Charmenability of arithmetic groups of product type
aa r X i v : . [ m a t h . G R ] N ov CHARMENABILITY OF ARITHMETIC GROUPS OF PRODUCT TYPE
URI BADER, R´EMI BOUTONNET, CYRIL HOUDAYER, AND JESSE PETERSON
Abstract.
We discuss special properties of the spaces of characters and positive definite func-tions, as well as their associated dynamics, for arithmetic groups of product type. Axiomatizingthese properties, we define the notions of charmenability and charfiniteness and study their ap-plications to the topological dynamics, ergodic theory and unitary representation theory of thegiven groups. To do that, we study singularity properties of equivariant normal ucp maps be-tween certain von Neumann algebras. We apply our discussion also to groups acting on productof trees. Introduction and statements of the main results
For a countable discrete group Γ, we consider the convex set PD (Γ) ⊂ ℓ ∞ (Γ) consisting ofnormalized positive definite functions and endow it with the weak ∗ -topology (which coincideswith the topology of pointwise convergence) and the Γ-action associated with the conjugationaction of Γ on itself. This is a compact convex Γ-space. Its compact convex subset consistingof Γ-fixed points is denoted by Char(Γ) and its elements are called characters of Γ. TheGNS representation ( π, H, ξ ) associated with a character φ ∈ Char(Γ) generates a tracial vonNeumann algebra M = π (Γ) ′′ . Then φ ∈ Char(Γ) is an extremal character if and only if M = π (Γ) ′′ is a factor, that is, a von Neumann algebra with trivial center.The problem of the classification of characters of higher rank lattices has seen important progressin the last fifteen years. It has also attracted a lot of attention because of its connection withthe theory of Invariant Random Subgroups (IRS) (see e.g. [7s12, AGV12, Ge14]). Bekka [Be06]obtained a complete classification of characters of SL n ( Z ) for n ≥
3. This result was laterextended by Peterson [Pe14] to all higher rank lattices with property (T). Recently, Boutonnet-Houdayer [BH19] strengthened these results and obtained a complete classification of stationarycharacters of higher rank lattices in simple Lie groups. We refer to [CP13, PT13, Be19, BeF20,LL20] for other classification results for characters.Before stating our main theorems, we first introduce some terminology.
Definition 1.1.
A character φ on Γ is called amenable if the corresponding GNS representation( π, H ) is amenable in the sense of [Be89], that is, π ⊗ π weakly contains the trivial representation.It is called von Neumann amenable if π (Γ) ′′ is moreover an amenable von Neumann algebra. Itis called finite if H is finite dimensional.Note that if Γ is amenable, then any Γ-invariant compact convex subset of PD (Γ) containsa character, and every character of Γ is von Neumann amenable. Conversely, a non-amenablegroup always contains a character that is not von Neumann amenable, namely the regularcharacter δ e . In fact, if Γ is non-amenable, any character supported on the amenable radical ofΓ is not amenable. Mathematics Subject Classification.
Key words and phrases.
Arithmetic groups; Characters; Irreducible lattices; Poisson boundaries; Semisimplealgebraic groups; Tree automorphism groups; Unitary representations; von Neumann algebras.UB is supported by ISF Moked 713510 grant number 2919/19.RB is supported by a PEPS grant from CNRS and ANR grant AODynG, 19-CE40-0008.CH is supported by Institut Universitaire de France.JP is supported by NSF Grant DMS Beware that in some texts the term “character” is reserved for an extreme point in this set.
Definition 1.2.
The group Γ is said to be charmenable if it satisfies the following two properties:(1) Every compact convex Γ-invariant subset of PD (Γ) contains a character.(2) Every extremal character of Γ is either supported on the amenable radical Rad(Γ) orvon Neumann amenable.Moreover, Γ is said to be charfinite if it also satisfies the following properties:(3) Rad(Γ) is finite.(4) Γ has a finite number of isomorphism classes of unitary representations in each givenfinite dimension.(5) Every amenable extremal character of Γ is finite.As we will see in §
3, charmenable and charfinite groups enjoy remarkable properties pertainingto the structure of C ∗ -algebras associated with their unitary representations and the stabilizerstructure of their ergodic and topological actions. In particular, we will see the following (see[GW14] for the notion of URS): • For any charmenable group Γ with trivial amenable radical, any non-amenable unitaryΓ-representation weakly contains the left regular representation and any URS carries aΓ-invariant Borel probability measure. • Furthermore, for any charfinite group Γ, all URS and all ergodic IRS are finite.Our main result deals with arithmetic groups of product type.
Definition 1.3.
Let K be a global field and G a connected non-commutative K -almost simple K -algebraic group. Let S be a (possibly empty, possibly infinite) set of non-archimedean in-equivalent absolute values on K , let O < K be the ring of integers and let O S the correspondinglocalization, that is, O S = { α ∈ K | ∀ s ∈ S, s ( α ) ≤ } . Fix an injective K -representation ρ : G → GL n and denoteΛ S = ρ − (GL n ( O S )) ≤ G ( K ) . The triple ( K, G , S ) is said to be • of a compact type if for every absolute value v on K , the image of Λ S in G ( K v ) isbounded, • of a simple type if there exists a unique absolute value v on K such that the image ofΛ S in G ( K v ) is unbounded • and of a product type otherwise.The triple ( K, G , S ) is said to be of higher rank if it is either of a product type or of a simpletype and rank K v ( G ) ≥ ≤ G ( K ) is called S -arithmetic if it is commensurable with Λ S . It is called arithmetic if it is S -arithmetic for some S as above and we regard its type as the type of( K, G , S ). Example.
Let K = Q , G = SL n for n ≥ S ⊂ P a (possibly empty, possibly infinite) setof primes. If S = ∅ , then SL n ( Z S ) ≤ SL n ( Q ) is an S -arithmetic group of product type. Theorem A.
Let K be a global field and G a connected non-commutative K -almost simple K -algebraic group. If Γ ≤ G ( K ) is an arithmetic subgroup of a product type then Γ is charmenable.Assume further that there exists an absolute value v on K such that G ( K v ) has property (T) andfor which the image of Γ in G ( K v ) is unbounded. If either S is finite or G is simply connectedthen Γ is charfinite. HARMENABILITY OF ARITHMETIC GROUPS OF PRODUCT TYPE 3
The proof of Theorem A will be given in § Theorem B.
For every non-empty set of primes S , the group SL ( Z S ) is charfinite. The proof of Theorem B will be given in § ( Q ) (that is, where S in the above theorem is the set ofall primes) is particularly interesting, as this group has no non-trivial finite dimensional unitaryrepresentations. It follows that the only extremal characters on this group are the regular andthe trivial characters and that every Γ-invariant compact convex subset of PD (Γ) contains aconvex combination of these two characters. However, Γ is not finitely generated. It will bevery interesting to find a finitely generated charfinite simple infinite group. We expect certainKac-Moody groups to satisfy all of these properties.When Γ is of a simple type and the corresponding absolute value is archimedean (e.g. Γ =SL n ( Z )), the conclusion of Theorem A still holds under the assumption that Γ is of higher rank(e.g. n ≥ Theorem C.
For any n ≥ , the group SL n ( Z ) ⋉ Z n is charmenable. The proof of Theorem C will be given in § G, N )-von Neumann algebra M , whichis a choice of an equivariant normal ucp map M → N , where G is an lcsc group and M, N are G -von Neumann algebras. In § singularity of a ( G, N )-structure. The proofs of all theorems presented above willrely on these charmenability criteria. The proofs of Corollary 7.7 and Theorem C will also relyheavily on [BH19, Theorem B] which forms a noncommutative Nevo-Zimmer structure theoremfor stationary actions on von Neumann algebras. However, as pointed out in [NZ97, NZ00],such a structure theorem cannot hold for semisimple Lie groups admitting a rank one factor andtherefore the method of [BH19] could not be applied for proving our main theorem, Theorem A.To overcome this conceptual difficulty we develop a new strategy which applies in the settingof lattices with dense projections.
Definition 1.4.
Let I be a finite set and G i be an lcsc group for each i ∈ I . Let G = Q i ∈ I G i and Γ ≤ G be a lattice. We say that Γ has dense projections if its image in Q i = i G i is dense,for every i ∈ I .For such a lattice with dense projections Γ ≤ G we will consider in Theorem 5.8 the structure of(Γ , N )-von Neumann algebras, where N is the L ∞ -algebra of the Furstenberg-Poisson boundaryof G . This theorem will allow us to shift the discussion on Γ-dynamical systems to G i -dynamicalsystems, where G i is one of the simple factors. From there we will use the special form of G i ,its parabolic subgroups and Mautner phenomenon to deduce the desired singularity property.This second half is based on Proposition 4.14.In fact, for the groups considered in this paper, the combination of Theorem 5.8 and Proposition4.14 implies condition (a) from Proposition 4.16, which is our replacement of [BH19, TheoremB]. Note that in the setting of [BH19], the non-commutative Nevo-Zimmer theorem implies thiscondition (a), although the proof is different.Note that in the setting of Theorem A, if S is finite and G is simply connected then the groupΓ is a lattice with dense projections in Q v ∈ V G ( K v ), where V denotes the set of places of K under which Γ is unbounded. This indeed holds by the strong approximation theorem, see[Ma91, Theorem II.6.8]. In the proof of Proposition 6.1, which is a main step towards provingTheorem A, we will explain how to reduce the general case to the case above, finite S and simply URI BADER, R´EMI BOUTONNET, CYRIL HOUDAYER, AND JESSE PETERSON connected G , in which Theorem 5.8 is applicable. We emphasize that, apart of invoking thestrong approximation theorem, our work here does not rely at all on arithmeticity properties ofthe groups under consideration and our choice of presenting Theorem A for arithmetic latticesrather then lattices with dense projections in the first place is a matter of taste more thananything else.Our next theorem deals with a geometric situation which generalizes a product of rank onegroups over non-archimedean fields. Theorem D.
For n ≥ and i = 1 , . . . , n , let T i be a bi-regular tree and let G i be a closedsubgroup of Aut + ( T i ) , the group of the bicoloring preserving automorphisms of T i , which acts -transitively on its boundary. Let Γ < G × · · · × G n be a cocompact lattice with dense projections.Then Γ is charmenable. The proof of Theorem D will be given in § G i in lieu of semisimplicity. Remark 1.5.
It seems reasonable to expect that some or all of the groups in Theorem D areactually charfinite, but so far we are unable to prove property (5) in Definition 1.2. The fact thatthese groups satisfy property (3) is well known and property (4) is established in Proposition 7.3below.
Remark 1.6.
In a sequel work, we will show that in Theorem A, the assumption that Γ is ofproduct type could be replaced by a higher rank assumption. We will do this by combiningthe techniques developed in the current paper with the ones developed in [BH19]. This willcompletely settle the question of charmenability for lattices in semisimple algebraic groups.
Acknowledgments.
We wish to thank Yair Glasner and Pierre-Emmanuel Caprace for providingus with the proof of Proposition 6.4. We thank Pierre-Emmanuel Caprace also for findinga flawed claim we made in an earlier version of this paper and for various suggestions forimprovements of our presentation. We are grateful to Adrian Ioana and Narutaka Ozawa fortheir valuable remarks.
Contents
1. Introduction and statements of the main results 12. Preliminaries 53. Charmenable and charfinite groups 114. (
G, N )-structures, singularity and criteria for charmenability 165. (
G, N )-structures, lattices with dense projections and induction 256. Proofs of charmenability 307. Proofs of charfiniteness 34References 37
HARMENABILITY OF ARITHMETIC GROUPS OF PRODUCT TYPE 5 Preliminaries
In this section we collect various preliminary definitions and results. § § § § Positive definite functions.
In this subsection we consider positive definite functions ona locally compact group G . The L p -spaces over G will be considered with respect to the Haarmeasure µ G .Recall that a function φ ∈ L ∞ ( G ) is said to be positive definite if R G ( f ∗ ∗ f ) φ d µ G ≥
0, forevery f ∈ L ( G ). A positive definite function is necessarily continuous, that is, agrees a.e. witha continuous function. The set of all positive definite functions on G is denoted PD( G ) and wedenote by PD ( G ) the subset of functions φ satisfying φ ( e ) = 1. We endow it with the subspacetopology inherited from the weak*-topology on L ∞ ( G ). It becomes a compact convex G -space.The compact convex subset consisting of G -invariant points in PD ( G ) is denoted Char( G )and its elements are called characters. The extreme points of Char( G ) are called extremalcharacters. Definition 2.1.
Let φ ∈ PD( G ). By definition, the associated GNS triple ( π φ , H φ , ξ φ ) is thedata of a unitary representation π φ of G on the Hilbert space H φ , together with a cyclic vector ξ φ ∈ H φ satisfying h π φ ( g ) ξ φ , ξ φ i = φ ( g ), for all g ∈ G . Such a triple is unique up to conjugation(i.e. up to an isomorphism of the Hilbert spaces, which intertwines the representations, andmaps cyclic vector to cyclic vector).For φ ∈ Char( G ), π φ extends to a unitary representation e π φ of G × G on H φ whose restrictionto the left factor is π φ and for which ξ φ is invariant under the diagonal subgroup in G × G .Every lcsc group has at least one character: the trivial character, namely the constant function 1.Every non-trivial discrete group has at least one more character, the regular character δ e . Ingeneral the trivial character might be the only character. Proposition 2.2.
Let k be a local field and G a connected simply connected k -isotropic k -almostsimple k -algebraic group and denote G = G ( k ) . Then Char( G ) = { } . This Proposition is essentially due to Segal and von Neumann who proved it for non-compactsimple Lie groups, see [SN50] . Proof.
Given a character φ = 1 of G we get an isometric action of G × G on H φ where the factorgroups (which are the only non-compact normal subgroups) have no non-zero fixed point andthe vector ξ φ is invariant under the diagonal subgroup in G × G , contradicting the Howe-Mooretheorem, see e.g [BG14, Theorem 6.2]. (cid:3) The assumption that G is k -isotropic is equivalent to the non-compactness of G ( k ) and it isessential. Indeed, non-trivial compact groups always admit non-trivial characters. The followingis well known. Proposition 2.3.
Let K be a compact group. Then the extremal characters of K are in one toone correspondence with its irreducible representations; the correspondence is given by assigningto the irreducible representation π the character g tr π ( π ( g )) , where tr π is the normalizedtrace associated with π . The GNS construction associated with this character is the direct sumof dim( π ) copies of π .In particular, every character of K is obtained by a summable convex combination of countablymany extremal characters. See the discussion in p. 2 of [BG14] for some history of ideas.
URI BADER, R´EMI BOUTONNET, CYRIL HOUDAYER, AND JESSE PETERSON
Definition 2.4.
Let φ ∈ PD( G ). We say that φ is • Compact if π φ is a compact representation, i.e. π φ ( G ) is relatively compact in U ( H φ ) forthe strong operator topology. • Amenable if π φ is amenable, i.e. there is a state Φ on B ( H ) which is invariant underAd( π φ ( g )), g ∈ G or equivalently, π ⊗ π weakly contains the trivial representation. • von Neumann amenable if π φ ( G ) ′′ is an injective von Neumann algebra, i.e. there is aconditional expectation E : B ( H φ ) → π φ ( G ) ′′ .A compact positive definite function is von Neumann amenable, hence amenable. Note that acompact character is a character which factorizes through a character on the Bohr compact-ification of G . So by the previous proposition, any compact character is a countable convexcombination of countably many extremal characters.In the case where φ is a character, φ is von Neumann amenable if and only if there exists anAd( π φ ( G ))-central state on B ( H φ ) such that Φ( x ) = h xξ φ , ξ φ i for every x ∈ π φ ( G ) ′′ . In otherwords, Φ is an extension of φ , which is normal on π φ ( G ) ′′ . Indeed, the existence of such astate implies the amenability property of π φ ( G ) ′′ , which is known to be equivalent to injectivity.On the other hand, if π φ ( G ) ′′ is injective, then we can compose the conditional expectation E : B ( H φ ) → π φ ( G ) ′′ with the trace h · ξ φ , ξ φ i on π φ ( G ) ′′ to get the desired state extension.We point out that the spaces of compact or (von Neumann) amenable PD-functions are notclosed in general. For example, if G is a non-amenable residually finite discrete group thenthe regular character, which is not amenable, lies in the closure of the compact characters.Nevertheless these sets are easily checked to be Borel sets. Moreover, we have the followingconvexity property. Lemma 2.5.
Let ν ∈ Prob(Char( G )) and t := ν ( { von Neumann amenable characters } ) . De-note by φ := Bar( ν ) , by ( H, π, ξ ) the corresponding GNS triple, by M := π ( G ) ′′ and by τ = h · ξ, ξ i the unique normal trace on M that extends φ .Then there exists a projection p ∈ M with trace at least t such that pM p is amenable. Inparticular, if ν is supported on the set of von Neumann amenable characters (i.e. t = 1 ) then Bar( ν ) is von Neumann amenable.Proof. For simplicity, we denote by X = Char( G ) and by X the subset of von Neumannamenable characters. Denote φ = Bar( ν ) and identify ( π, H, ξ ) with (the cyclic subspace of)( e π, e H, e ξ ) := Z ⊕ X ( π ψ , H ψ , ξ ψ ) d ν ( ψ ) . It is shown in [AB18, Lemma 4.1] that e π ( G ) ′′ ≃ π ( G ) ′′ = M (and we observe that this identifi-cation preserves the trace). So in the sequel we will rather denote by M = e π ( G ) ′′ .Denote by p = X ∈ B ( e H ) the orthogonal projection onto the subspace R ⊕ X H ψ d ν ( ψ ) ⊂ e H .Then this projection lies inside e π ( G ) ′ = M ′ and satisfies h p e ξ, e ξ i = t . Moreover M p is containedin the amenable tracial von Neumann algebra R ⊕ X π ψ ( G ) ′′ d ν ( ψ ), so M p is amenable as well.Denote by p ∈ Z ( M ) = Z ( M ′ ) the central support of p ∈ M ′ . Then pM is amenable and wehave τ ( p ) = h p e ξ, e ξ i ≥ h p e ξ, e ξ i = t , as desired. (cid:3) However the above convexity property doesn’t hold for compact characters. For example, if G = Z , then the regular character is not compact, but by Fourier transform, it is the Lebesgueaverage of the compact characters φ z : n ∈ Z e i πnz , z ∈ [0 , Group actions on operator algebras.
In this paper, we will consider groups actionson C ∗ -algebras and von Neumann algebras. Let G be an lcsc group.By a G -C*-algebra we mean a C ∗ -algebra A endowed with a continuous map G × A → A , calledthe action map, which induces an action of G on A , to be denoted G y A , by C*-algebra HARMENABILITY OF ARITHMETIC GROUPS OF PRODUCT TYPE 7 automorphisms. Such an action G y A induces a weak ∗ continuous affine action of G on thestate space S ( A ) defined by the formula gφ := φ ◦ g − , for all g ∈ G , φ ∈ S ( A ). In particular,every probability measure µ ∈ Prob( G ) defines a convolution operator φ ∈ S ( A ) µ ∗ φ := Z G gφ d µ ( g ) ∈ S ( A ) . A fixed point for this convolution operator is called a µ -stationary state on A . We will denoteby S µ ( A ) ⊂ S ( A ) the closed convex subset of µ -stationary states.By a G -von Neumann algebra we mean a von Neumann algebra M endowed with a map G × M → M which is continuous with respect to the ultraweak topology on M and which induces an actionof G on M by von Neumann algebra automorphisms. Recall that the ultraweak topology is theweak-* topology when M is identified with the dual of its pre-dual M = ( M ∗ ) ∗ and note thatin general a G -von Neumann algebra is not a G -C*-algebra. Again, such an action defines byduality an affine action G y M ∗ , which is continuous for the norm topology on M ∗ . As in theC*-case, any probability measure µ ∈ Prob( G ) gives rise to a convolution operator on M ∗ , anda normal state φ on M fixed by this operator is called a µ -stationary state on M . We say thatthe action G y M is ergodic if the fixed point algebra M G := { x ∈ M | gx = x, ∀ g ∈ G } istrivial.To avoid notational misinterpretation, unless otherwise specified, we will generically use thenotation σ to denote our actions.By a regularization argument, any G -von Neumann algebra M admits an ultraweakly denseC*-subalgebra A on which the action is norm continuous, see the proof of [Ta03b, PropositionXIII.1.2]). Since we assume G to be second countable, if M has separable predual we maychoose A to be a separable C*-subalgebra. This passage to a G -C*-algebra parallels the choiceof a compact model in classical ergodic theory.In the other direction, given a G -C*-algebra A , we may extend the G -action on A to a G -actionon A ∗∗ but unfortunately this action is not continuous in general. However, when one restrictsto certain corners of A ∗∗ it may be continuous. Proposition 2.6.
Let A be a G -C*-algebra and N be a G -von Neumann algebra. Consider a G -equivariant unital completely positive (ucp) map E : A → N .Extend E to a normal ucp map on A ∗∗ and extend the G -action on A to a (non-continuous)action on A ∗∗ . Denote by z ∈ A ∗∗ the central support projection of E , i.e. the smallest projectionin Z ( A ∗∗ ) such that E ( z ) = 1 . Then z is G -invariant and the G -action on zA ∗∗ is a continuousvon Neumann algebraic action.Proof. One can prove this fact by identifying zA ∗∗ with the “Stinespring von Neumann al-gebra” of E , and checking that the action can be unitarily implemented in the Stinespringrepresentation, and hence extends to a continuous action on zA ∗∗ .The more concise argument we provide was communicated to us by Narutaka Ozawa. Denoteby M := zA ∗∗ . By Hahn-Banach theorem, the set L of all normal linear functionals of the form x ∈ M ( ψ ◦ E )( axb ) ∈ C , for ψ ∈ N ∗ , a, b ∈ A , spans a norm dense subspace in the predual M ∗ . Since the G -actions on N ∗ and on A are norm continuous, one checks that for every φ ∈ L ,the map g ∈ G gφ ∈ M ∗ is norm continuous. So the action G y M ∗ is norm continuous, andthis is known to be equivalent to the action G y M being a continuous von Neumann algebraicaction. (cid:3) More generally, G -invariant corners of A ∗∗ on which the action is continuous have been studiedby Ikunishi, see [Ik87].Let us now give two results about fixed point algebras in stationary von Neumann algebras. Werecall that a probability measure µ on an lcsc group G is generating if its support generates adense semi-group of G . URI BADER, R´EMI BOUTONNET, CYRIL HOUDAYER, AND JESSE PETERSON
Proposition 2.7.
Let G be an lcsc group with a generating probability measure µ ∈ Prob( G ) and M be a G -von Neumann algebra with a faithful µ -stationary state φ ∈ M ∗ . The followingfacts hold true.(1) There exists a unique φ -preserving normal conditional expectation E µ : M → M G .(2) Every µ -stationary normal state ψ ∈ M ∗ satisfies ψ = ψ ◦ E µ . In particular, if the action G y M is ergodic, then φ is the only µ -stationary normal state on M .(3) Let A ⊂ M be any ultraweakly dense unital G - C ∗ -subalgebra. Then G y M is ergodicif and only if φ | A ∈ S µ ( A ) is an extreme point.Proof. Given (
G, µ ) and M as in the statement, define the convolution ucp map T µ : x ∈ M ˇ µ ∗ x = Z G σ − g ( x ) d µ ( g ) ∈ M. Since µ ∗ φ = φ , we have φ ◦ T µ = φ . Since φ ∈ M ∗ is faithful, this implies that T µ : M → M isa faithful normal ucp map. Next, choose a non-principal ultrafilter ω ∈ β ( N ) \ N and define E µ : x ∈ M lim n → ω n n X k =1 T nµ ( x ) ∈ M. Here the limit is meant for the ultra-weak topology. Observe that E µ is a ucp map on M , it isidempotent and its image is the set of elements invariant under T µ .(1) Let x ∈ M such that E µ ( x ) = x . Then T µ ( x ) = x and since φ = φ ◦ T µ , we find that Z G k x − σ − g ( x ) k φ d µ ( g ) = Z G (cid:0) φ ( x ∗ x ) − ℜ ( φ ( x ∗ σ − g ( x ))) + φ ( σ − g ( x ∗ x )) (cid:1) d µ ( g )= k x k φ − ℜ ( φ ( x ∗ T µ ( x ))) + φ ◦ T µ ( x ∗ x ) = 0 . This implies that σ − g ( x ) = x for µ -almost every g ∈ G . Since µ is generating and G y M iscontinuous, it follows that x ∈ M G . Therefore, E µ : M → M G is a conditional expectation.Since φ ◦ E µ = φ and φ is a faithful normal state this implies that E µ is also faithful and normal;it is the unique φ -preserving condition expectation onto M G .(2) For every µ -stationary normal state ψ ∈ M ∗ , we have ψ = ψ ◦ T µ . So the formula ψ = ψ ◦ E µ follows from the concrete formula defining E µ . If the action is ergodic then ψ ( x )1 = E µ ( x ) = φ ( x )1 for every x ∈ M , showing that ψ = φ .(3) Assume first that the action is ergodic. If ψ ∈ S µ ( A ) is a positive linear functional suchthat ψ ≤ φ , then ψ extends continuously to a normal linear functional on M , which must beproportional to φ thanks to (2). This implies that φ | A ∈ S µ ( A ) is an extreme point.Conversely, assume that φ | A ∈ S µ ( A ) is an extreme point. Take an element p ∈ M G , with0 ≤ p ≤
1. Denote by ( L ( M ) , L ( M ) + , J ) the standard form of M , and by ξ ∈ L ( M ) + theunique positive vector implementing φ (see [Ha73]). Define a linear functional ψ ∈ M ∗ by theformula ψ ( x ) = h xJ pJ ξ, ξ i , for every x ∈ M. We claim that ψ is µ -stationary as well. In fact, since φ ◦ E µ = φ we have e µ ( ξ ) = ξ , where e µ isthe orthogonal projection L ( M ) → L ( M G ) corresponding to the conditional expectation E µ .Indeed this equivalence follows from the fact that e µ maps positive vectors to positive vectors,and e µ ( ξ ) implements φ ◦ E µ (thanks to the formula e µ xe µ = E µ ( x ) e µ , for all x ∈ M ).The projection e µ commutes with J and since p is G -invariant, we have e µ p = pe µ . So for every x ∈ M , ψ ( x ) = h xJ pJ e µ ( ξ ) , e µ ( ξ ) i = h e µ xe µ J pJ ξ, ξ i = h E µ ( x ) e µ J pJ ξ, ξ i = ψ ◦ E µ ( x ) . This shows that indeed ψ is µ -stationary. Moreover, it is obvious that 0 ≤ ψ ≤ φ , so byextremality of φ | A ∈ S µ ( A ), ψ must be proportional to φ on A , and hence on M (by ultraweakcontinuity): ψ = cφ for some c ∈ [0 , h J pJ xξ, yξ i = c h xξ, yξ i for every x, y ∈ M . Since φ is faithful, ξ is a cyclic vector and hence J pJ = c ∈ C M G = C (cid:3) HARMENABILITY OF ARITHMETIC GROUPS OF PRODUCT TYPE 9
Lemma 2.8.
Let G = G × G be the product of two lcsc groups. Choose generating measures µ ∈ Prob( G ) , µ ∈ Prob( G ) , and denote by µ = µ ⊗ µ ∈ Prob( G ) the product measureon G .Let M be a G -von Neumann algebra with a faithful normal µ -stationary state φ ∈ M ∗ . Choose i = j ∈ { , } . The following facts hold true.(1) φ is µ i -stationary.(2) The unique φ -preserving normal conditional expectation E i : M → M G i is G j -equivariant.(3) If φ is not G j -invariant, φ | M Gi is not G j -invariant. In particular, we have M G i = C .Proof. (1) Note that µ ∗ µ j = µ j ∗ µ . Then we have µ j ∗ φ = µ j ∗ µ ∗ φ = µ ∗ ( µ j ∗ φ ) . This shows that µ j ∗ φ is a µ -stationary normal state. Since µ j ∗ φ | M G = φ | M G , Proposition2.7(2) implies that µ j ∗ φ = φ .(2) The existence and uniqueness of E i follows from Proposition 2.7. The equivariance propertyfollows from uniqueness: for g ∈ G j , gE i g − must equal E i .(3) follows trivially from (2). (cid:3) Metric ergodicity and the Mautner property.
Metric ergodicity is an important toolthat we use. We recall its definition.
Definition 2.9.
Let G be an lcsc group and B a G -Lebesgue space. The action of G on B iscalled metrically ergodic if every measurable G -map from B into a separable metric G -space X on which G acts continuously by isometries is essentially constant, equal to a G -fixed point. Wewill say that B is G -metrically ergodic.We refer to [BF14, Section 2] for examples and further extensions of this notion. For homoge-neous spaces, metric ergodicity is closely related to the Mautner phenomenon. Definition 2.10.
Let P be an lcsc group and A ≤ P a closed subgroup. The pair ( P, A ) issaid to have the
Mautner property if for every continuous action of P on a metric space X byisometries, every point x ∈ X which is A -invariant is P -invariant.The following is an immediate consequence of [BG14, Lemma 6.3]. Lemma 2.11.
Let P be an lcsc group and A ≤ P a closed subgroup. Endow P/A with theunique P -invariant measure class. Then the action of P on P/A is metrically ergodic if andonly if the pair ( P, A ) has the Mautner property. Definition 2.12.
Let G be a topological group and P ≤ G a closed subgroup. We will say that • P has the relative Mautner property in G if for every g ∈ G , the pair ( P, P ∩ gP g − )has the Mautner property. • P is stably self normalizing in G if every intermediate closed subgroup P ≤ Q ≤ G isits own normalizer in G .Note that both the relative Mautner property and the stably self normalizing property of P in G are conjugation invariant: if P has it then also gP g − for any g ∈ G . Lemma 2.13.
Let G be an lcsc group and P ≤ G a closed subgroup. Assume that P is stablyself normalizing and it has the relative Mautner property in G . Let H be a (not necessarilyclosed) subgroup of G which contains P . Then P is stably self normalizing and it has therelative Mautner property in H , where H is taken with the topology induced from G , and thepair ( H, P ) has the Mautner property. Moreover, for every g ∈ H , the pair ( H, P ∩ gP g − ) hasthe Mautner property. Proof.
The fact that P has the relative Mautner property in H is immediate. To see that P isstably self normalizing in H we fix an intermediate closed subgroup P ≤ Q ≤ H and g ∈ H normalizing Q . Since g normalizes ¯ Q in G we get that g ∈ ¯ Q , as ¯ Q is self normalizing in G . Itfollows that indeed g ∈ H ∩ ¯ Q = Q . In view of the above we assume as we may H = G .We are only left to prove that the pair ( G, P ) has the Mautner property, the moreover part willthen follow from the obvious fact that if (
G, P ) and (
P, P ∩ gP g − ) have the Mautner property,then so does ( G, P ∩ gP g − ).Let X be a metric space on which G acts continuously by isometries and let x ∈ X be a P -fixed point. Denote by S the stabilizer of x . Note that ( gP g − , P ∩ gP g − ) has the Mautnerproperty, so x must be fixed by gP g − , for every g ∈ G . So the closed group Q generated by S g ∈ G gP g − is contained in S . Since Q is a normal subgroup of G which contains P , the stablyself-normalizing condition implies that Q = G . Hence S = G , as desired. (cid:3) Example 2.14.
Let k be a local field and G a connected non-commutative k -isotropic k -almost simple k -algebraic group. Let P be a minimal k -parabolic subgroup. We consider thesubgroup G ( k ) + discussed in [Ma91, Sections I.1.5 and I.2.3] and recall that it is the imageof ˜ G ( k ) under the covering map ˜ G → G , where ˜ G is the simply connected cover of G , see[Ma91, Theorem I.2.3.1(a) and Proposition I.1.5.5]. Let G be an intermediate closed subgroup G ( k ) + ≤ G ≤ G ( k ) and set P = G ∩ P ( k ). Note that by [Bo91, Proposition 20.5] we havea natural identification G ( k ) / P ( k ) = G / P ( k ). We claim that G acts transitively on G / P ( k )with stabilizer P and P is stably self normalizing and it has the relative Mautner property in G . We claim further that for every intermediate subgroup P ≤ H ≤ G , not necessarily closeda priori, we have H = G ∩ Q ( k ) for some intermediate k -parabolic subgroup P ≤ Q ≤ G .The fact that G acts transitively on G ( k ) / P ( k ) follows from the fact that already G ( k ) + does.Indeed, G ( k ) + P ( k ) = G ( k ) by [Ma91, Proposition I.1.5.4(vi)]. It is now also clear that thestabilizer of the base point is P .Next we show that P has the relative Mautner property in G . We fix g ∈ G and consider thecorresponding conjugation of P , P g . Using [Bo91, Corollary 20.7(i)] we find a maximal k -splittorus T ⊂ P ∩ P g and using [Bo91, Theorem 22.6(i),(ii)] we lift P and T to a k -parabolic ˜ P ⊂ ˜ G and a maximal k -split torus ˜ T ⊂ ˜ P . We let U and ˜ U be the corresponding unipotent radicals of P and ˜ P , which are defined over k by [Bo91, Theorem 20.5]. We set ˜ T = ˜ T ( k ) and ˜ U = ˜ U ( k ).It is a standard fact, the classical Mautner phenomenon, that the pair ( ˜ T ˜ U , ˜ T ) has the Mautnerproperty. Let us explain why this implies that ( P, P ∩ P g ) has the Mautner property. Thanksto Lemma 2.11, it is enough consider the equivariant map ˜ T ˜ U / ˜ T → P/P ∩ P g , induced by thenatural homomorphism ˜ T ˜ U → P , and to observe that ˜ T ˜ U acts transitively on P/P ∩ P g . Infact, we may consider further the map P/P ∩ P g → P ( k ) / P ( k ) ∩ P g ( k ) and note that U = U ( k )acts transitively on the latter by the Bruhat decomposition [Bo91, Theorem 21.15] and the map˜ U → U is surjective by [Bo91, Proposition 22.4(ii)]. Thus indeed, P has the relative Mautnerproperty in G .We now let H be a (not necessarily closed a priori) intermediate subgroup P ≤ H ≤ G . Welet ˜ H be its preimage in ˜ G ( k ) and note that ˜ P ( k ) ≤ ˜ H . By [Bo91, Theorem 21.15] we havethat ( ˜ G ( k ) , ˜ P ( k ) , N ˜ G ( ˜ T )( k ) , S ) is a Tits system where S is the associated set of generators ofthe corresponding Weyl group. We conclude that ˜ H = ˜ Q ( k ) for some k -parabolic subgroup ˜ Q in ˜ G and it is self normalizing. Since ˜ G ( k ) acts transitively on G/H , as it acts transitivelyon
G/P , we get that there is a unique ˜ G ( k )-equivariant isomorphism between ˜ G ( k ) / ˜ Q ( k ) and G/H . By [Bo91, Theorem 22.6(i)] there exists a k -parabolic subgroup Q in G correspondingto ˜ Q , thus ˜ G ( k ) / ˜ Q ( k ) and G ( k ) / Q ( k ) are isomorphic as ˜ G ( k ) spaces, and we conclude havinga unique G ( k ) + -equivariant isomorphism between G ( k ) / Q ( k ) and G/H . As G ( k ) + is normalin G , we get that G acts by conjugation on the set of all such G ( k ) + -equivariant isomorphisms,thus this unique isomorphism must be G -invariant for the conjugation action, equivalently it is G -equivariant. It follows that indeed, H = G ∩ Q ( k ). Moreover, it follows that G/H has nonon-trivial G -equivariant self maps, thus H is self normalizing. HARMENABILITY OF ARITHMETIC GROUPS OF PRODUCT TYPE 11
Example 2.15.
Consider a thick simplicial tree T and a group G <
Aut( T ) acting co-compactlyon T . Denote by ∂T the visual boundary of the tree, and assume that the action of G on ∂T is 2-transitive.Fix ξ ∈ ∂T , and denote by P the stabilizer of ξ in G . Then G/P is homeomorphic with ∂T .Since G acts 2-transitively on G/P , we have a decomposition G = P ⊔ P gP , for any g ∈ G \ P .In particular, P is a maximal subgroup in G . So its normalizer is either equal to P or to G ,and the later case is impossible under the 2-transitivity assumption. In conclusion P is stablyself normalizing. The next lemma ensures the Mautner condition. Lemma 2.16.
Keep the notation from Example 2.15. Then ( G, P ) has the relative Mautnerproperty.Proof. We first point out that the 2-transitivity assumption implies that G has no fixed pointin T . Indeed, otherwise its closure in Aut( T ) is compact, thus so is its unique non-diagonalorbit in ∂T × ∂T and it follows that the diagonal is open, thus ∂T is discrete, contradicting thethickness assumption. It follows that G contains a hyperbolic element.Fix g ∈ G and take a continuous isometric action P y X on a metric space X . Assume that x ∈ X is a P ∩ gP g − -invariant point. We need to prove that it is P -invariant. Obviously, wemay assume that g / ∈ P . In this case we observe that the P -orbit of gξ is open in ∂T , equal to ∂T \ ξ . In particular, the orbit map p ∈ P pgξ ∈ ∂T is open.Take h ∈ P and ε >
0. Consider the neighborhood U := { pgξ | p ∈ P, d ( px, x ) < ε } of gξ in ∂T .Since G contains a hyperbolic element, it contains a hyperbolic element k with axis [ gξ, ξ ], by2-transitivity. Then k ∈ P ∩ gP g − , and we may find n ∈ Z such that k n hgξ ∈ U . By definitionof U , there exists p ∈ P such that d ( px, x ) < ε and pgξ = k n hgξ . In this case, the element p − k n h belongs to P ∩ gP g − , and therefore fixes x . We may then compute: d ( hx, x ) = d ( hx, k − n x ) = d ( k n hx, x ) ≤ d ( k n hx, px ) + d ( px, x ) < ε. Since ε is arbitrarily small, h must fix x . (cid:3) Charmenable and charfinite groups
In this section, we discuss charmenable and charfinite groups as defined in Definition 1.2. In § § § § § Observation 3.1.
Every amenable group is charmenable and every finite group is charfinite.Further, an amenable group is charfinite if and only if it is finite.
Properties of charmenable and charfinite groups.
The following lemma will be oftenused without mention.
Lemma 3.2.
Every character of a charmenable group Γ is a convex combination of a vonNeumann amenable character and a character supported on Rad(Γ) . In particular, the set ofcharacters supported on
Rad(Γ) is a face of
Char(Γ) and its complement set consists of amenablecharacters.Furthermore, if Γ is charfinite, then the GNS representation of an amenable character containsa finite dimensional subrepresentation.Proof. The lemma holds trivially if Γ is amenable, thus we assume that this is not the case. Inparticular, the characters supported on Rad(Γ) are non-amenable. We denote the set of suchcharacters Char
Rad (Γ).Given φ ∈ Char(Γ) we get by Choquet’s representation theorem ν ∈ Prob(Char(Γ)) supportedon the extreme points such that φ = Bar( ν ). By the assumption that Γ is charmenable we have that ν is a convex combination of ν and ν , where ν is supported on Char Rad (Γ) and ν issupported on the set of von Neumann amenable characters. We conclude that φ is a convexcombination of φ and φ , where φ i = Bar( ν i ). Clearly, φ ∈ Char
Rad (Γ) and by Lemma 2.5, φ is von Neumann amenable. This proves the first claim.If φ = 0 then φ = φ ∈ Char
Rad (Γ). Otherwise, the GNS representation of φ contains the GNSrepresentation of φ and is thus amenable. We get that φ is amenable if and only if it is notin Char Rad (Γ) and conclude that, indeed, Char
Rad (Γ) is a face whose complement consists ofamenable characters.Finally, if Γ is charfinite and φ is amenable then the GNS representation of φ , thus also of φ ,contains a finite dimensional subrepresentation as ν is atomic and its atoms consist of finitecharacters. Indeed, ν is supported on a countable set of finite characters. (cid:3) The next proposition is a special case of Proposition 3.4 and Proposition 3.5. Nevertheless westate it here for its importance and the clarity of its proof.
Proposition 3.3.
Every normal subgroup N ⊳ Γ of a charmenable group is amenable or co-amenable in Γ . If further Γ is a charfinite group then N is finite or of finite index in Γ .Proof. If N is non-amenable, then φ := χ N is not supported on Rad(Γ). Thus π φ is an amenablerepresentation and we get that Γ /N is an amenable group. If Γ is charfinite and N is infinitethen also φ is not supported on Rad(Γ), which is finite, thus again π φ is amenable, hence itcontains a finite dimensional subrepresentation. However, π φ is the regular representation ofΓ /N and so it follows that indeed, Γ /N is finite. (cid:3) We denote by Sub(Γ) the space consisting of all subgroups of Γ and endow it with the Chabautytopology. This is a compact space on which Γ acts by conjugation. An IRS of Γ is a Γ-invariantprobability measure on Sub(Γ) (see [AGV12]).
Proposition 3.4.
Let Γ be a charfinite group and assume that Γ acts ergodically on the proba-bility space ( X, µ ) preserving the measure µ . Then either X is essentially finite or the stabilizerof a.e. point of X is contained in Rad(Γ) . In particular, every ergodic IRS of Γ is finite.Proof. We assume X is not essentially finite and consider the character φ ( g ) = µ (Fix( g )). Wenote that the GNS representation associated with φ is a sub-representation of L ( R ) where R ⊂ X × X is the orbit equivalence relation endowed with the µ -integration of the countingmeasures on the fibers of the first coordinate projection. By [PT13, Proposition 3.1], thisrepresentation is weakly mixing and we deduce by charfiniteness that φ is indeed supported onRad(Γ). We note that the IRS associated with X via the stabilizer map X → Sub(Γ) is finite,as there are only finitely many subgroups in Rad(Γ). The last bit of the proposition follows bythe fact that every IRS of Γ can be obtained as the image such a stabilizer map. (cid:3)
Recall that a URS of Γ is a minimal Γ-invariant subset of Sub(Γ) (see [GW14]).
Proposition 3.5.
Let Γ be a charmenable group and X a URS of Γ . Then either every H ∈ X is contained in Rad(Γ) or X carries a Γ -invariant probability measure. Furthermore, if Γ ischarfinite then X is finite.Proof. The map θ : H ∈ Sub(Γ) H ∈ PD (Γ) is continuous and Γ-equivariant. The push-forward of measures, together with the barycenter map Prob(PD (Γ)) → PD (Γ) further yielda continuous affine Γ-map e θ : Prob( X ) → Prob(PD (Γ)) → PD (Γ) . By charmenability, the closed convex Γ-subset e θ (Prob( X )) ⊂ PD (Γ) contains a fixed point φ = e θ ( µ ). By definition of e θ , we have φ ( g ) = µ ( { H ∈ X | g ∈ H } ). We observe that the HARMENABILITY OF ARITHMETIC GROUPS OF PRODUCT TYPE 13
GNS representation π φ of φ is a sub-representation of the direct integral representation e π onthe space K := Z ⊕ X ℓ (Γ /H ) d µ ( H ) . If φ is supported on Rad(Γ) then µ -almost every H ∈ X is contained in Rad(Γ). Since X isa URS, we find in this case that every H ∈ X is contained in Rad(Γ). We now assume thisis not the case. Thus π φ is amenable by Lemma 3.2 and we get that the representation e π on K is amenable as well: there exists a state Φ on B ( K ) which is invariant under conjugacy byelements e π ( g ). Observe moreover that there is a Γ-equivariant *-homomorphism α : f ∈ C ( X ) Z ⊕ X f H d µ ( H ) ∈ B ( K ) , where f H ∈ ℓ ∞ (Γ /H ) is defined by f H : ¯ g ∈ Γ /H f ( gHg − ). The composition Φ ◦ α is aΓ-invariant state on C ( X ), i.e. a Γ-invariant Borel probability measure on X .Finally, if Γ is charfinite then both possibilities imply that X is finite: the first one by thefiniteness of Rad(Γ) and the second by Proposition 3.4. (cid:3) Unitary representations of charmenable and charfinite groups.
The fundamentalfact that any positive definite function on a group could be viewed as a state on its universalC*-algebra is indispensable in this work. Using it we get easily the following.
Proposition 3.6.
For a charfinite group, every unitary representation either weakly contains afinite dimensional subrepresentation or weakly contains a representation which is induced froma finite normal subgroup.Proof.
Let Γ be a charfinite group and π a unitary representation. We let C ⊂ PD (Γ) bethe Γ-invariant compact convex subset consisting of positive definite functions that extend tostates on C ∗ π (Γ). By charmenability there exists a character φ ∈ C which we now fix. Weget that the GNS representation π φ is weakly contained in π . If φ is supported on Rad(Γ)then π φ is induced from Rad(Γ) which is finite. Otherwise, π φ is amenable thus it contains afinite dimensional subrepresentation, by Lemma 3.2. This subrepresentation is therefore weaklycontained in π . (cid:3) Let Γ be a discrete group. Any character of Γ can be viewed as a trace on the universal C*-algebra C ∗ (Γ). Naturally, for the regular character δ e , the corresponding trace on C ∗ (Γ) iscalled the regular trace . Note that if π is a unitary representation of Γ then π weakly containsthe regular representation of Γ if and only if the regular trace on C ∗ (Γ) factors through theprojection C ∗ (Γ) → C ∗ π (Γ). We still call “regular trace” the trace obtained on C ∗ π (Γ) throughthis factorization. Proposition 3.7.
Assume Γ is a charmenable discrete group with trivial amenable radical. Let π be a unitary representation of Γ . Denote by A := C ∗ π (Γ) . If π is non-amenable then thefollowing facts are true.(1) π weakly contains the regular representation, the regular trace τ is the unique trace on A , and every proper ideal of A is contained in I τ := { x ∈ A | τ ( x ∗ x ) = 0 } .(2) The regular trace τ on A = C ∗ π (Γ) satisfies a Powers property: for every x ∈ A , τ ( x ) ∈ conv( { π ( g ) xπ ( g ) ∗ | g ∈ Γ } ) . (3) There exists µ ∈ Prob(Γ) whose support generates Γ and such that the only µ -stationarystate on A is τ .Proof. (1) We let C ⊂ PD (Γ) be the Γ-invariant compact convex subset consisting of positivedefinite functions that extend to states on A = C ∗ π (Γ). By charmenability there exists a charac-ter φ ∈ C which we now fix. We get that the GNS representation π φ is weakly contained in π .Since π is non-amenable we have that π φ is non-amenable, and φ must be the regular character. Thus the GNS representation associated with φ is λ , and we conclude that λ is weakly containedin π .Further, if I is a proper ideal in A , then represent faithfully A/I on a Hilbert space H . Thecomposed representation Γ → A → A/I → B ( H ) is still non-amenable, and thus must weaklycontain the regular representation. So the canonical map A → C ∗ λ (Γ), whose kernel is precisely I τ , factors through A/I . This shows that I ⊂ I τ .(2) Arguing as in the previous point, every non-empty closed convex Γ-subset of S ( A ) mustcontain a trace, which must be equal to τ . Claim.
For every n ≥
1, every non-empty closed convex Γ-subset of S ( A ) n contains the fixedpoint τ ( n ) := ( τ, . . . , τ ).We prove this by induction. We just observed it was true for n = 1. Assume it is true for some n ≥ C ⊂ S ( A ) n +1 . Then image of C under thefirst projection map contains τ by the n = 1 step. Now the set C ∩ { τ } × S ( A ) n is non-empty,and identifies with a closed convex Γ-subset of S ( A ) n , which contains τ ( n ) by our inductionassumption. Thus C contains τ ( n +1) .Using the above claim and a diagonal argument, we may find a sequence ( µ n ) n ≥ ∈ (Prob(Γ)) N such that ( µ n ∗ φ ) n converges weakly to τ for every state φ ∈ S ( A ). In particular µ n ∗ x convergesweakly to τ ( x )1 for every x ∈ A . Thus τ ( x )1 belongs to the weak closure of the convex hull of { π ( g ) xπ ( g ) ∗ | g ∈ Γ } . By Hahn-Banach theorem, it belongs to its norm closure, which is thedesired Powers property.(3) This follows from (2) by the proof of [HK17, Theorem 5.1]. Note that [HK17, Theorem5.1] essentially states that (2) is equivalent to (3) for the regular representation λ , but its proofapplies verbatim for an arbitrary unitary representation, showing that (2) implies (3). (cid:3) For µ ∈ Prob(Γ), we say that ψ ∈ PD (Γ) is a µ - character if ψ is a µ -stationary state on C ∗ (Γ)with respect to the conjugation action. The following is an easy consequence. Proposition 3.8.
Assume Γ is a charmenable discrete group with trivial amenable radical.Then Γ is C*-simple. If in addition Γ has property (T) then it is charfinite. Further, in thiscase we may find µ ∈ Prob(Γ) whose support generates Γ such that every µ -character on Γ is acharacter.Proof. We assume as we may that Γ is non-trivial, thus non-amenable. The C*-simplicity ofΓ follows at once from Proposition 3.7(1), applied to the regular representation which is non-amenable.Assume now Γ has property (T) and consider Definition 1.2. By assumption, Γ satisfies property(3). It satisfies property (4) by [Wa74, Theorem 2.6] and property (5) by [BV91, Theorem 1.1].It follows that Γ is indeed charfinite.Denote by π the universal weakly mixing representation of Γ, i.e. the direct sum of all cyclicweakly mixing representations of Γ. Then since Γ has property (T), π is non-amenable and anyrepresentation weakly contained in π is again weakly mixing. Let µ ∈ Prob(Γ) be a measurewhose support generates Γ and such that the only µ -stationary state on A = C ∗ π (Γ) is τ , where τ is the regular trace on A = C ∗ π (Γ), as guaranteed by Proposition 3.7(3).Let φ be a µ -character. We consider the compact convex subset of Γ-invariant positive definitefunctions which are dominated by φ ; S = { ψ ∈ PD(Γ) Γ | φ − ψ ∈ PD(Γ) } ⊂
PD(Γ)and claim that φ ∈ S , thus φ is a character. We assume that this is not the case and argue toshow a contradiction. If S = { } we set φ = φ . Otherwise we let ψ be a non-zero extremepoint of S and we set φ = ( φ − ψ ) / (1 − ψ ( e )). In both cases, φ is a µ -character whichdominates no non-zero Γ-invariant positive definite function. HARMENABILITY OF ARITHMETIC GROUPS OF PRODUCT TYPE 15
Claim. φ is a compact positive definite function.By definition of π , a positive definite function on Γ factorizes through a state on A = C ∗ π (Γ) ifand only if its GNS representation is weakly mixing, and conversely. Denote by C ⊂ PD(Γ) theclosed convex subset consisting of such positive definite functions (including the null function).Then a positive definite function on Γ is compact if and only if it does not dominate a positivedefinite function in C . To prove the claim, we therefore consider the compact space C = { ψ ∈ C | φ − ψ ∈ PD(Γ) } ⊂ C and prove that C = { } . Note that this set is µ -invariant. Take ψ ∈ C . By an averagingprocedure, we may find ψ ∈ C which is µ -stationary, and ψ (1) = ψ (1). By definition of C , ψ may be viewed as a multiple of a µ -stationary state on C , and thus must be Γ-invariant,thanks to our choice of µ . This forces ψ = 0, since φ does not dominate non-zero invariantPD-functions. We thus find ψ = 0, and C = { } , proving our claim.Since φ is compact, M := π φ (Γ) ′′ is a tracial von Neumann algebra (contained in the directsum of countably many finite dimensional algebras). Denote by Tr a normal faithful tracialstate on M . Then Tr and φ are two µ -stationary normal states on M . Since Tr is faithfuland invariant, Proposition 2.7(2) tells us that in fact, φ must be invariant as well. This is thedesired contradiction. (cid:3) Permanence properties.Proposition 3.9.
Let Γ be charmenable group. Then for any normal subgroup N ⊳ Γ , Γ /N ischarmenable. Moreover, if Γ is charfinite then so is Γ /N .Proof. The result is clear if Γ /N is amenable. We thus may assume by Proposition 3.3 that N is amenable, thus N ≤ Rad(Γ). We view PD (Γ /N ) as a Γ-invariant closed subset of PD (Γ) inthe obvious way. The fact that every closed convex Γ /N -invariant subset of PD (Γ /N ) containsa fixed point clearly follows from the corresponding property of Γ. Note that Char(Γ /N ) is infact the subset of characters on Γ that are equal to 1 on N . Hence it is a face of Char(Γ). inparticular, extreme points of Char(Γ /N ) are also extremal in Char(Γ). So an extremal characteron Γ /N which is not supported on Rad(Γ /N ) may be viewed as an extremal character of Γ,which is not supported on Rad(Γ). In turn it must be von Neumann amenable, as a characterof Γ, hence as a character of Γ /N .The moreover part follows easily. (cid:3) The following proposition will be important for us when discussing charmenability of lattices ininfinite restricted products.
Proposition 3.10.
Let
Γ = S n ∈ N Γ n be an ascending union. If for every n ∈ N , Γ n ischarmenable, then so is Γ .Proof. Let C ⊂ PD(Γ) be a compact convex Γ-invariant subset. For every n , let C n be thepreimage of the Γ n -invariants in the image of the restriction map PD(Γ) → PD(Γ n ). Then( C n ) n is a descending sequence of compact sets, hence has non-trivial intersection. This showsthat the set of Γ-invariants in C is non-empty. Fix an extremal character φ ∈ Char(Γ) andassume the support of φ is not contained in Rad(Γ). We argue to show that φ is von Neumannamenable. We note that Rad(Γ) = ∞ [ k =1 ∞ \ n = k Rad(Γ n ) . Indeed, the inclusion ⊂ is clear and ⊃ follows from the fact that g ∈ Rad(Γ) iff for everyfinite set F ⊂ Γ, the group generated by { g f | f ∈ F } is amenable. We fix g ∈ Γ \ Rad(Γ)such that φ ( g ) = 0. By passing to a subsequence we assume as we may that for every n , g ∈ Γ n \ Rad(Γ n ). For every n ∈ N , we let φ n = φ | Γ n ∈ Char(Γ n ). Claim.
For every n , φ n is a von Neumann amenable character of Γ n . Fix an index n , and denote by N n the GNS von Neumann algebra associated with (Γ n , φ n ), anddenote by τ the normal trace on N n extending φ n . We want to prove that N n is an amenablevon Neumann algebra. Apply Lemma 3.2 to φ m , m ≥ n , to get a decomposition φ m = t m φ m + (1 − t m ) φ m , for some t m ∈ [0 ,
1] and characters φ m , φ m ∈ Char(Γ m ) such that φ m is von Neumann amenableas a character of Γ m , and φ m is supported on Rad(Γ m ).Note that such a decomposition may be restricted to Γ n , and that the restriction φ m to Γ n isstill von Neumann amenable. By Lemma 2.5, we may find a projection p ∈ N n , with trace atleast t m such that pN n p is an amenable von Neumann algebra. So the claim will follow oncewe prove that t m tends to 1 as m goes to infinity. This later fact relies on the extremalityassumption.For every m ≥ n , denote by ψ m and ψ m the positive definite extensions to Γ of φ m and φ m ,respectively, obtained by assigning the value 0 outside of Γ m . Passing to a subsequence ifnecessary, we assume as we may that the sequences ψ m , ψ m and t m all converge and we observethat the limit functions are characters of Γ. Since the sequence t m ψ m + (1 − t m ) ψ m convergesto φ which is an extremal character, since φ ( g ) = 0 while for every m , ψ m ( g ) = 0, it followsthat t m →
1, as desired. This concludes the proof of the claim.Let us now deduce that φ is a von Neumann amenable character on Γ. Denote by ( H, π, ξ ) theGNS triple associated with φ , and for every n , denote by ( H n , π n , ξ n ), the GNS triple of φ n .Naturally H n can be viewed as a subspace of H , in such a way that ξ n coincides with ξ , and π n is the restriction of π to H n . Since Γ = S n Γ n , we find that the increasing union of the spaces H n is dense in H .Define M = π (Γ) ′′ and M n := π (Γ n ) ′′ , for each index n . Denoting by p n : H → H n theorthogonal projection, we find that p n ∈ M ′ n and p n M n ≃ π n (Γ n ) ′′ is amenable for all n . Inparticular p m M n ⊂ p m M m is also amenable for all m ≥ n . Now p m converges strongly to 1, sowe find that M n is amenable, and M = ( S n M n ) ′′ follows amenable as well. (cid:3)
4. (
G, N ) -structures, singularity and criteria for charmenability Throughout this section G denotes an lcsc group. The first three subsections will be devoted tothe study of ( G, N )-von Neumann algebras with a focus on their singularity properties. Thisstudy will be used to develop charmenability criteria in § Definition and examples of ( G, N ) -structures. The setting of µ -stationary actions isquite general, but it is sometimes too loose for our purposes. This is because Furstenberg-Poisson boundaries don’t always behave well when passing to subgroups, even to lattices. Incontrast the amenability of an action remains when restricting to any closed subgroup.For this reason, we want to study the more general data of a G -action G y A on a C*-algebra A together with a G -map θ : B → S ( A ) from an amenable G -space ( B, ν ). In the commutativecase, such boundary maps θ : B → Prob( X ) naturally give rise to a measure class Bar( θ ∗ ν ) on X . Then measurable notions on X , such as ergodicity, are discussed.In the non-commutative case, it is the same: a boundary map θ : ( B, ν ) → S ( A ) naturallycomes with a state φ = Bar( θ ∗ ν ) and we want to study “measurable” aspects of the GNS vonNeumann algebra M = π φ ( A ) ′′ (such as ergodicity). In fact, we can keep track of θ purely interms of M , as follows. By duality, the map θ gives rise to a G -ucp map E : A → L ∞ ( B ). If wedenote by z ∈ A ∗∗ the central support projection of the normal extension E : A ∗∗ → L ∞ ( B ),then z is G -invariant (with respect to the normal extension of the action), and M is naturallyisomorphic with zA ∗∗ . Moreover Proposition 2.6 shows that M is indeed a G -von Neumannalgebra. The map E can thus be viewed as a normal G -ucp map M → L ∞ ( B ).Note that in this picture, if A is separable, we recover the initial map θ form the composition A → M → L ∞ ( B ) by duality. So the two points of view are equivalent, but the advantage of HARMENABILITY OF ARITHMETIC GROUPS OF PRODUCT TYPE 17 expressing things in terms of M is that this will allow us to change the compact model of theaction. Definition 4.1.
Let N be a G -von Neumann algebra. A ( G, N ) -von Neumann algebra will bethe data ( M, E ) of a G -von Neumann algebra M and a normal G -ucp map E : M → N . We willsometimes refer to E as the ( G, N )-structure map. We say that (
M, E ) is faithful or extremal if E satisfies the corresponding properties. If E ( M ) ⊂ N G , we say that ( M, E ) is G - invariant .Classically, stationary states give rise to boundary maps. This is our first example. Proposition 4.2.
Fix a generating probability measure µ ∈ Prob( G ) , denote by ( B, ν ) thecorresponding Furstenberg-Poisson boundary and set N = L ∞ ( B, ν ) . We view ν as the state on N given by integration w.r.t. the measure ν . Let M be a G -von Neumann algebra.Then a ( G, N ) -structure map E : M → N gives rise to a unique µ -stationary normal state ϕ = ν ◦ E on M . Conversely, a normal µ -stationary state ϕ on M gives rise to a normal G -ucpmap E : M → Har µ ( G ) ≃ L ∞ ( B ) , defined by E ( x )( g ) = ϕ ( g − ( x )) , for all x ∈ M , g ∈ G .These two maps are inverse of one another. Moreover, • E is faithful if and only if ϕ is faithful; • E is extremal if and only if ϕ is extremal; • E is invariant if and only ϕ is G -invariant.Proof. The fact that the two maps E ϕ and ϕ E are inverse of each other follows fromthe definition of the Poisson transform Har ν ( G ) ≃ L ∞ ( B, ν ) (see e.g. [BS04, § (cid:3) Let us give now a purely non-commutative example.
Example 4.3.
Assume that G is discrete and take an arbitrary G -algebra N , whose actionis denoted by σ . Let M be any tracial factor with separable predual and π : G → U ( M ) anyunitary representation such that π ( G ) ′′ = M . In other words, M is the GNS von Neumannalgebra of an extremal character on G . We denote by ( L ( M ) , L ( M ) + , J ) the standard formof M .The group G acts on N ⊗ B ( L ( M )) via the automorphisms σ g ⊗ Ad(
J π ( g ) J ), g ∈ G . Thefixed point algebra M = ( N ⊗ B ( L ( M ))) G admits another action e σ : G y M , given byAd(1 N ⊗ π ( g )), for g ∈ G .Denote by ξ ∈ L ( M ) + the unique cyclic vector implementing the trace τ and by Φ = h · ξ, ξ i ∈ B ( L ( M )) ∗ the corresponding normal vector state. Since τ is a trace, we know that aξ = ξa = J a ∗ J ξ for every a ∈ M . Then for every g ∈ G , the normal ucp map E = id N ⊗ Φ :
M → N satisfies E ◦ e σ g = id N ⊗ (Φ ◦ Ad( π ( g ))) = id N ⊗ (Φ ◦ Ad(
J π ( g ) ∗ J ))= (id N ⊗ Φ) ◦ (Ad(1 N ⊗ J π ( g ) ∗ J ))= (id N ⊗ Φ) ◦ ( σ g ⊗ id)= σ g ◦ E. We used the invariance property for elements in M to obtain the third line above. This showsthat ( M , E ) is a ( G, N )-von Neumann algebra.
Lemma 4.4.
In the above example, the structure map E is faithful.Proof. We keep the notation from the previous example. Let x ∈ M be such that E ( x ∗ x ) = 0.Then for every g ∈ G , we get E ((1 ⊗ π ( g ) ∗ ) x ∗ x (1 ⊗ π ( g ))) = σ − g ( E ( x ∗ x )) = 0.Now, viewed as a ucp map on N ⊗ B ( L ( M )), the support of E = id N ⊗ Φ is 1 ⊗ p ξ , where p ξ is the rank one projection onto C ξ . We thus find that x (1 ⊗ π ( g ) p ξ ) = 0 for every g ∈ G . Since π ( g ) ξ , g ∈ G , spans a dense subspace of L ( M ), this indeed implies x = 0. (cid:3) In our last example we explain how to induce structure maps from a lattice to the ambientgroup G . Example 4.5.
Let Γ < G be a lattice, N be a Γ-von Neumann algebra and ( M, E ) any (Γ , N )-von Neumann algebra. Equally denote by σ the Γ actions on M and N . Denote by λ and ρ thetranslation actions of G on L ∞ ( G ) on the left and right, respectively. Define the fixed pointvon Neumann algebras f M := ( L ∞ ( G ) ⊗ M ) ( ρ ⊗ σ )(Γ) and e N := ( L ∞ ( G ) ⊗ N ) ( ρ ⊗ σ )(Γ) . Since E is Γ-equivariant, the map e E = id ⊗ E maps f M into e N . Moreover, this map clearlyintertwines the induced G -actions λ ⊗ id on f M and e N . We call ( f M , e E ) the induced ( G, e N )-vonNeumann algebra. Note that it is faithful if E is.In the special case where N is already a G -von Neumann algebra, we further have a faithful G -equivariant normal ucp map E N : e N → N . Indeed, in this case the induced G -action on e N is conjugate with the diagonal G -action on L ∞ ( G/ Γ) ⊗ N . Then E N is given by integrating onthe first component, E N = m G/ Γ ⊗ id. So in this case we also get a ( G, N ) structure E N ◦ e E on f M . Lemma 4.6.
Keep the setting of the above example, and assume that N is a G -algebra on which Γ acts ergodically . The following are equivalent:(1) E is Γ -invariant.(2) e E ranges into ( L ∞ ( G ) ⊗ ( ρ ⊗ σ )(Γ) ≃ L ∞ ( G/ Γ) ;(3) E N ◦ e E is G -invariant.Proof. The implications (1) ⇒ (2) ⇒ (3) are clear thanks to our ergodicity assumption.(2) ⇒ (1). Fix x ∈ M . We want to prove that E ( x ) is a scalar operator. Choose a fundamentaldomain F ⊂ G for the right Γ-action on G . Let f ∈ f M be the Γ-equivariant function G → M which is equal to σ − γ ( x ) on the translate F γ of F , for all γ ∈ Γ. Then e E ( f ) ∈ e N is constanton F , equal to E ( x ). By (2), this function has scalar values, so E ( x ) ∈ C ⇒ (2). Note that L ∞ ( G/ Γ) may be viewed as a subalgebra of both f M and e N , and that itlies in the multiplicative domain of e E . Hence the image of e E is an L ∞ ( G/ Γ)-module.Let us describe explicitly the map E N : e N → N . Identify e N with the algebra of Γ-equivariantfunctions from G to N (with respect to the right action of G on itself). For f ∈ e N , define θ ( f ) : G → N by the formula θ ( f )( g ) = σ g ( f ( g )), for every g ∈ G . Since f is Γ-equivariant, θ ( f ) is right Γ-invariant and thus defines an N -valued function on G/ Γ, that is, an element of L ∞ ( G/ Γ) ⊗ N . The map θ : e N → L ∞ ( G/ Γ) ⊗ N defined this way is an onto ∗ -isomorphism,which intertwines the induced action on e N with the diagonal G -action on L ∞ ( G/ Γ) ⊗ N . Then E N = ( m G/ Γ ⊗ id) ◦ θ : e N → N. Observe that θ maps the algebra ( L ∞ ( G ) ⊗ ( ρ ⊗ σ )(Γ) ⊂ e N onto the subalgebra L ∞ ( G/ Γ) ⊗ f ∈ e N in the image of e E . By (3), the element θ ( f ) ∈ L ∞ ( G/ Γ) ⊗ N is such that( m G/ Γ ⊗ id)( aθ ( f )) ∈ C , for every a ∈ L ∞ ( G/ Γ) . This implies that the essential range of θ ( f ), viewed as an N -valued function on G/ Γ, is con-tained in C
1. Hence θ ( f ) ∈ L ∞ ( G/ Γ) ⊗
1, as desired. (cid:3)
Remark 4.7.
Let Γ be a lattice in G , and µ ∈ Prob( G ) be a generating measure. Assume that( B, ν ) is the (
G, µ )-Furstenberg-Poisson boundary and set N = L ∞ ( B, ν ). Assume moreoverthat (
B, ν ) is the (Γ , µ )-Furstenberg-Poisson boundary for some admissible probability measure µ ∈ Prob(Γ). Let M be a Γ-von Neumann algebra with a µ -stationary state ϕ . By Example4.2, there is a unique normal ucp Γ-map E : M → N so that ν ◦ E = ϕ . Using the previous In fact the assumption that N Γ = N G would also imply the equivalence between (1) and (3). HARMENABILITY OF ARITHMETIC GROUPS OF PRODUCT TYPE 19 observation, the normal state ϕ = ν ◦ E N ◦ e E on f M is faithful and µ -stationary. This providesa more conceptual view of [BH19, Theorem 4.3].4.2. Singular states and singular structures.Definition 4.8.
Two states φ, ψ on a C*-algebra A are called singular , denoted by φ ⊥ ψ , if k φ − ψ k = 2. Remark 4.9.
There are several equivalent formulations of this notion: φ ⊥ ψ if and only if theirsupport projections in A ∗∗ are perpendicular, if and only if there exists a sequence ( a n ) n ∈ A N ,0 ≤ a n ≤
1, such that lim n φ ( a n ) = 0 and lim n ψ ( a n ) = 1.Observe that singularity passes to larger algebras: if A ⊂ B is an inclusion of C*-algebras and φ, ψ ∈ S ( B ) have their restrictions to A that are singular, then φ and ψ themselves are singular.The following proposition extends this definition to boundary maps, and proves independenceon the choice of a compact model. Proposition 4.10.
Consider a separable von Neumann algebra M and a probability space ( B, ν ) and two normal ucp maps E , E : M → L ∞ ( B ) . For every separable C ∗ -subalgebra A ⊂ M ,the restriction maps E k : A → L ∞ ( B ) , dualize to measurable maps θ Ak : B → S ( A ) for all k ∈ { , } . The following are equivalent.(1) For every weakly dense, separable unital C*-subalgebra A ⊂ M , we have θ A ( b ) ⊥ θ A ( b ) ,for almost every b ∈ B .(2) There exists a separable unital C*-algebra A ⊂ M such that θ A ( b ) ⊥ θ A ( b ) , for almostevery b ∈ B .(3) For every ε > , there exist finitely many projections p , . . . , p N ∈ L ∞ ( B ) , such that P Nn =1 p n = 1 and k φ ,n − φ ,n k ≥ − ε , for all ≤ n ≤ N , where φ ,n , φ ,n ∈ M ∗ arethe normal states defined by φ k,n : x ∈ M ν ( p n ) ν ( p n E k ( x )) , for all k ∈ { , } , ≤ n ≤ N. Definition 4.11.
Two normal ucp maps E , E satisfying the above equivalent conditions arecalled singular . Remark 4.12.
Note that by (2), if M ≤ M is a von Neumann subalgebra and E , E : M → L ∞ ( B ) are normal ucp maps whose restrictions to M are singular normal ucp maps then E and E are singular normal ucp maps. Proof of Proposition 4.10. (1) ⇒ (2) follows from the separability of M .(2) ⇒ (3). Assume that (2) is true and take ε >
0. Observe that statement (3) has someflexibility. Instead of looking for a true partition of unity we may only look for pairwise orthog-onal projections p , . . . , p n such that ν (1 − P i p i ) < ε and which satisfy the conclusion of (3).Indeed, once this is achieved, we may distribute the remaining mass (of size < ε ) proportionallyto each p i , to form new projections q i which actually add up to 1, and still satisfy the rest ofthe conclusion (up to inflating ε in a non-essential way).Since A is separable, we may find a sequence ( x n ) n ∈ N in A , which is dense in A [0 , := { x ∈ A | ≤ x ≤ } . For each n ∈ N , we define B n := (cid:8) b ∈ B | θ A ( b )( x n ) ≤ ε/ θ A ( b )( x n ) ≥ − ε/ (cid:9) . By density of the sequence ( x n ) n in A [0 , , condition (2) tells us that S n ∈ N B n is co-null in B .Let us pick pairwise disjoints sets ( B n ) n ∈ N of B such that B n ⊂ B n for every n ∈ N , and still S n B n is co-null in B . Fix N large enough so that ν (1 − S Nn =1 B n ) < ε . We emphasize that we are talking here about the support projections and not the central support projections.
Then the projections p i = B i , i = 1 . . . N , satisfy the desired conclusion. Indeed, for every i ≤ N , we have k − x i k ≤
1, while p i E (1 − x i ) − p i E (1 − x i ) ≥ p i (1 − ε/ − p i (1 − − ε/ − ε ) p i . (3) ⇒ (1). Fix a separable dense C*-subalgebra A ⊂ M , and ε >
0. We claim that the set B ε := { b ∈ B | k θ A ( b ) − θ A ( b ) k ≥ − ε } has measure at least 1 − ε . This claim clearly implies (1).Take projections p , . . . , p N ∈ L ∞ ( B ) as in condition (3), with respect to some δ < ε . Notethat the corresponding states φ ,n , φ ,n , n = 1 , . . . , N are normal on M . So by Kaplanskydensity theorem, we have k φ ,n − φ ,n k = sup x ∈ A [ − , φ ,n ( x ) − φ ,n ( x ) . So for every n = 1 , . . . , N , we may find a self-adjoint element x n ∈ A , k x n k ≤
1, such that φ ,n ( x n ) − φ ,n ( x n ) ≥ − ε . Fix n ≤ N , and define p n := B n , f n = p n ( E ( x n ) − E ( x n )) ∈ L ∞ ( B ) and B n,ε := { b ∈ B n | (2 − f n )( b ) ≤ ε } . Note that 2 − f n has non-negative real values. By Markov inequality, we have ν ( B n \ B n,ε ) ε ≤ Z B n (2 − f n )( b ) d ν ( b )= 2 ν ( B n ) − ν ( B n )( φ ,n ( x n ) − φ ,n ( x n )) ≤ ν ( B n ) ε . So we find ν ( B n,ε ) ≥ (1 − ε ) ν ( B n ). Observe that B n,ε ⊂ B n ∩ B ε . So adding up over n , we get ν ( B ε ) ≥ − ε , as desired. (cid:3) Definition 4.13.
Let G be an lcsc group. Let ( B, ν ) be a G -space and set N = L ∞ ( B ). Let( M, E ) be a separable (
G, N )-von Neumann algebra. We say that (
M, E ) is a singular ( G, N ) -von Neumann algebra if the normal ucp maps E g : x ∈ M E ( gx ) ∈ L ∞ ( B ), g ∈ G , arepairwise singular.4.3. Singularity criterion.
Our next goal is giving a criterion for singularity of (
G, N )-algebras for some G and N . It is inspired by [BF14, Section 2, Theorem 2.5]. Proposition 4.14 (Singularity criterion) . Let G be an lcsc group and P ≤ G a closed subgroup.Assume that P is stably self normalizing and that it has the relative Mautner property in G asdefined in Definition 2.12. Denote by N := L ∞ ( G/P, ν ) , where ν is a G -quasi-invariant Radonmeasure.(1) Let A be a separable G -C*-algebra and φ an extremal P -invariant state on A . For every g ∈ G , either gφ = φ or gφ ⊥ φ .(2) Let ( M, E ) be an extremal separable ( G, N ) -von Neumann algebra which is not G -invariant. If g ∈ G has a null set of fixed points in G/Q for any intermediate properclosed subgroup
P < Q < G , then E and E g are singular.In particular, if G acts essentially freely on G/Q for any intermediate proper closedsubgroup
P < Q < G , then ( M, E ) is a singular ( G, N ) -von Neumann algebra.Proof. (1) We fix g ∈ G such that φ and gφ are not singular and argue to show that φ is g -fixed. We denote by H the subgroup of G generated by P and g and endow it with theinduced topology. Thus, noting that φ is P -invariant, we are arguing to show that φ is in fact H -invariant.Extend the H -action on A to a non-continuous action on A ∗∗ . We still denote by φ the normalextension of φ to A ∗∗ and we denote by z ∈ Z ( A ∗∗ ) the central support projection of φ . Note HARMENABILITY OF ARITHMETIC GROUPS OF PRODUCT TYPE 21 that z is P -invariant since φ is P -invariant. We get by Lemma 2.6 that P acts continuously on zA ∗∗ and note that the pair ( P, P ∩ gP g − ) has the Mautner property.We consider the central projection σ g ( z ) ∈ Z ( A ∗∗ ) and the normal positive linear functional φ g : a ∈ zA ∗∗ φ ( σ g ( z ) a ) = φ ( zσ g ( z ) a ) . We observe that φ g is P ∩ gP g − -invariant and deduce that it is also P -invariant. Clearly, φ g ≤ φ so by extremality of φ , φ g must be proportional to φ . In terms of the central supportprojection, this tells us that zσ g ( z ) is either null (in case φ g = 0) or it is equal to z . We assumedthat φ and gφ are not singular, so zσ g ( z ) = 0 is excluded and we get zσ g ( z ) = z . We concludethat z ≤ σ g ( z ).Considering similarly the functional φ g − and using the fact that g − φ and φ are not singular,we also get the z ≤ σ g − ( z ), thus σ g ( z ) ≤ z . We conclude that z = σ g ( z ). Since z is also P -invariant and H is generated by P and g we get that z is H -invariant. Using again Lemma 2.6we now get that H acts continuously on zA ∗∗ . By Lemma 2.13 the pair ( H, P ) has the Mautnerproperty. Since φ is P -invariant we conclude that it is indeed H -invariant.(2) The extremality assumption means that E is extremal among the (normal) G -ucp maps M → N . In particular, it implies that the restriction of E to any weakly dense G -C*-subalgebra A is extremal. Choose such a C*-subalgebra A ⊂ M , and assume that A is separable. Thenthe restriction of E to A corresponds to a G -equivariant map θ : G/P → S ( A ). Since G actstransitively on G/P , we may assume that θ is everywhere defined and everywhere equivariant. Inother words, there exists a P -invariant state φ on A such that θ ( gP ) = gφ for every gP ∈ G/P .The extremality condition implies that φ is extremal among P -invariant states on A . So by (1),we find that for every g ∈ G , either gφ = φ or gφ ⊥ φ .Assume that E is not G -invariant, which implies that φ is not G -invariant. Denote by Q < G thestabilizer of φ . This is a proper closed subgroup and for every g, h ∈ G , we have gθ ( hP ) ⊥ θ ( hP )as soon as ghQ = hQ , or equivalently, as soon as hQ is not in the fixed point set of g inside G/Q .If g ∈ G has a null set of fixed points in G/Q , then for almost every hP ∈ G/P , gθ ( hP ) ⊥ θ ( hP ).This is exactly the singularity condition E ⊥ E g . (cid:3) We make the following observation about the extremality condition in Proposition 4.14(2).
Lemma 4.15.
Take an lcsc group G with a generating measure µ ∈ Prob( G ) and denote by ( B, ν ) the corresponding Furstenberg-Poisson boundary and by N = L ∞ ( B, ν ) .If M is an ergodic G -von Neumann algebra, then there exists at most one ( G, N ) -structure map E : M → N . In particular it is extremal.Proof. By Example 4.2, a structure map E is the same data as the normal µ -stationary state φ = ν ◦ E . But we saw in Proposition 2.7 that if M is ergodic, there exists at most one normal µ -stationary state on M . (cid:3) Charmenability criteria.
The proof of our main results will rely on the following crite-rion. It is an abstract version of the techniques used in [BH19]. Recall Definition 2.9 of metricergodicity.
Proposition 4.16.
Let Γ be a discrete group with trivial amenable radical. Let ( B, ν ) be aseparable, amenable and metrically ergodic Γ -space and set N = L ∞ ( B ) . The existence of suchan amenable space ( B, ν ) with the following property implies charmenability of Γ . (a): Every separable, ergodic, faithful (Γ , N ) -von Neumann algebra ( M, E ) is either in-variant or Γ -singular.Proof. We have two statements to verify.
Fixed point property.
Let C ⊂ PD (Γ) be a closed convex Γ-subset. We need to show that C contains a character φ , i.e. a Γ-fixed point. Denote by A = C ∗ (Γ) the universal C*-algebra of Γ, endowed with the conjugacy Γ-action by the unitaries u g , g ∈ Γ. We may view C as acompact convex Γ-subset of S ( A ).By amenability of ( B, ν ), we may find a measurable Γ-map θ : B → C . We claim that the state φ := Bar( θ ∗ ν ) ∈ C is Γ-invariant. In fact, we claim that this holds for every measurable Γ-map θ : B → S ( A ).The set e C of such maps θ is a convex set, and since A is separable, there is an affine bijectionbetween e C and the convex set of Γ-equivariant ucp maps E : A → L ∞ ( B ). When endowed withthe topology of pointwise ultraweak convergence, this later convex set of ucp maps is compact.This induces on e C a structure of compact convex space. So by Krein-Milman it is the closedconvex hull of its extremal points. Moreover, the map θ ∈ e C Bar( θ ∗ ν ) ∈ S ( A ) is affine andcontinuous. So it suffices to prove our claim under the assumption that θ is an extremal mapin e C .Let θ : B → S ( A ) be an extremal map, and denote by E : A → L ∞ ( B ) the correspondingΓ-equivariant ucp map. We may extend the Γ-action on A to an action on A ∗∗ , and we mayalso extend E to a normal Γ-ucp map A ∗∗ → L ∞ ( B ). We denote by z ∈ Z ( A ∗∗ ) the centralsupport of E , and set M = zA ∗∗ , so that ( M, E ) is a (Γ , L ∞ ( B ))-von Neumann algebra. Claim.
The map E is faithful and the Γ-action on M is ergodic.Denote by p ∈ M the support projection of E (so p ∈ A ∗∗ , p ≤ z , and z is the central supportof p ). Assume that x ∈ M is such that E ( x ∗ x ) = 0. Since E is equivariant and the actionΓ y A is a conjugacy action, we also have E ( u ∗ g x ∗ xu g ) = σ − g ( E ( x ∗ x )) = 0, for every g ∈ Γ.This implies that pu ∗ g x ∗ xu g p = 0, and further xu g p = 0 for every g ∈ Γ. Since the image of A in M is ultraweakly dense, we thus get xyp = 0 for every y ∈ M , and since the central supportof p in M is 1, this implies that x = 0. So E is indeed faithful.Let q ∈ M Γ be a Γ-invariant projection. Assume by contradiction that q / ∈ { , } . Then q ∈ Z ( M ), and E ( q ) ∈ L ∞ ( B ) Γ = C
1. Set t ∈ [0 ,
1] so that E ( q ) = t
1. Since E is faithful, wefind that t ∈ (0 , E , E : M → L ∞ ( B ) by the formulae E ( x ) = 1 t E ( xq ) and E ( x ) = 11 − t E ( x (1 − q )) , for all x ∈ M. By construction, E = tE + (1 − t ) E . By extremality of E | A , we find that E , E and E coincide on the image of A in M . Now, since E is normal on M and tE ≤ E , (1 − t ) E ≤ E , wefind that E and E are also normal on M . Thus these three maps coincide, which contradicts E ( q ) = 1, E ( q ) = 0. This finishes the proof of the claim.Thanks to the claim, we may apply condition ( a ). We find that either φ is invariant, or foralmost every b ∈ B , for every g ∈ Γ, the states gθ ( b ) ∈ S ( A ) are pairwise singular. Let us provethat this later case is impossible.In fact, we will check that a state ψ on A which is singular with respect to all its translates gψ , g ∈ Γ \ { e } is the regular trace. In particular such a ψ is Γ-invariant, so it cannot be singular.Extend ψ to a normal state on A ∗∗ , and denote by q its support projection. By assumption, ψ ( q ) = 1 and ψ ( u g qu ∗ g ) = ( g − ψ )( q ) = 0 for every g ∈ Γ \ { e } . Therefore,(4.1) | ψ ( u g ) | = | ψ ( u g q ) | = | ψ ( u g qu ∗ g u g ) | ≤ ψ ( u g qu ∗ g ) / ψ (1) / = 0 . This shows that ψ is the regular trace, as wanted. Classification of characters.
Set N = L ∞ ( B ). Take an extremal character τ on Γ, and denoteby M the corresponding GNS von Neumann algebra, which is a tracial factor. We consider thecorresponding (Γ , N )-von Neumann algebra ( M , E ) = (( N ⊗ B ( L ( M ))) Γ , id N ⊗ Φ) as definedin Example 4.3. By Lemma 4.4, E is faithful. Claim.
The Γ-action on M is ergodic.This is where we use the condition that ( B, ν ) is metrically ergodic. By definition M Γ is thecommutant of 1 ⊗ π τ (Γ) inside M so it is equal to ( L ∞ ( B ) ⊗ J M J ) Γ . This later algebra can HARMENABILITY OF ARITHMETIC GROUPS OF PRODUCT TYPE 23 be viewed as the algebra of Γ-equivariant measurable functions B → J M J , where the Γ-actionon
J M J is simply given by conjugacy by the unitaries
J π τ ( g ) J , g ∈ Γ. Since M is a tracialfactor, J M J can be viewed as a subspace of its L -space, on which Γ acts isometrically. So anyequivariant function B → J M J must be constant, equal to a scalar operator. Hence M Γ = C a ). We find that either the structure map E isinvariant, or it is Γ-singular. We treat these two cases separately.If E is invariant, then M = 1 ⊗ M . Indeed, assume that E is invariant and take f ∈ M , whichwe view as a Γ-equivariant function B → B ( L ( M )). Given x, y ∈ M , we have 1 ⊗ x, ⊗ y ∈ M ,and hence E ((1 ⊗ y ) ∗ f (1 ⊗ x ))( b ) = Φ( y ∗ f ( b ) x ) = h f ( b ) xξ, yξ i does not depend on b ∈ B . Since M is separable and ξ is an M -cyclic vector, this implies that f is essentially constant. Thus wefind that M = M ∩ (1 ⊗ B ( L ( M ))) = 1 ⊗ M , as claimed. Since the action Γ y B is amenable, M is amenable and so is M . In this case, τ is a von Neumann amenable character.If E is singular, we claim that τ is the regular character. Indeed, take a separable weakly denseC*-subalgebra A ⊂ M containing 1 ⊗ π (Γ). Denote by θ : B → S ( A ) the measurable Γ-mapcorresponding to E | A . Then computation (4.1) tells us that for almost every b ∈ B , for every g ∈ Γ, θ ( b )(1 ⊗ π ( g )) = δ g,e and so θ ( b ) ◦ (1 ⊗ π ) is the regular character on Γ. In this case, thebarycenter of these characters, which is exactly ν ◦ E ◦ (1 ⊗ π ) = τ is also the regular character,as claimed. (cid:3) We can also use condition (a) in Proposition 4.16 to strengthen Proposition 3.5.
Proposition 4.17.
Let Γ be a discrete group with trivial amenable radical. Let ( B, ν ) be anamenable ergodic Γ -space for which condition (a) in Proposition 4.16 is satisfied.Then any minimal action Γ y X on a compact space is either topologically free or carries a Γ invariant Borel probability measure.Proof. As in the proof of Proposition 4.16, we may choose an extremal measurable Γ-map θ : B → Prob( X ). Set η = Bar( θ ∗ ν ) ∈ Prob( X ). Then η is Γ-quasi-invariant and by minimalityof Γ y X , the topological support of η equals X . The Γ-ucp map C ( X ) → L ∞ ( B ) coming from θ extends to a well-defined faithful normal Γ-ucp map F : L ∞ ( X, η ) → L ∞ ( B ). By extremalityof θ , the nonsingular action Γ y ( X, η ) is ergodic. Note that η = ν ◦ F .By condition (a), F is either Γ-invariant or Γ-singular. The former case implies that η is aΓ-invariant Borel probability measure. Let us assume that F is singular and argue that theaction is topologically free. By definition, singularity of F exactly means that θ ( b ) ⊥ gθ ( b ),for every g ∈ Γ \ { e } , for almost every b . Fixing g ∈ Γ \ { e } , this condition further impliesthat θ ( b )(Fix( g )) = 0, for almost every b ∈ B . Integrating this quantity w.r.t. ν , we get η (Fix( g )) = 0. Since η has full support, this forces Fix( g ) has empty interior. So indeed theaction is topologically free. (cid:3) The criterion above can be adapted also for groups with a non-trivial amenable radical, but weneed an extra stiffness assumption.
Proposition 4.18.
Let Λ be a countable group. Take a separable, metrically ergodic amenable Λ -space ( B, ν ) and write N = L ∞ ( B ) . The following conditions together imply that Λ is char-menable. (a’): For every separable, ergodic, faithful (Λ , L ∞ ( B )) -von Neumann algebra ( M, E ) , ei-ther E is invariant or the maps E g , E h given in Definition 4.13 are singular for every g, h ∈ Λ such that h − g / ∈ Rad(Λ) . (b): Every measurable Λ -map B → PD (Rad(Λ)) is essentially constant.Proof. The proof follows the lines of the previous proposition. Let us explain the changes thatcome up.
In the fixed point property, condition ( a ′ ) ensures that for every Λ-map, θ : B → S ( C ∗ (Λ)),the state φ := Bar( θ ∗ ν ) is either invariant or for almost every b ∈ B , for every g ∈ Λ \ Rad(Λ), θ ( b ) ⊥ θ ( gb ). In the later case, computation (4.1) tells us that θ ( b )( π ( g )) = 0 for almost every b ∈ B , for every g ∈ Λ \ Rad(Λ). Further, θ ( b ) is supported on C ∗ (Rad(Λ)) for almost every b ∈ B . So in this case we may view θ as a Λ-map from B into PD (Rad(Λ)). By condition (b)such a map must be constant, and hence its essential image must be a single Λ-invariant state.In particular, it cannot be singular with respect to its translates. So the second possibility isimpossible, and φ is invariant.The second part of the proof about classification of characters follows exactly the proof ofProposition 4.16. (cid:3) The following is a version of Proposition 4.18 which is somewhat easier to manage.
Proposition 4.19.
Let Λ be a countable group and denote Γ = Λ / Rad(Λ) . Take a separable,metrically ergodic amenable Γ -space ( B, ν ) and write N = L ∞ ( B ) . The following conditionstogether imply that Λ is charmenable. (a): Every separable, ergodic, faithful (Γ , L ∞ ( B )) -von Neumann algebra ( M, E ) is eitherinvariant or Γ -singular. (b): Every measurable Λ -map B → PD (Rad(Λ)) is essentially constant.Proof. Seeing B as a Λ-space, it is still metrically ergodic and amenable, thus we only needto verify condition (a’) of Proposition 4.18. We let ( M, E ) be a separable, ergodic, faithful(Λ , L ∞ ( B ))-von Neumann algebra for which E is not invariant and claim that E is not invari-ant on the (Γ , L ∞ ( B ))-von Neumann algebra ( M Rad(Λ) , E ). This will finish the proof, usingRemark 4.12 and condition (a). To prove this claim, it suffices to find a conditional expectation E : M → M Rad(Λ) which is Λ-equivariant, and such that E = E ◦ E .Fix a faithful normal state ν on L ∞ ( B ), and consider the faithful normal and Rad(Λ)-invariantstate φ = ν ◦ E on M . Take a generating probability measure µ on Rad(Λ) and note that φ is µ -stationary. Consider the normal conditional expectation E µ : M → M Rad(Λ) givenin Proposition 2.7(1). Then E µ is the unique φ -preserving conditional expectation E onto M Rad(Λ) . In particular, it does not depend on the choice of µ .Fix g ∈ Λ, and denote by α g ∈ Aut(Rad(Λ)) the automorphism obtained by restricting theconjugation action of g . Then denote by µ g := ( α g ) ∗ µ the push forward measure. Using theexplicit construction of E µ , a direct computation shows that E = E µ g = σ g E µ σ − g = σ g E σ − g .This proves that E is Γ-equivariant.By Proposition 2.7(2), for every normal state ν ′ on L ∞ ( B ), ν ′ ◦ E = ν ′ ◦ E ◦ E . It follows that E = E ◦ E , finishing the proof of the claim. (cid:3) Corollary 4.20.
Let Λ be a countable group and assume that Rad(Λ) is either finite or centralin Λ . Denote Γ = Λ / Rad(Λ) and let ( B, ν ) be a separable, amenable and metrically ergodic Γ -space and set N = L ∞ ( B ) . If condition (a) is satisfied then Λ is charmenable.Proof. We need to verify condition (b) of Proposition 4.19. In case Rad(Λ) is central this followsat once from the ergodicity of B . In case Rad(Λ) is finite, PD (Rad(Λ)) is finite dimensionaland this follows from the metric ergodicity of B . (cid:3) The rest of the paper is devoted to prove that these conditions (a) and (b) are satisfied in thecases of interest.
HARMENABILITY OF ARITHMETIC GROUPS OF PRODUCT TYPE 25
5. (
G, N ) -structures, lattices with dense projections and induction In this section, we are interested in the following problem. Assume that σ : Γ y X is an action ofa discrete countable group on a topological vector space X with some extra structure (typically X is a Hilbert space or a von Neumann algebra). Let ι : Γ → G be a group homomorphisminto an lcsc group G with dense image. Then we want to give an algebraic description of theset of elements x ∈ X such that the orbit map Γ → X : γ σ γ ( x ) factors to a map defined on ι (Γ), which extends continuously to a map G → X .In our setting, Γ will be a lattice in an lcsc group G and the morphism ι extends to a continuoushomomorphism G → G . In this case, we shall identify this continuity space with a fixed pointset in the induced action.5.1. Continuity vectors for unitary representations.
Let Γ < G be a lattice in an lcscgroup G , let G be a quotient of G with kernel G . Denote by ι : G → G the quotient mapand assume that ι (Γ) is dense in G .Let π : Γ → U ( H ) be any unitary representation, and denote by ( e π, e H ) the induced unitaryrepresentation of G .We say that a vector v ∈ H is ι - continuous if lim n k π ( γ n ) v − v k = 0 for any sequence ( γ n ) n ∈ N in Γsuch that ι ( γ n ) → e in G . We denote by H ι the set of ι -continuous vectors. Because the actionof Γ on H is isometric, one checks that H ι is a closed Γ-invariant subspace of H . Moreover,for any v ∈ H ι , there exists a unique continuous map c v : G → H such that π ( γ ) v = c v ( ι ( γ ))for every γ ∈ Γ. In other words, we may extend π : Γ → U ( H ι ) to a continuous unitaryrepresentation π : G → U ( H ι ) that factors through G and that satisfies π ( g ) v = c v ( ι ( g )) forevery g ∈ G and every v ∈ H ι . Proposition 5.1.
We keep the notation as above. There is a G -equivariant surjective isometry κ : H ι → ( e H ) G . Proof.
Let us view e H as the Hilbert space of measurable maps f : G → H such that(i) For every γ ∈ Γ and almost every g ∈ G , f ( gγ ) = π ( γ − ) f ( g ).(ii) k f k = R G/ Γ k f ( g ) k d m G/ Γ ( g Γ) < + ∞ .In this description, let us check that the map κ : H ι → e H defined by κ ( v )( g ) := π ( g − ) v , forevery v ∈ H ι , g ∈ G , suits us. It is indeed isometric and G -equivariant, and it indeed rangesinto e H G by definition of H ι . It remains to prove that κ is surjective. Fix f ∈ e H G . Claim.
Every essential value of f is an element of H ι .Indeed, let v be any essential value of f and take a sequence ( γ n ) n ∈ N in Γ such that ι ( γ n ) → e in G . We want to check that lim n k π ( γ n ) v − v k = 0. We may find elements h n ∈ G suchthat γ n h n → e in G . Take ε >
0. By assumption, the set A ε = { x ∈ G | k f ( x ) − v k < ε } has positive measure in G . Since γ n h n → e in G , we may find n ∈ N large enough so that A ε ∩ ( A ε · ( γ n h n ) − ) has positive measure. As an element of e H G , the function f : G → H isleft G -invariant (so right G -invariant as well since G is normal in G ) and right Γ-equivariant.Thus for every g ∈ G and every n ∈ N , we have f ( g ( γ n h n )) = f ( gγ n ) = π ( γ − n ) f ( g ). So for n ∈ N large enough, choosing g ∈ A ε ∩ ( A ε · ( γ n h n ) − ), we have k v − π ( γ n ) v k ≤ k v − f ( g ) k + k f ( g ) − π ( γ n ) v k ≤ k v − f ( g ) k + k f ( gγ n h n ) − v k ≤ ε. As ε > f on a null set if necessary to view it as an H ι -valued map.Then the measurable function G → H ι : g π ( g )( f ( g )) is well-defined, it is G -invariant andalso right Γ-invariant. Since the product set G Γ is dense in G , this implies that the abovemeasurable function is essentially constant. If we denote by v ∈ H ι its essential value, we findthat f = κ ( v ). (cid:3) Remark 5.2.
In fact a similar result holds for more general metric Γ-spaces and L p -induction,for arbitrary p ∈ [1 , ∞ ). We will not elaborate on this further here, as we will only make use ofthe above setting.5.2. Continuity points in (Γ , N ) -algebras. We now investigate the case of von Neumannalgebras. We start with the following general terminology.
Definition 5.3.
Consider a countable discrete group Γ, an lcsc group G and a group homo-morphism ι : Γ → G with dense range. Let M be a Γ-von Neumann algebra. We say that anelement x ∈ M is ι -continuous if σ γ n ( x ) → x ∗ -strongly in M for any sequence ( γ n ) n ∈ N in Γsuch that ι ( γ n ) → e in G .When the map ι is obviously understood from G , we will also use the terminology G -continuous, instead of ι -continuous.From now on, we denote by G = G × G a product of two lcsc groups and Γ < G a latticewith dense projections. For every i ∈ { , } , we denote by p i : G → G i the factor map and forconsistency of notation with the previous paragraphs, we denote by ι the restriction of p to Γ.If a Γ-von Neumann algebra M carries a Γ-invariant faithful normal state, then we can usemetric considerations as in the previous subsection to identify the set of ι -continuous elementswith a fixed point subalgebra in the induced von Neumann algebra. This was observed in [Pe14](see the comment after Proposition 3.1 of that paper). Unfortunately in the cases of interestto us, no such state is assumed to exist. Instead we have a specific stationary state, comingfrom a Furstenberg-Poisson boundary of G . We aim to provide the analogous conclusion in thisweaker setting.For i = 1 ,
2, choose an admissible Borel probability measure µ i ∈ Prob( G i ) and denote by( B i , ν i ) the Furstenberg-Poisson boundary of ( G i , µ i ). Then the product G -space ( B, ν ) :=( B , ν ) × ( B , ν ) is the Furstenberg-Poisson boundary of G with respect to the product measure µ := µ ⊗ µ ∈ Prob( G ) (see [BS04, Corollary 3.2]). We will write N = L ∞ ( B ), N = L ∞ ( B )and N = L ∞ ( B ) = N ⊗ N .Observe that if ( M, E ) is a (Γ , N )-von Neumann algebra then E maps ι -continuous elementsin M to ι -continuous elements in N . We can therefore take advantage of the fact that N isalready a G -algebra. The following lemma will play an essential role. Lemma 5.4.
The set of ι -continuous elements in N is equal to N ⊗ .Proof. Let f ∈ N be a ι -continuous element in N . We view f as an N -valued function on B , f ∈ L ∞ ( B , N ) and we choose an essential value y ∈ N of f .For ε >
0, the set E ε = { b ∈ B | k f ( b ) − y k η < ε } has positive measure in B . By [Pe14,Lemma 5.1], there exists a sequence ( γ n ) n ∈ N in Γ, so that ι ( γ n ) → e in G and η ( p ( γ n ) E ε ) → f is ι -continuous, we find k f − y ⊗ k ν = lim n k σ γ n ( f ) − y ⊗ k ν = lim n Z B k σ p ( γ n ) ( f ( p ( γ n ) − b )) − y k ν d ν ( b )= lim n Z B k f ( p ( γ n ) − b ) − σ − ι ( γ n ) ( y ) k ν ◦ σ ι ( γn ) d ν ( b )= lim n Z B k f ( p ( γ n ) − b ) − y k ν d ν ( b ) . This latter integral can split into two parts: the integral over p ( γ n ) E ε , where the integrand isless than ε , and the integral over the complementary set, whose measure goes to 0 as n goesto infinity (and where the integrand is bounded by (2 k f k ) ). So we find that k f − y ⊗ k ν ≤ ε .Since ε > f = y ⊗ ∈ N ⊗ (cid:3) HARMENABILITY OF ARITHMETIC GROUPS OF PRODUCT TYPE 27
Theorem 5.5.
Let ( M, E ) be a faithful (Γ , N ) -von Neumann algebra. Denote by M ⊂ M thesubset of G -continuous elements with respect to ι : Γ → G .Then M ⊂ M is a globally Γ -invariant von Neumann subalgebra and the action Γ y M extends to a continuous action G y M such that G acts trivially.Proof. One easily checks that M is Γ-invariant. Moreover, M is a ∗ -subalgebra of M simplybecause the multiplication map M × M → M : ( x, y ) xy is ∗ -strongly continuous on uniformlybounded sets. The following claims prove the remaining statements. Claim 1.
For any x ∈ M , the orbit map Γ → M : γ σ γ ( x ) extends to a continuous map G → M , which only depends on the first variable and takes values in M .The extension map is constructed by the classical extension argument for uniformly continuousmaps into complete spaces, but we do it by hand. Take x ∈ M . Let g ∈ G and take a sequence( γ n ) n ∈ N in Γ such that ι ( γ n ) → ι ( g ) in G . We prove that ( σ γ n ( x )) n ∈ N converges ∗ -stronglyin M . For this, consider the faithful normal state φ = ν ◦ E ∈ M ∗ and recall that the strongtopology on bounded sets of M is given by the norm k · k φ . For all m, n ∈ N , we have k σ γ n ( x ) − σ γ m ( x ) k φ = ν ◦ E (( σ γ n ( x ) − σ γ m ( x )) ∗ ( σ γ n ( x ) − σ γ m ( x )))= ν ◦ σ γ m ◦ E ( y n,m ) , where y n,m = ( σ γ − m γ n ( x ) − x ) ∗ ( σ γ − m γ n ( x ) − x ). Since x ∈ M and E is normal, E ( y n,m )converges ultraweakly to 0 as n, m → ∞ . Moreover, since M is a Γ-invariant *-subalgebraof M , y n,m ∈ M and thus Lemma 5.4 implies that E ( y n,m ) ∈ N ⊗
1, for all n, m ∈ N . Inparticular, we find σ γ m ◦ E ( y n,m ) = σ ι ( γ m ) ◦ E ( y n,m ) . Since ι ( γ m ) → ι ( g ) in G and since the action map G × N → N is ultraweakly continuous, weconclude that σ γ m ◦ E ( y n,m ) → σ g (0) = 0, ultraweakly in N as m → ∞ and n → ∞ .This shows that the uniformly bounded sequence ( σ γ n ( x )) n ∈ N is k·k φ -Cauchy and hence stronglyconverges to some y ∈ M . Applying the same argument with x ∗ instead of x , we see that thesequence ( σ γ n ( x )) n ∈ N is ∗ -strongly convergent to y ∈ M . The above computation also appliesto show that the ∗ -strong limit y ∈ M does not depend on the choice of the sequence ( γ n ) n ∈ N but only on ι ( g ) ∈ G . Therefore, we may define σ g ( x ) = y , which thus only depends on thefirst variable ι ( g ) ∈ G . The independence on the sequence ( γ n ) n ∈ N also implies that the orbitmap g ∈ G σ g ( x ) is strongly continuous.Let us check that for every g ∈ G , σ g ( x ) ∈ M . Indeed, let g ∈ G and ( γ n ) n ∈ N any sequencein Γ such that ι ( γ n ) → e in G . We have to show that σ γ n ( σ g ( x )) → σ g ( x ) ∗ -strongly. For any ε >
0, we may find a neighborhood U ⊂ G of ι ( g ) such that k σ γ ( x ) − σ g ( x ) k φ < ε for all γ ∈ Γsuch that ι ( γ ) ∈ U . Take a neighborhood U ⊂ G of e and a neighborhood U ⊂ G of ι ( g )such that U U ⊂ U . Fix n ∈ N large enough so that ι ( γ n ) ∈ U . By definition of σ g ( x ), wemay find γ ∈ Γ such that ι ( γ ) ∈ U and k σ γ n ( σ γ ( x ) − σ g ( x )) k φ = k σ γ ( x ) − σ g ( x ) k φ ◦ σ γn < ε. Then we have k σ γ n ( σ g ( x )) − σ g ( x ) k φ ≤ k σ γ n ( σ g ( x ) − σ γ ( x )) k φ + k σ γ n ( σ γ ( x )) − σ g ( x ) k φ ≤ ε + k σ γ n γ ( x ) − σ g ( x ) k φ . But since ι ( γ n γ ) ∈ U , the last term above is also bounded by ε , and hence for all n ∈ N largeenough, we get k σ γ n ( σ g ( x )) − σ g ( x ) k < ε. This proves that σ γ n ( σ g ( x )) → σ g ( x ) strongly. Applying the same reasoning to x ∗ ∈ M , weobtain σ γ n ( σ g ( x )) → σ g ( x ) ∗ -strongly. This proves that σ g ( x ) ∈ M and finishes the proof ofClaim 1. Claim 2. M ⊂ M is a von Neumann subalgebra and σ : G y M is a continuous action. Indeed, let x ∈ ( M ) ′′ and take a sequence ( γ n ) n in Γ such that ι ( γ n ) → e in G . Fix ε > x ∈ M such that k x − x k φ < ε . Since ( x − x ) ∗ ( x − x ) is in the weak closure of M ,Lemma 5.4 implies that E (( x − x ) ∗ ( x − x )) ∈ N ⊗
1, i.e. this element is ι -continuous in N .In particular, lim n σ γ n ( E (( x − x ) ∗ ( x − x ))) = E (( x − x ) ∗ ( x − x )). Applying ν , we findlim sup n k σ γ n ( x − x ) k φ = lim sup n ν ◦ σ γ n ◦ E (( x − x ) ∗ ( x − x )) = k x − x k φ < ε . This allows to computelim sup n k σ γ n ( x ) − x k φ ≤ lim sup n ( k σ γ n ( x − x ) k φ + k σ γ n ( x ) − x k φ + k x − x k φ ) < ε. As ε > σ γ n ( x ) → x strongly. Applying the samereasoning to x ∗ ∈ M , we obtain that σ γ n ( x ) → x strongly. So x ∈ M and thus M is indeed avon Neumann algebra. The fact the action σ : G y M is continuous follows from Claim 1 and[Ta03a, Proposition X.1.2]. (cid:3) Theorem 5.6.
Keep the notation Γ < G = G × G , ι , N = N ⊗ N as above. Let ( M, E ) be a faithful (Γ , N ) -von Neumann algebra ( M, θ ) . Denote by ( f M , e E ) the induced ( G, e N ) -algebraas defined in Example 4.5.The algebra M ⊂ M of ι -continuous elements identifies with the fixed point algebra f M G .More precisely, there is a G -equivariant surjective isomorphism κ : M → f M G such that ( E N ◦ e E ) ◦ κ = E , where E N : e N → N is as defined in Example 4.5.Proof. We view f M = ( L ∞ ( G ) ⊗ M ) ( ρ ⊗ σ )(Γ) as the algebra of Γ-equivariant functions from G to M (with respect to the right Γ-action on G ). We then define the map κ : M → f M by theformula κ ( x )( g ) = σ − g ( x ) ∈ M, for all x ∈ M , g ∈ G. This map is clearly G -equivariant, so it must range into f M G . It is also obvious that κ isinjective; let us prove that it is surjective.Let f ∈ f M G . Proceeding as in the proof of Proposition 5.1, in order to show that f is in therange of κ , it suffices to show that any essential value y of f is an element of M .Lemma 5.7 below implies that e N G ⊂ L ∞ ( G ) ⊗ N ⊗
1, so e E ( f ) ∈ L ∞ ( G ) ⊗ N ⊗
1. Since e E isequal to id ⊗ E , we deduce that E ( f ( g )) ∈ N ⊗
1, for almost every g ∈ G . In particular, E ( y ) ∈ N ⊗
1. We may apply the same reasoning to f ∗ f ∈ ( f M ) G and deduce that E ( y ∗ y ) ∈ N ⊗ Claim.
For almost every g, h ∈ G , we have E ( f ( g ) ∗ f ( h )) ∈ N ⊗ F : G × G → N : ( g, h ) E ( f ( g ) ∗ f ( h )) is G × G -invariant and it is Γ-equivariant in the sense that F ( gγ, hγ ) = σ − γ ( F ( g, h )), for allalmost all g, h ∈ G and all γ ∈ Γ. The claim now follows from Lemma 5.7 below.Using this claim and the observations preceding it, we find that for almost every g ∈ G , E (( f ( g ) − y ) ∗ ( f ( g ) − y )) ∈ N ⊗ E (( f ( g ) − y ) ∗ ( f ( g ) − y )) is G -invariant. Let ( γ n ) n ∈ N be any sequence in Γ such that ι ( γ n ) → e in G . We now show that σ γ n ( y ) → y ∗ -strongly in M . Let ε > A = { g ∈ G | k f ( g ) − y k φ < ε } . Choose n ∈ N large enough so that the intersection A ∩ ( A · ι ( γ n )) has positive measure andpick an element g ∈ A ∩ ( A · ι ( γ n )) such that E (( f ( g ) − y ) ∗ ( f ( g ) − y )) is G -invariant. We mayalso assume that n is large enough so that k ν − ν ◦ σ ι ( γ n ) k · (2 k f k ) < ε .Then on the one hand, we have gι ( γ n ) − ∈ A , and k f ( gγ − n ) − y k φ = k f ( gι ( γ n ) − ) − y k φ < ε .On the other hand, we have k f ( gγ − n ) − σ γ n ( y ) k φ = k σ γ n ( f ( g ) − y ) k φ = ν ◦ σ γ n ◦ E (( f ( g ) − y ) ∗ ( f ( g ) − y )) . HARMENABILITY OF ARITHMETIC GROUPS OF PRODUCT TYPE 29
By our choice of g , E (( f ( g ) − y ) ∗ ( f ( g ) − y )) is G -invariant and hence we may continue ourcomputation k f ( gγ − n ) − σ γ n ( y ) k φ = ν ◦ σ ι ( γ n ) ◦ E (( f ( g ) − y ) ∗ ( f ( g ) − y )) ≤ k f ( g ) − y k φ + k ν − ν ◦ σ ι ( γ n ) k · (2 k f k ) < ε . In conclusion, we see that k y − σ γ n ( y ) k φ ≤ k y − f ( gγ − n ) k φ + k f ( gγ − n ) − σ γ n ( y ) k φ < (1 + √ ε. This proves that σ γ n ( y ) → y strongly in M . Applying the same reasoning to y ∗ ∈ M which isan essential value of f ∗ ∈ ( f M ) G , we obtain σ γ n ( y ) → y ∗ -strongly in M . So y ∈ M , as desired.Finally, the equality E N ◦ e E ◦ κ = E can be verified by making the map E N explicit. (cid:3) We used the following technical result.
Lemma 5.7.
Let N = L ∞ ( G ) ⊗ L ∞ ( G ) ⊗ N and define the action σ : G × G × Γ y N by σ ( g,h,γ ) = λ g ρ γ ⊗ λ h ρ γ ⊗ σ γ for g, h ∈ G , γ ∈ Γ . Then we have N G × G × Γ ⊂ L ∞ ( G ) ⊗ L ∞ ( G ) ⊗ N ⊗ B . In particular e N G ⊂ L ∞ ( G ) ⊗ N ⊗ B .Proof. Set P = L ∞ ( G/ Γ) ⊗ L ∞ ( G ) ⊗ N and define the action β : G × G y P by β ( g,h ) = λ g ⊗ λ h ρ g ⊗ σ g for g, h ∈ G . Define the unital ∗ -isomorphism Ξ : N Γ → P by the formulaΞ( F )( g Γ , h ) = σ g ( F ( g, hg )) , for every F ∈ N Γ , almost every ( g, h ) ∈ G × G, One checks that the isomorphism Ξ is onto and intertwines the action G × G y N Γ with theaction β : G × G y P .Let now F ∈ N G × G × Γ . Then Ξ( F ) ∈ L ∞ ( G/ Γ) ⊗ L ∞ ( G ) ⊗ N and Ξ( F ) invariant underthe automorphisms λ g ⊗ id G ⊗ σ g for all g ∈ G . Since Γ < G × G is a lattice with denseprojections, the pmp action G y G/ Γ is ergodic and [BS04, Corollary 2.18] implies that thediagonal action G y G/ Γ × B is ergodic. This further implies that Ξ( F ) ∈ L ∞ ( G/ Γ) ⊗ L ∞ ( G ) ⊗ N ⊗ B which in turn implies that F ∈ L ∞ ( G ) ⊗ L ∞ ( G ) ⊗ N ⊗ B . (cid:3) Combining the above result with Lemma 2.8 we obtain the following key theorem.
Theorem 5.8.
Take k ≥ and a product G = G × · · · × G k of k lcsc groups. For every ≤ i ≤ k , choose an admissible Borel probability measure µ i ∈ Prob( G i ) and denote by ( B i , ν i ) the Furstenberg-Poisson boundary of ( G i , µ i ) . Then the product G -space ( B, ν ) := ( B , ν ) ×· · · × ( B k , ν k ) is the Furstenberg-Poisson boundary of G with respect to the product measure µ := µ ⊗ · · · ⊗ µ k ∈ Prob( G ) (see [BS04, Corollary 3.2] ). Set N = L ∞ ( B, ν ) .Take a lattice with dense projections Γ < G and a faithful (Γ , N ) -von Neumann algebra ( M, E ) .If E is not Γ -invariant, then there exists ≤ i ≤ k , such that the von Neumann subalgebra M i of G i -continuous elements in M is nontrivial, E | M i is not G i -invariant and its image is in L ∞ ( B i ) ≤ L ∞ ( B ) .Proof. Following Example 4.5, denote by ( f M , e E ) the induced ( G, e N )-structure and view E N ◦ e E as a ( G, N )-structure. If E is not Γ-invariant, Lemma 4.6 implies that E N ◦ e E is not G -invariant.Since ( B, ν ) is the Furstenberg-Poisson boundary of G , Example 4.2 further implies that thefaithful µ -stationary state φ = ν ◦ E N ◦ e E is not G -invariant on f M .In particular, there exists i such that φ is not G i -invariant. Gather the factors of G to writeit as a product of two groups G i × H i . By Lemma 2.8, we find that φ is not G i -invariant on f M H i . Thanks to the observations in Example 4.2, this amounts to saying that E N ◦ e E is not G i -invariant on f M H i . By Theorem 5.6, this exactly means that E is not invariant on the algebra of G i -continuous elements M i . We saw in Lemma 5.4 and the comment preceding it that indeed E maps M i into L ∞ ( B i ). (cid:3) Proofs of charmenability
In this section, we prove Theorem C and Theorem D, as well as Proposition 6.1 which consistsof the first half of Theorem A.6.1.
Arithmetic groups.
The main result of this subsection is the following proposition.
Proposition 6.1.
Let K be a global field and G a connected non-commutative K -almost simple K -algebraic group. If Γ ≤ G ( K ) is an S -arithmetic subgroup of a product type then Γ ischarmenable. For the proof we need to establish the following freeness result.
Lemma 6.2.
Let k be a local field and G a connected k -almost simple k -algebraic group. Let H (cid:12) G be a proper k -subgroup and let G = G ( k ) , H = H ( k ) . We endow G/H with the unique G -invariant class of Radon measures. Then for every g ∈ G \ Z ( G ) , for almost every w ∈ G/H ,we have gw = w . The proof of Lemma 6.2 in turn relies on the following preliminary result.
Lemma 6.3.
Let k be a local field and ¯ k an algebraically closed field extension of k . Let G bea connected k -algebraic group and denote G = G ( k ) . Let H ≤ G be a k -algebraic subgroup anddenote H = H ( k ) . We endow G/H with the unique G -invariant class of Radon measures. Welet U be a closed proper subvariety of G / H , U ( G / H . Then, considering G/H as a subset of G / H ( k ) ⊂ G / H (¯ k ) , we have that U (¯ k ) ∩ G/H is a null set in
G/H .Proof.
Denote by π : G (¯ k ) → G / H (¯ k ) the natural map and by π k : G → G/H ⊂ G / H (¯ k ) itsrestriction to the k -points. It is a general fact about lcsc groups that a subset of G/H is null ifand only if its preimage in G is null. Let us check that indeed π − k ( U (¯ k )) is null in G .We denote by V the preimage of U in G and observe that this is a closed proper subvariety of G satisfying V ( k ) = π − k ( U (¯ k )). By [Bo91, Theorem AG.14.4], the Zariski closure V of V ( k )in G is a k -subvariety of G , contained in V . So in particular V is a proper k -subvariety of G , which satisfies V ( k ) = V ( k ). Since G is connected, [Ma91, Proposition I.2.5.3(ii)] impliesthat V ( k ) is indeed a null set in G . (cid:3) Proof of Lemma 6.2.
Fix g ∈ G \ Z ( G ). Note that g acts non-trivially on G/H , otherwise g would belong to the normal subgroup T x ∈ G xHx − , the Zariski closure of which is a propernormal k -subgroup N of G . Since G is k -almost simple, we have N ⊂ Z ( G ), forcing g ∈ Z ( G ),which we excluded. Hence the subvariety U of fixed points of g in G / H is proper, and weconclude by applying Lemma 6.3. (cid:3) We now have set up all the tools we need to prove Proposition 6.1.
Proof of Proposition 6.1.
By Proposition 3.10, we assume as we may that the set S is finite.We consider the finite set I of places v of K such that the image of Γ in G ( K v ) is unbounded.For each i ∈ I , we denote by • k i the completion of K with respect to the place i ; • G i the algebraic group G viewed as a k i -group; • P i a minimal k i -parabolic subgroup of G i ; • G i < G i ( k i ) the closure of the image of Γ in G i ( k i ), and P i := P i ( k i ) ∩ G i . HARMENABILITY OF ARITHMETIC GROUPS OF PRODUCT TYPE 31
Note that G i ( k i ) + ≤ G i ≤ G i ( k i ), by the strong approximation theorem (see [Ma91, The-orem II.6.8]). Therefore we may apply Example 2.14, and find that G i acts transitively on G i ( k i ) / P i ( k i ) with stabilizer P i and the pair ( G i , P i ) is stably self-normalizing and it has therelative Mautner property. By [BS04, Corollary 5.2], for every i ∈ I , there exists a generatingmeasure µ i on G i such that ( G i /P i , ν i ) is the Furstenberg-Poisson boundary of ( G i , µ i ), where ν i is the (unique) µ i -stationary measure on G i /P i and it is G i -quasi-invariant. We denote B i = G i /P i and endow it with the quasi-invariant measure ν i . We let B = Q I B i , endow it withthe measure ν = Q I ν i and set µ = Q I µ i . By [BS04, Corollary 3.2], ( B, ν ) is the Furstenberg-Poisson boundary of (
G, µ ) and by [BF14, Theorem 2.7] and [BF18, Lemma 3.5] it is amenableand metrically ergodic as a G -space and as a Γ-space.We will prove that Γ satisfies condition (a’) and (b) from Proposition 4.18. By Margulis normalsubgroup theorem [Ma91, VIII(A), p. 259], the amenable radical of Γ is its center, so condition(b) is automatically fulfilled, as was observed in the proof of Corollary 4.20. Set N := L ∞ ( B ),and take a non-invariant (Γ , N )-von Neumann algebra ( M, E ). We need to argue that E and E g are singular, for every g ∈ Γ \ Z (Γ). For the sake of clarity, we first give the proof in thesimply connected setting, and then explain the modifications to make in the general case. Special case: G is simply connected. In this case the strong approximation theorem (see [Ma91, Theorem II.6.8]) gives that G i = G i ( k i ) and Γ is a lattice with dense projections in G := Q i ∈ I G i . By Theorem 5.8, there exists i ∈ I such that the G i -algebra M i in M is non-trivial, E | M i is not G i -invariant and its image isin N i := L ∞ ( B i ) ⊂ N .Since the action Γ y M is ergodic, we note that G i y M i is also ergodic. By Lemma 4.15,we find that E | M i is an extremal ( G i , N i )-structure map. Proposition 4.14 then gives that forevery g ∈ G i \ Z ( G i ), E | M i and ( E | M i ) g are singular. Observe that the projection map Γ → G i is injective, indeed it coincides with the injectionΓ ֒ → G ( K ) → G ( k i ) = G i ( k i ) . Therefore, the image of g ∈ Γ \Z (Γ) is in G i \Z ( G i ), thus E and E g are singular when restrictedto M i and by Remark 4.12, it follows that they are singular on M . This is the desired conclusion. General case.
In general unfortunately we don’t know that Γ is with dense projections, so we may not applyTheorem 5.8 as such. Nevertheless we show that we can still get the conclusion of this theorem.Once we arrive there, we will just continue the proof as in the simply connected case.Denote by ( f M , e E ) the induced ( G, e N )-structure, as in Example 4.5. Since E is not invariant,Lemma 4.6 implies that E N ◦ e E is not G -invariant. Since ( B, ν ) is the Furstenberg-Poissonboundary of G , Example 4.2 further implies that the faithful µ -stationary state φ = ν ◦ E N ◦ e E is not G -invariant on f M . In particular, there exists i ∈ I such that φ is not G i -invariant. Gatherthe factors of G to write it as the product of two groups G i × H i . By Lemma 2.8, we find that φ is not G i -invariant on f M H i . Thanks to the observations in Example 4.2, this amounts to sayingthat E N ◦ e E is not G i -invariant on f M H i .At this stage, we don’t know a priori that f M H i identifies with the G i -algebra in M , becausewe don’t know that the projection of Γ into H i is dense. Fortunately, this algebra f M H i can beexpressed without reference to H i , as the algebra of Γ-equivariant L ∞ -functions G i → M . Weclaim that the structure map E N ◦ e E on this algebra may also be described without appealingto the specific group H i , provided H i acts metrically ergodically on the Lebesgue space B ′ i := Q j = i B j . In fact, given such an equivariant function f ∈ f M H i , the Γ-invariant function g ∈ G i σ g ( E ( f ( g )) ∈ N is essentially constant, by density of the image of Γ in G i . We denote by F ( f ) its essential value. Claim. F ( f ) = E N ◦ e E ( f ), for every f ∈ f M H i . By definition E N ◦ e E ( f ) is obtained by viewing the function f ′ : g ∈ G σ g ( E ( f ( g )) ∈ N as a right Γ-invariant function and integrating it against the G -invariant probability measureon G/ Γ. View f ′ as an element of L ∞ ( G/ Γ) ⊗ L ∞ ( B i ) ⊗ L ∞ ( B ′ i ), which is invariant underthe diagonal H i -action (where H i acts trivially on B i and metrically ergodically on B ′ i ). SinceΓ H i is dense in G , H i acts ergodically on G/ Γ, and hence, by metric ergodicity, we find that f ′ ∈ ⊗ L ∞ ( B i ) ⊗
1. So f ′ is essentially constant; its integral over G/ Γ is equal to its essentialvalue y ∈ L ∞ ( B i ), i.e. E N ◦ e E ( f ) = y . Moreover, since f ′ is essentially constant when viewed asa function over G , we find that for almost every g ∈ G i , h ∈ H i , σ hg ( E ( f ( g ))) = y . In particular,for almost every g ∈ G i , σ g ( E ( f ( g )) is an H i -invariant element in N , equal to y = E N ◦ e E ( f ).We thus conclude that F ( f ) = E N ◦ e E ( f ), as claimed.Thanks to these observations we will replace H i at our advantage to get the dense projectionsassumption, and verify that we are still in a situation where N is the Poisson boundary of the(new) ambient group. Define H ′ i < H i to be the closure of the image of Γ in H i and view Γas a lattice with dense projections inside G i × H ′ i . It is important to observe that H ′ i containsthe group Q j = i G j ( k j ) + , thanks to the strong approximation theorem (see [Ma91, TheoremII.6.8]). Thus we may apply [BS04, Corollary 5.2], and find a generating measure µ ′ i on H ′ i suchthat the Poisson boundary of ( H ′ i , µ ′ i ) can be identified with B ′ i , as a Lebesgue H ′ i -space. By[BS04, Corollary 3.2], the Lebesgue space B = B i × B ′ i , is the Furstenberg-Poisson boundaryof G i × H ′ i , for the measure µ i ⊗ µ ′ i . We can now apply Theorem 5.6 to Γ < G i × H ′ i with the(Γ , N )-structure ( M, E ). We obtain an identification between the G i -algebra M i and the algebraof H ′ i -invariant elements L ∞ ( G i × H ′ i , M ) H ′ i × Γ which intertwines the natural ( G i , N i )-structuremaps. By the previous paragraph, we know that the later algebra L ∞ ( G i × H ′ i , M ) H i × Γ togetherwith its ( G i , N i )-structure map identifies with ( f M H i , E N ◦ e E ). Since E N ◦ e E is not G i -invarianton f M H i , we conclude that E is not G i -invariant on M i .As announced we thus conclude that there is an index i and a Γ-invariant von Neumann sub-algebra M i ⊂ M on which the Γ-action extends to a continuous action G y M i that factorsthrough the projection map G → G i , and on which E is not Γ-invariant. We can now finish theproof as in the simply connected case. (cid:3) We end this subsection by proving Theorem C.
Proof of Theorem C.
We denote Λ = SL n ( Z ) ⋉ Z n and Γ = SL n ( Z ). We let B the flag manifoldassociated with G = SL n ( R ) and check that (a) and (b) of Proposition 4.19 are satisfied. UsingFourier transform we identify Char(Rad(Λ)) with Prob( T n ). Then (b) follows from the mainresult of [Fu98]. The proof of property (a) follows from [BH19, Theorem B] by combiningProposition 4.14 with Lemma 6.2 as above. (cid:3) Lattices in product of trees.
This subsection is devoted to the proof of Theorem D.
Proposition 6.4.
Fix n ≥ and natural numbers p , . . . , p n , q , . . . , q n > . For each ≤ i ≤ n ,let T i be a ( p i + 1 , q i + 1) -biregular simplicial tree and let Γ < Aut + ( T ) × · · · × Aut + ( T n ) be acocompact lattice. We look at a projection onto a factor, say the first factor. Endow ∂T withits unique Aut + ( T ) -invariant measure class, and look at the action of Γ on ∂T . If the firstprojection is injective on Γ , then for every g ∈ Γ \ { e } , the fixed point set in ∂T has measure . To prove the proposition we need the following lemma.
Lemma 6.5.
Fix integers p, q > , a ( p +1 , q +1) -biregular simplicial tree T and a proper subtree T ′ ( T . Assume that the subgroup H ≤ Aut( T ) which stabilizes T ′ acts on it cocompactly. Then ∂T ′ is a null set of ∂T , where ∂T is endowed with the unique Aut( T ) -invariant measure class.In particular, this measure class on ∂T is non-atomic. which really is the algebra of all G i -continuous elements in M HARMENABILITY OF ARITHMETIC GROUPS OF PRODUCT TYPE 33
Proof.
We fix a vertex o ∈ T ′ and consider the space R consisting of rays in T emanating at o and the natural surjection π : R → ∂T . Endowing R with the unique probability measure µ which is invariant under Stab( o ) < Aut( T ), this map is measure class preserving. We thus needto show that the subset R ′ = π − ( ∂T ′ ) is a null set in R .By the assumption that T ′ = T there exist adjacent vertices u, v ∈ T such that u ∈ T ′ , v / ∈ T ′ .Without loss of the generality we assume that the degree of u is p + 1. Setting A = { x ∈ R | x ( k ) ∈ Hu for infinitely many values of k } we easily see that A ⊂ R ′ . By the fact that H acts cocompactly on T ′ and the law of largenumbers we also have that A is conull in R ′ , thus µ ( A ) = µ ( R ′ ). Writing A as the descendingintersection A = T n ∈ N A n of A n = { x ∈ R | x ( k ) ∈ Hu for at least n values of k } , we have that µ ( A ) = lim n µ ( A n ). Since for all n ≥ µ ( A n +1 ) ≤ (cid:16) p − p (cid:17) µ ( A n ) we get that µ ( A n ) ≤ (cid:16) p − p (cid:17) n − , thus indeed ∂T ′ is a null set in ∂T .The last sentence of the proposition follows by considering the special case where T ′ is a geodesicin T . (cid:3) Proof of Proposition 6.4.
We fix a non-trivial element g ∈ Γ and set F = Fix( g ). We assumeas we may that F has at least three points. Let T ′ be the convex hull in T of F . Then T ′ isnon-empty, it coincides with the set of fixed points of g in T and F = ∂T ′ . Let Z < G be thecentralizer of g and note that T ′ is Z invariant. We claim that the Z -action on T ′ is cocompact.From this claim we will get by Lemma 6.5 that F = ∂T ′ is a null set in ∂T , thus proving theproposition.We endow X = T × · · · × T n with the L -product metric and note that this is a CAT(0) space.We consider the displacement function D : X → [0 , ∞ ), D ( x ) = d ( gx, x ) and let Y ⊂ X beits minset, that is setting d = min x ∈ X d ( gx, x ), Y = D − ( d ). Note that the image of D isdiscrete in [0 , ∞ ), as the Γ action on X is simplicial, thus D attends its minimum d and Y is a closed convex subset of X . By a result of Kim Ruane, [Ru99, Theorem 3.2 and Remark1], the action of Z on Y is cocompact. In particular, the Z -action on the image of Y underthe projection X → T is cocompact. We are done by observing that this image is exactly theminset of g in T , that is the tree of g -fixed points T ′ . (cid:3) Proof of Theorem D.
By [BM00, Lemma 3.1.1, Proposition 3.1.2], the 2-transitivity assumptionimplies that for every i , every closed normal subgroup of G i is co-compact. This 2-transitivityalso implies that G i is non-amenable, and hence it has no non-trivial amenable normal closedsubgroup. So Γ has trivial amenable radical.Let us argue that each projection map Γ → G i is injective. Indeed, the kernel of such aprojection map is equal to Γ ∩ H i , where H i = Q j = i G j . It is a closed subgroup of H i , whichis normalized by the projection of Γ on H i . So by the dense projection assumption, Γ ∩ H i isa normal closed subgroup of H i . Since every non-trivial normal subgroup of each factor G j , j = i , is co-compact in G j , this normal subgroup is either trivial or it contains a co-compactnormal closed subgroup G ′ j of some G j , j = i . In this case, Γ contains G ′ j . Further, Γ /G ′ j isa lattice with dense projections inside ( G j /G ′ j ) × Q i = j G i . The only way this can happen is ifthe compact factor G j /G ′ j is trivial. In this case, G j = G ′ j is discrete, which contradicts the2-transitivity assumption, and the fact that T i is thick.We set for every i ∈ I , B i = ∂T i endowed with the unique G i -invariant measure class and let B = Q B i . For every i ∈ I , we fix a point in ∂T i and let P i be its stabilizer. By this weidentify B i = G i /P i . By Example 2.15, P i is stably self-normalizing and it has the relativeMautner property in G i . By [BS04, Theorem 5.1] we have that B i is the Furstenberg-Poissonboundary of G i for some generating measure µ i on G i and by [BS04, Corollary 3.2], ( B, ν ) is the Furstenberg-Poisson boundary of G for the measure µ = Q µ i . By [BF14, Theorem 2.7], B is amenable and metrically ergodic G -space and by [BF18, Lemma 3.5] it is amenable andmetrically ergodic Γ-space. Therefore, by Proposition 4.16, it is enough to verify condition (a)of Proposition 4.19.We now fix a separable, ergodic, faithful (Γ , L ∞ ( B ))-von Neumann algebra ( M, E ) which is notΓ-invariant and argue to show that it is Γ-singular. By Theorem 5.8, we find an index i ∈ I suchthat the G i -algebra M i in M is non-trivial, and such that E | M i is not G i -invariant. Since theaction Γ y M is ergodic, we note that G i y M i is also ergodic. By Lemma 4.15, we find that E | M i is an extremal ( G i , L ∞ ( B i ))-structure map. We combine Proposition 4.14 with our freenessresult Proposition 6.4 and find that E | M i is Γ i -singular, where Γ i is the projection of Γ into G i .As the projection map Γ → G i is injective, E | M i is Γ-singular. From the characterizations ofsingular ucp maps we gave, this implies that E is Γ-singular, as desired. (cid:3) Proofs of charfiniteness
In this section, we prove the second half of Theorem A and Theorem B.7.1.
Finite dimensional unitary representations.
In this subsection, we prove the follow-ing proposition, which is well known to experts.
Proposition 7.1.
Let K be a global field and G a connected non-commutative K -almost simple K -algebraic group. Let Γ ≤ G ( K ) be an S -arithmetic subgroup of higher rank. If either S isfinite or G is simply connected then Γ has a finite number of isomorphism types of unitaryrepresentation at each finite dimension. We will use heavily the results of [Ma91, Chapter VIII] and also rely on [Sh99, Section 5]. Forthe terminology regarding arithmetic groups used in the proof, see Definition 1.3.
Proof.
We first note that if Γ is of a simple type and of higher rank then it has property (T),which clearly implies the result. Thus we assume as we may that Γ is of a product type. Next,we observe that if Λ has the property of having a finite number of isomorphism types of unitaryrepresentation at each finite dimension and Λ → Γ is a homomorphism with a finite kernel andfinite index image then also Γ has this property. Therefore we assume as we may that G issimply connected even in case S is finite. Indeed, in this case letting ˜ G be the simply connectedcover of G and letting Λ be the preimage of Γ under the covering map ˜ G ( K ) → G ( K ), we havethat Λ ≤ ˜ G ( K ) is an S -arithmetic subgroup of higher rank and Λ → Γ is a homomorphism witha finite kernel and finite index image. We fix n and argue to show that Γ has a finite numberof U( n )-conjugacy classes of homomorphisms into U( n ). This fact would easily follow from[Sh99, Theorem 5.7] in the case where S is finite. However, such a statement is badly behavedunder inductive limits. For this reason we need to be more accurate and invoke superrigiditytechniques of Margulis.Let us say that such a homomorphism Γ → U ( n ) is finite if it has a finite image. Claim 7.2.
Γ has a finite number of U( n )-conjugacy classes of finite homomorphisms into U( n ).We will prove the claim later, first finishing the proof assuming the claim. We first note thatwe may assume that K is of characteristic zero, as if K is of positive characteristic then everyhomomorphism ρ : Γ → U( n ) is finite by [Ma91, VIII(C), p. 259]. Indeed, if ρ had an infiniteimage, upon setting ℓ = R and denoting by H the identity component of the Zariski closure of ρ (Γ), we would get a field extension K → ℓ , thus a contradiction. We thus may apply [Ma91,VIII(B)(iii), p. 258] for ℓ = R and letting H be the n -dimensional real unitary group, thus H ( ℓ ) = U( n ). It follows that any ρ : Γ → U( n ) is of the form ρ = φ · ν where φ : Γ → U( n ) isobtained by a compositionΓ → G ( K ) ≃ R K/ Q G ( Q ) → R K/ Q G ( ℓ ) → H ( ℓ ) = U( n ) , HARMENABILITY OF ARITHMETIC GROUPS OF PRODUCT TYPE 35 where R K/ Q G ( ℓ ) → H ( ℓ ) is the ℓ -points evaluation of an ℓ -algebraic morphism R K/ Q G → H and ν : Γ → U( n ) is a finite homomorphism whose image commutes with φ (Γ). Since thenumber of ℓ -algebraic morphisms R K/ Q G → H is finite, we are done by the claim.Next we prove the claim. By [Ma91, VIII(A), p. 259] it is enough to show that there existsa non-central element g ∈ Γ which is in the kernel of all finite morphisms Γ → U( n ). Indeed,letting N ≤ Γ be the normal subgroup generated by g , we have that Γ /N is finite and since everyfinite morphism Γ → U( n ) factors via Γ /N there is only a finite number of isomorphism typesof those. We can find a finite subset S ⊂ S such that the S -arithmetic subgroup Γ = Γ ∩ Λ S (see Definition 1.3) is still of a product type. We thus assume as we may that S is finite. By thefact that G is simply connected, the strong approximation theorem (see [Ma91, Theorem II.6.8])implies that Γ is a lattice with dense projections in G = Q i ∈ I G i where for each i ∈ I , G i is the k i -points of the K -algebraic group G for some local field extension k i of K . By Proposition 2.2,the groups G i have no non-trivial characters and it follows that they have no non-trivial finitedimensional unitary representations. It then follows by [Sh99, Theorem 5.7] that Γ has a finitenumber of isomorphism types of homomorphism into U( n ), and in particular a finite numberof finite ones. It follows that all finite homomorphisms into U( n ) factor via one finite quotient,thus we can pick a non-central element g ∈ Γ which is in its kernel. This finishes the proof. (cid:3)
A similar statement also holds for lattices in product of groups of automorphisms of trees.
Proposition 7.3.
For n ≥ and i = 1 , . . . , n , let T i be a bi-regular tree and let G i be aclosed subgroup of Aut + ( T i ) , the group of the bicoloring preserving automorphism of T i , whichacts -transitively on its boundary. Let Γ < G × · · · × G n be a cocompact lattice with denseprojections. Then Γ has a finite number of isomorphism types of unitary representation at eachfinite dimension.Proof. We note that G = G × · · · × G n is totally disconnected, hence it has an open kernel forevery continuous homomorphism into a Lie group. It is a standard fact that the groups G i arejust non-compact, thus we get that every open normal subgroup of G is of finite index. It followsthat every continuous homomorphism of G into the Lie group U( d ) ⋉ C d has a finite image,thus the corresponding cohomology group H ( G, C d ) vanishes. Since for a finite dimensionalunitary representation Γ → U( d ) we have H (Γ , C d ) = ¯ H (Γ , C d ) (for the definition of thereduced cohomology ¯ H , see [Sh99, Definition 1.7]), we conclude by [Sh99, Theorem 5.7] that H (Γ , C d ) = 0.By the fact that Γ acts properly and cocompactly on a locally finite simplicial complex (aproduct of trees), it is finitely generated. We may thus consider the compact representationvariety Hom(Γ , U( d )). The previous paragraph allow us to conclude that the U( d ) action onit by postcomposition with conjugation has open orbits, see [We63]. It follows by compactnessthat there are only finitely many such orbits, thus indeed, Γ has a finite number of isomorphismtypes of unitary representation at dimension d . (cid:3) We are now ready to prove Theorem B.
Proof of Theorem B.
Fix a non-empty set of primes S and set Γ = SL ( Z S ). We have seen inProposition 6.1 that Γ is charmenable so we are left to verify properties (3)-(5) in Definition 1.2.Property (3) follows from [Ma91, VIII(A), p. 259], property (4) was verified in Proposition 7.1and property (5) follows from [PT13, Theorem 2.6]. So indeed, Γ is charfinite. (cid:3) We note that the assumption that G is simply connected is missing in this reference. This is certainly a typo,as this assumption is used in its proof. Of course when S is finite this doesn’t matter, but when S is infinite thisassumption is necessary, as can be seen for example by the natural morphism PGL ( Q ) → Q ∗ / ( Q ∗ ) determinedby the determinant morphism. Char-(T) and the proof of Theorem A.
Recall that an lcsc group G has property (T) ifand only if every amenable representation of G contains a finite dimensional sub-representation,see [BV91, Theorem 1.1]. Definition 7.4.
An lcsc group G is said to have property char- (T) if for every amenablecharacter the associated GNS representation contains a finite dimensional sub-representation. Proposition 7.5.
Let Γ be a lattice with dense projections in G = G × G . Assume that G has property (T) and Char( G ) = { } . Then Γ has char- (T) .Proof. Let φ be an amenable character of Γ, and denote by ( π, H, ξ ) the corresponding GNStriple. We need to prove that H contains a non-zero finite dimensional invariant subspace. Wewill argue that ( H ⊗ H, π ⊗ π ) has non-zero invariant vectors. Note that π ⊗ π is a tracialrepresentation of Γ, in the sense that ( π ⊗ π )(Γ) ′′ has a normal faithful trace, implemented bythe vector ξ ⊗ ξ . Moreover π ⊗ π has almost invariant vectors.Denote by ( e H, e π ) the induced unitary representation of ( H ⊗ H, π ⊗ π ) to G . Then this represen-tation has almost invariant vectors, and since G has property (T), e H must contain non-trivial G -invariant vectors. By assumption, ι (Γ) is dense in G , where ι : Γ → G is the restriction ofthe projection map. We observed in Proposition 5.1 that e H G is naturally identified with the ι -continuity space of H ⊗ H . In other words, we have found a non-trivial subspace K ⊂ H ⊗ H on which the representation π ⊗ π extends to a continuous representation of G . Since therepresentation of Γ on H ⊗ H is tracial, so is the restricted representation on K . Thus thecontinuous extension ρ : G → U ( K ) is a tracial representation. But Char( G ) is trivial so ev-ery continuous group homomorphism from G into the unitary group of a tracial von Neumannalgebra is trivial. Therefore ρ is trivial, which implies that H ⊗ H contains invariant vectors,as desired. (cid:3) Proposition 7.6.
Let K be a global field and G a connected non-commutative K -almost simple K -algebraic group. Let Γ ≤ G ( K ) be an S -arithmetic subgroup of a product type. Assumefurther that there exists an absolute value v such that G ( K v ) has property (T) . If either S isfinite or G is simply connected then Γ has char- (T) .Proof. We begin as in the proof of Proposition 7.1. We first note that if Γ is of a simple typeand of higher rank then it has property (T), which clearly implies the result. Thus we assumeas we may that Γ is of a product type. Next, observe that if Λ has char-(T) and Λ → Γ is ahomomorphism with a finite kernel and finite index image then also Γ has char-(T). Thereforewe assume as we may that G is simply connected even in case S is finite. Indeed, in thiscase letting ˜ G be the simply connected cover of G and letting Λ be the preimage of Γ underthe covering map ˜ G ( K ) → G ( K ), we have that Λ ≤ ˜ G ( K ) is an S -arithmetic subgroup of ahigher rank and Λ → Γ is a homomorphism with a finite kernel and finite index image. Asusual, view Γ as a lattice in the corresponding restricted product of all almost simple factorsover all local completions of K in which the image of Γ is unbounded. We set G = G ( K v )and denote the restricted product of all other factors by G . By the strong approximationtheorem (see [Ma91, Theorem II.6.8]), Γ is a lattice with dense projections in G = G × G . ByProposition 2.2, the group G has no non-trivial characters and by assumption G has (T). Itfollows by Proposition 7.5 that Γ has char-(T). (cid:3) The proof of Theorem A now follows similarly to the proof Theorem B.
Proof of Theorem A.
We have seen in Proposition 6.1 that Γ is charmenable so we are left toverify properties (3)-(5) in Definition 1.2. Property (3) follows from [Ma91, VIII(A), p. 259],property (4) was verified in Proposition 7.1 and property (5) follows from Proposition 7.6. Soindeed, Γ is charfinite. (cid:3)
Combining [BH19, Theorem B] with Corollary 4.20 and using property (T), we also get thefollowing result.
HARMENABILITY OF ARITHMETIC GROUPS OF PRODUCT TYPE 37
Corollary 7.7.
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