is the image of ν under Φ (see Equation(5.6) in [Mau51]). The regular representation λ Γ is of type II if andonly if infinitely many G n ’s are non abelian (see loc.cit. , Theorem 1 orTheorem E below).6.2. Infinite dimensional Heisenberg group.
Let F p be the fieldof order p for an odd prime p and let V = ⊕ i ∈ N F p be a vector spaceover F p of countable infinite dimension. Denote by ω the symplecticform on V ⊕ V given by ω (( x, y ) , ( x ′ , y ′ )) = X i ∈ N ( x i y ′ i − y i x ′ i ) for ( x, y ) , ( x ′ , y ′ ) ∈ V ⊕ V. LANCHEREL FORMULA FOR COUNTABLE GROUPS 19
The “infinite dimensional” Heisenberg group over F p is the group Γwith underlying set V ⊕ V ⊕ F p and with multiplication defined by( x, y, z )( x ′ , y ′ , z ′ ) = ( x + x ′ , y + y ′ , z + z ′ + ω (( x, y ) , ( x ′ , y ′ )))for ( x, y, z ) , ( x ′ , y ′ , z ′ ) ∈ Γ.The group Γ is an FC-group; since p ≥ , its center Z coincideswith [Γ , Γ] and consists of the elements of the form (0 , , z ) for z ∈ F p . Observe that Γ is not virtually abelian.Let z be a generator for the cyclic group Z of order p . The uni-tary dual b Z consists of the characters defined by χ ω ( z j ) = ω j for j ∈ { , , . . . , p − } and ω ∈ C p , where C p is the group of p -th rootsof unity in C . For ω ∈ C p , the subspace H ω = { f ∈ ℓ (Γ) | f ( z x ) = ωf ( x ) for every x ∈ Γ } . is left and right translation invariant and we have an orthogonal de-composition of ℓ (Γ) = L ω ∈ C p H ω . The orthogonal projection P ω on H ω belongs to the center of L (Γ) and is given by P ω ( f )( x ) = 1 p p − X i =0 ω − i f ( z i x ) for all f ∈ ℓ (Γ) , x ∈ Γ . One checks that k P ω ( δ e ) k = 1 /p. Let π ω be the restriction of λ Γ to H ω . Observe that H can beidentified with ℓ (Γ /Z ) and π with λ Γ /Z .For ω = 1, the representation π ω is factorial of type II and the cor-responding character is f χ ω (for more details, see the proof of Theorem7.D.4 in [BH]).The integral decomposition of δ e is δ e = 1 p Z d Γ /Z χdν ( χ ) + 1 p X ω ∈ C p \{ } f χ ω , with the corresponding Plancherel formula given for every f ∈ C [Γ] by k f k = 1 p Z d Γ /Z |F ( P ( f ))( χ ) | dν ( χ ) + 1 p X ω ∈ C p \{ } p − X j =0 ( f ∗ ∗ f )( z j ) ω j , where ν is the normalized Haar measure of the compact abelian group d Γ /Z and F the Fourier transform. In particular, λ Γ (Γ) ′′ is a direct sumof an abelian von Neumann algebra and p − . Fora more general result, see [Kap51, Theorem 2]. An example involving SL d ( Z ) . Let Λ = SL d ( Z ) for an oddinteger d. Fix a prime p and for n ≥ , let G n = SL ( Z /p n Z ), viewedas (finite) quotient of Λ . Let Γ be the semi-direct product Λ ⋉ Q ′ n ≥ G n ,where Λ acts diagonally in the natural way on the restricted directproduct G := Q ′ n ≥ G n of the G n ’s.Since Λ is an ICC-group, it is clear that Γ fc = G. The group K Γ asin Theorem B can be identified with the projective limit of the groups G n ’s, that is, with SL d ( Z p ) , where Z p is the ring of p -adic integers.Since Λ acts trivially on Ch( G ) , the same is true for the action of K Γ on Ch( G ) . Let λ G = R ⊕ Ch( G ) π t dν ( t ) be the Plancherel decomposition of λ G (seeExample 6.1). It follows from Theorem B that the Plancherel decom-position of λ Γ is λ Γ = Z ⊕ Ch( G ) Ind Γ G π t dν ( t ) . Appendix A. Proof of Theorem E
A.1.
Easy implications.
The implications ( iv ′ ) ⇒ ( iv ) and ( i ) ⇒ ( iii ) are obvious; if ( iv ) holds then Γ is a so-called CCR group and so( i ) holds, by a general fact (see [Dix77, 5.5.2]).We are going to show that ( ii ) ⇒ ( iv ′ ) , Assume that Γ contains anabelian normal subgroup N of finite index. Let ( π, H ) be an irreduciblerepresentation of Γ.Denote by B the set of Borel subsets of the dual group b N and byProj( H ) the set of orthogonal projections in L ( H ) . Let E : B ( b N ) → Proj( H ) be the projection-valued measure on b N associated with therestriction π | N by the SNAG Theorem (see [BHV08, D.3.1]); so, wehave π ( n ) = Z b G χ ( n ) dE ( χ ) for all n ∈ N. The dual action of Γ on b N , given by χ γ ( n ) = χ ( γ − nγ ) for χ ∈ b N and γ ∈ Γ , factorizes through Γ /N . Moreover, the following covariancerelation holds π ( γ ) E ( B ) π ( γ − ) = E ( B γ ) for all B ∈ B ( b N ) , where B γ = { χ γ | χ ∈ B } .Let S ∈ B ( b N ) be the support of E, that is, S is the complementof the largest open subset U of b N with E ( U ) = 0. We claim that S consists of a single Γ-orbit.Indeed, let χ ∈ S and let ( U n ) n ≥ be a sequence of open neigh-bourhoods of χ with T n ≥ U n = { χ } . Fix n ≥ . The set U Γ n is Γ- LANCHEREL FORMULA FOR COUNTABLE GROUPS 21 invariant and hence E ( U Γ n ) ∈ π (Γ) ′ , by the covariance relation. Since π is irreducible and E ( U n ) = 0 , we have therefore E ( U Γ n ) = I H . By theusual properties of a projection-valued measure, this implies that E ( χ Γ0 ) = E ( \ n ≥ U Γ n ) = I H . and the claim is proved.Since S is finite, we have H = L χ ∈ S H χ , where H χ := { ξ ∈ H | π ( n ) ξ = χ ( n ) ξ for all n ∈ N } ;moreover, since N is a normal subgroup, we have π ( γ ) H χ = H χ γ forevery χ ∈ S and every γ ∈ Γ.Let H be the stabilizer of χ and let T ⊂ Γ be a set of representativesof the right T -cosets of H . Then H χ is invariant under π ( H ) and wehave H = M t ∈ T H χ t = M t ∈ T π ( t ) H χ . This shows that π is equivalent to the induced representation Ind Γ H σ ,where σ is the subrepresentation of π | H defined on H χ We claim that Ind Γ H σ is contained in Ind Γ N χ . Indeed, as is well-known (see [BHV08, E.2.5]), Ind HN ( σ | N ) is equivalent to the tensorproduct representation σ ⊗ λ H/N , where λ H/N is the quasi-regular rep-resentation on ℓ (Γ /H ) . Since
H/N is finite, 1 H is contained in λ H/N and therefore σ is contained in Ind HN ( σ | N ). Notice that σ | N is a mul-tiple nχ of χ , for some cardinal n . We conclude that π = Ind Γ H σ is contained in Ind Γ H (Ind HN nχ ) = n Ind Γ N χ . Since π is irreducible, itfollows that π is contained in Ind Γ N χ .Now, Ind Γ N χ has dimension [Γ : N ]; hence, dim π ≤ [Γ : N ] and so( iv ′ ) holds.A.2. Proof of the other implications.
We have to give the proof ofthe implication ( iii ) ⇒ ( i ) and the equivalence ( v ) ⇔ ( iv ) . In the sequel, Γ will be a countable group and λ Γ = R ⊕ Ch(Γ) π t dµ ( t )the direct integral decomposition given by the Plancherel Theorem A.Recall (see Section 3) that, if we write δ e = R ⊕ Ch(Γ) ξ t dµ ( t ), then ξ t isa cyclic vector in the Hilbert space H t of π t and t = h π t ( · ) ξ t | ξ t i , for µ -almost every t ∈ Ch(Γ) . A.2.1.
Case where the FC-centre of Γ has infinite index. We assumethat [Γ : Γ fc ] is infinite; we claim that λ Γ is of type II.By Remark 6.i, there exists a subset X of Ch(Γ) with µ ( X ) = 1 suchthat t = 0 outside Γ fc for every t ∈ X. Let t ∈ X . Then the factor π t (Γ) ′′ is infinite dimensional. In-deed, since [Γ : Γ fc ] is infinite, we can find a sequence ( γ n ) n ≥ in Γwith γ − m γ n / ∈ Γ fc for every m, n with m = n. Then h π t ( γ n ) ξ t | π t ( γ m ) ξ t i = h π t ( γ − m γ n ) ξ t | ξ t i = t ( γ − m γ n ) = 0 , for m = n ; so, ( π t ( γ n ) ξ t ) n ≥ is an orthonormal sequence in H t . Thisimplies that ( π t ( γ n )) n ≥ is a linearly independent sequence in π t (Γ) ′′ and the claim is proved.Observe that we have proved, in particular, that Γ is not of type I.A.2.2. Reduction to FC-groups.
Let Ch(Γ) fd be the set of t ∈ Ch(Γ)for which π t (Γ) ′′ is finite dimensional. ThenCh(Γ) fd = [ n ≥ Ch(Γ) n , where Ch(Γ) n is the set of t such that dim π t (Γ) ′′ = n . We claim thatCh(Γ) n and hence Ch(Γ) fd is a measurable subset of Ch(Γ).Indeed, let F be collection of finite subsets of Γ . For every F ∈ F , let C F be the set of t ∈ Ch(Γ) such that the family ( π t ( γ )) γ ∈ F is linearlyindependent, equivalently (see Remark 6.ii), such that ( π t ( γ ) ξ t ) γ ∈ F islinearly independent. Since t = h π t ( · ) ξ t | ξ t i , it follows that C F := (cid:8) t ∈ Ch(Γ) | det( t ( γ − γ ′ )) = 0 for all ( γ, γ ′ ) ∈ F × F (cid:9) and this shows that C F is measurable. Since( ∗ ) Ch(Γ) n = [ F ∈F : F = n C F \ [ F ′ ∈F : F ′ >n C F ′ ! and F is countable, it follows that Ch(Γ) n is measurable.Assume now that [Γ : Γ fc ] is finite. Observe that, for a cyclic rep-resentation π of Γ fc , the induced representation Ind ΓΓ fc π is cyclic andso (Ind ΓΓ fc π )(Γ) ′′ is finite dimensional if and only if π (Γ fc ) ′′ is finitedimensional.let ν be the Plancherel measure of Γ fc . It follows from Theorem B that µ (Ch(Γ) fd ) = ν (Ch(Γ fc ) fd ); in particular, we have µ (Ch(Γ) fd ) = 0 (or µ (Ch(Γ) fd ) = 1) if and only if ν (Ch(Γ fc ) fd ) = 0 (or ν (Ch(Γ fc ) fd ) = 1),that is, λ Γ is of type I (or of type II) if and only if λ Γ fc is of type I (orof type II).Observe also that Γ is virtually abelian if and only if Γ fc is virtuallyabelian. As a consequence, we see that it suffices to prove the impli-cation ( iii ) ⇒ ( i ) and the equivalence ( v ) ⇔ ( iv ) in the case whereΓ = Γ fc . LANCHEREL FORMULA FOR COUNTABLE GROUPS 23
A.2.3.
Case of an FC-group.
We will need the following lemma of in-dependent interest, which is valid for an arbitrary countable group Γ . Let r : Ch(Γ) → Tr([Γ , Γ]) be the restriction map. We will identifyCh(Γ / [Γ , Γ]) with the set { s ∈ Ch(Γ) | r ( s ) = 1 [Γ , Γ] } , that is, with theset of unitary characters of Γ . Observe that, for every s ∈ Ch(Γ / [Γ , Γ])and t ∈ Ch(Γ) , we have st ∈ Ch(Γ) . Lemma 8.
Let Γ be a countable group and t, t ′ ∈ Ch(Γ) be such that r ( t ) = r ( t ′ ) . Then there exists s ∈ Ch(Γ / [Γ , Γ]) such that t ′ = st. Proof.
The integral decomposition of [Γ , Γ] ∈ Tr(Γ) into characters isgiven by [Γ , Γ] = Z Ch(Γ / [Γ , Γ]) sdν ( s ) , where ν is the Haar measure of Γ / [Γ , Γ] . By assumption, we have t [Γ , Γ] = t ′ [Γ , Γ] and hence t [Γ , Γ] = Z Ch(Γ / [Γ , Γ]) tsdν ( s ) = Z Ch(Γ / [Γ , Γ]) t ′ sdν ( s ) . By uniqueness of integral decomposition, it follows that the images ν t and ν t ′ of ν under the maps Ch(Γ / [Γ , Γ]) → Ch(Γ) given respectivelyby multiplication with t and t ′ coincide. In particular, the supports of ν t and ν t ′ are the same, that is, t Ch(Γ / [Γ , Γ]) = t ′ Ch(Γ / [Γ , Γ]) andthe claim follows. (cid:3)
We assume from now on that Γ = Γ fc . Step 1
We claim that the regular representation λ Γ is of type II ifand only if [Γ , Γ] is infinite.We have to show that µ (Ch(Γ) fd ) > , Γ] is finite.Assume first that [Γ , Γ] is finite. The representation λ Γ / [Γ , Γ] , lifted toΓ, is a subrepresentation of λ Γ , since ℓ (Γ / [Γ , Γ]) can be viewed in anobvious way as Γ-invariant subspace of ℓ (Γ). As Γ / [Γ , Γ] is abelian, λ Γ / [Γ , Γ] is of type I and so µ (Ch(Γ) fd ) > . Conversely, assume that µ (Ch(Γ) fd ) > . Since Γ is an FC-group,it suffices to show that Γ has a subgroup of finite index with finitecommutator subgroup (see [Neu55, Lemma 4.1]).As µ (Ch(Γ) fd ) > fd = S n ≥ Ch(Γ) n , we have µ (Ch(Γ) n ) > n ≥ . It follows from (*) that there exists F ∈ F with | F | = n such that µ ( C F ∩ Ch(Γ) n ) > . Let Λ be the subgroup of Γ generated by F. Since Γ is an FC-groupand Λ is finitely generated, the centralizer H := Cent Γ (Λ) of Λ in Γhas finite index. Let t ∈ C F ∩ Ch(Γ) n and γ ∈ H . On the one hand, since ( π t ( γ )) γ ∈ F is a basis of the vector space π t (Γ) , we have π t (Λ) ′′ = π t (Γ) ′′ . On theother hand, as γ centralizes Λ , we have π t ( γ ) ∈ π t (Λ) ′ . Hence, π t ( γ )belongs to the center π t (Γ) ′ ∩ π t (Γ) ′′ of the factor π t (Γ) ′′ and so π t ( γ )is a scalar multiple of I H t . It follows in particular that π t is trivial on[ H, H ] . As a result, the subrepresentation R ⊕ C F ∩ Ch(Γ) n π t dµ ( t ) of λ Γ istrivial on [ H, H ] . Since the matrix coefficients of λ Γ vanish at infinity,it follows that [ H, H ] is finite and the claim is proved.In view of what we have shown so far, we may and will assume fromnow on that [Γ , Γ] is finite and that λ Γ is of type I. We are going toshow that Γ is a virtually abelian (in fact, a central) group and thiswill finish the proof of Theorem E.Set N := [Γ , Γ] and let r : Ch(Γ) → Tr( N ) be the restriction map. Step 2
We claim that there exist finitely many functions s , . . . , s m in Tr( N ) such that r ( t ) ∈ { s , . . . , s m } , for µ -almost every t ∈ Ch(Γ) . Indeed, since λ Γ is of type I, there exists a subset X of Ch(Γ) with µ ( X ) = 1 such that π t (Γ) ′′ is finite dimensional for every t ∈ X. Let t ∈ X. The Hilbert space H t of π t is finite dimensional and π t is a (finite) multiple of an irreducible representation σ t of Γ . As π t and σ t have the same normalized character, we may assume that π t isirreducible.Let K be an irreducible N -invariant subspace of H t and let ρ be thecorresponding equivalence class of representation of N. For g ∈ Γ , thesubspace π t ( g ) K is N -invariant with ρ g as corresponding representationof N . Since π t is irreducible, we have H t = P g ∈ Γ π t ( g ) K . Let L be the stabilizer of ρ ; observe that L has finite index in Γ , since L contains the centralizer of N and Γ is an FC-group. Let g , . . . , g r be a set of representatives for the coset space Γ /L . Then H t = ⊕ rj =1 π t ( g j ) K ρ , where K ρ is the sum of all N -invariant subspacesof H t with corresponding representation equivalent to ρ. The normal-ized trace of the representation of N on π t ( g j ) K ρ is χ g j ρ , where χ ρ is thenormalized character of ρ. It follows that, for every g ∈ N, we have t ( g ) = 1 r r X j =1 χ g j ρ ( g ) . Since [Γ , Γ] is finite, [Γ , Γ] has only finitely many equivalence classes ofirreducible representations and the claim follows.
LANCHEREL FORMULA FOR COUNTABLE GROUPS 25
Step 3
We claim that the center Z (Γ) has finite index in Γ.Indeed, by Step 2, there exists a subset X of Ch(Γ) with µ ( X ) = 1and finitely many t , . . . , t m ∈ X such that r ( t ) ∈ { r ( t ) , . . . , r ( t m ) } and such that H t is finite dimensional for every t ∈ X. It follows from Lemma 8 that, for every t ∈ X , there exists s ∈ Ch(Γ / [Γ , Γ]) such that t = st i for some i ∈ { , . . . , m } and hencedim π t (Γ) ′′ = dim π t i (Γ) ′′ . As a result, we can find a finitely generatednormal subgroup M of Γ such that dim π t (Γ) ′′ = dim π t ( M ) ′′ for every t ∈ X. Since the centralizer C of M in Γ has finite index, it suffices to showthat C contains Z (Γ) . For g ∈ C, γ ∈ Γ and x ∈ M, we have t ( x − gγg − ) = t ( x − γ ), thatis, h π t ( gγg − ) ξ t | π t ( x ) ξ t i = h π t ( γ ) ξ t | π t ( x ) ξ t i . Since dim π t (Γ) ′′ = dim π t ( M ) ′′ , the linear span of π t ( M ) ξ t is dense in H t and this implies that π t ( gγg − ) ξ t = π t ( γ ) ξ t for all g ∈ C, γ ∈ Γ , and t ∈ X. It follows that λ Γ ( gγg − ) δ e = λ Γ ( γ ) δ e and hence gγg − = γ forall g ∈ C and γ ∈ Γ; so, C ⊂ Z (Γ). References [BHV08] B. Bekka, P. de la Harpe, and A. Valette,
Kazhdan’s property (T) , NewMathematical Monographs, vol. 11, Cambridge University Press, Cambridge,2008. ↑
3, 12, 20, 21[BH] B. Bekka and P. de la Harpe,
Unitary representations of groups, duals, andcharacters , Graduate Studies in Mathematics, American Mathematical Society,Providence, RI. ↑
3, 5, 19[CPJ94] L. Corwin and C. Pfeffer Johnston,
On factor representations of discreterational nilpotent groups and the Plancherel formula , Pacific J. Math. (1994), no. 2, 261–275. ↑ C ∗ -algebras , North-Holland Publishing Co., Amsterdam-NewYork-Oxford, 1977. ↑
1, 2, 3, 9, 14, 20[Dix69] ,
Les alg`ebres d’op´erateurs dans l’espace hilbertien (alg`ebres de vonNeumann) , Gauthier-Villars ´Editeur, Paris, 1969 (French). Deuxi`eme ´edition,revue et augment´ee; Cahiers Scientifiques, Fasc. XXV. ↑
7, 9, 10, 14, 16[Gli61] J. Glimm,
Type I C ∗ -algebras , Ann. of Math. (2) (1961), 572–612. ↑ Der Typ der regul¨aren Darstellung diskreter Gruppen , Math.Ann. (1969), 334–339 (German). ↑ Group algebras in the large , Tohoku Math. J. (2) (1951),249–256. ↑ Borel structure in groups and their duals , Trans. Amer.Math. Soc. (1957), 134–165. ↑
1, 11[Mac52] G. W. Mackey,
Induced representations of locally compact groups. I , Ann.of Math. (2) (1952), 101–139. ↑ [Mac61] G.W. Mackey, Induced representations and normal subgroups , Proc. In-ternat. Sympos. Linear Spaces (Jerusalem, 1960), Jerusalem Academic Press,Jerusalem; Pergamon, Oxford, 1961, pp. 319–326. ↑ The structure of the regular representation of certain dis-crete groups , Duke Math. J. (1950), 437–441. ↑ The regular representation of a restricted direct product of finitegroups , Trans. Amer. Math. Soc. (1951), 531–548. ↑ Groups with finite classes of conjugate subgroups , Math.Z. (1955), 76–96. ↑ On a Plancherel formula for certain discrete, finitelygenerated, torsion-free nilpotent groups , Pacific J. Math. (1995), no. 2,313–326. ↑ Some rationality questions on algebraic groups , Ann. Mat.Pura Appl. (4) (1957), 25–50. ↑ C ∗ -algebras and W ∗ -algebras , Springer-Verlag, New York-Heidelberg, 1971. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band60. MR0442701 ↑ An extension of Plancherel’s formula to separable unimodulargroups , Ann. of Math. (2) (1950), 272–292. ↑ Invariant measures for algebraic actions, Zariski dense sub-groups and Kazhdan’s property (T) , Trans. Amer. Math. Soc. (1999), no. 8,3387–3412. ↑ ¨Uber unit¨are Darstellungen abz¨ahlbarer, diskreter Gruppen ,Math. Ann. (1964), 111–138 (German). ↑
3, 7[Tho68] ,
Eine Charakterisierung diskreter Gruppen vom Typ I , Invent.Math. (1968), 190–196. ↑ ¨Uber das regul¨are Mass im dualen Raum diskreter Gruppen , Math.Z. (1967), 257–271 (German). ↑ Ergodic theory and semisimple groups , Monographs in Math-ematics, vol. 81, Birkh¨auser Verlag, Basel, 1984. ↑ Bachir Bekka, Univ Rennes, CNRS, IRMAR–UMR 6625, Campus Beaulieu,F-35042 Rennes Cedex, France
Email address ::