Lecture notes on “Character Rigidity”∗

Jesse Peterson

February 3, 2016

Contents

1 Introduction 21.1 Previous results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 General strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 The Howe-Moore property for SLn(R) 4

3 Property (T) 7

4 Amenability 11

5 Finite von Neumann algebras 145.1 Group characters and tracial von Neumann algebras . . . . . . . . . . . . . . . . . . 16

6 Completely positive maps 176.1 Conditional expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

7 Amenable von Neumann algebras 207.1 Amenable finite von Neumann algebras . . . . . . . . . . . . . . . . . . . . . . . . . 21

8 Property (T) for finite von Neumann algebras 23

9 Induced actions and ergodicity 24

10 Contracting automorphisms 26

11 Character rigidity for lattices 30

∗From the minicourse given at the workshop on Measurable Group Theory, at The Erwin Schrodinger InternationalInstitute for Mathematics and Physics

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12 Appendix: Background material 3212.1 Bounded linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3212.2 Topologies on the space of operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312.3 The double commutant theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3512.4 The spectral theorem and Borel functional calculus . . . . . . . . . . . . . . . . . . . 3812.5 Locally compact groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.6 Positive definite functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4312.7 Duality and Bochner’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4412.8 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

1 Introduction

Given a discrete group Γ, a character on Γ is a function τ : Γ→ C which satisfies:

1. τ(e) = 1;

2. τ is positive definite, i.e., for each γ1, . . . , γn ∈ Γ the matrix (τ(γ−1j γi))i,j is non-negative

definite;

3. τ(γ1γ2) = τ(γ2γ1) for all γ1, γ2 ∈ Γ.

The space of characters Char(Γ) forms a weak∗-closed convex subset of `∞Γ. A characterτ ∈ Char(Γ) is almost periodic if the set of translates {x 7→ τ(γx)}γ∈Γ forms a uniformly pre-compact set in `∞Γ.

The purpose of these notes is to discuss the following result and its corollaries:

Theorem A (P [Pet15a]). Let Γ be a lattice in a simple real Lie group G which has trivial center,and real rank at least 2. If τ ∈ Char(Γ) is an extreme point then either

1. τ is almost periodic, or else

2. τ = δe.

Corollary B (Stuck-Zimmer [SZ94]). Let Γ be as in Theorem A, then any probability measure-preserving ergodic action of Γ on a standard Lebesgue space is essentially free.

Corollary C (Margulis [Mar79]). Let Γ be as in Theorem A, then any non-trival normal subgroupof Γ is of finite index.

Characters on groups naturally correspond to representations of the group in a tracial vonNeumann algebra. In the latter setting Theorem A is equivalent (by Theorem 5.7 below) to thefollowing:

Theorem D (P [Pet15a]). Let Γ be as in Theorem A, if M is a finite factor, and π : Γ → U(M)is a representation such that π(Γ)′′ = M , then either

1. M is finite dimensional, or else

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2. π extends to an isomorphism LΓ∼−→M .

An example where the above theorem applies is the case G = PSLn(R) (and e.g., Γ = PSLn(Z))for n ≥ 3. For simplicity we will consider this case only in these notes. For the general case onemust adapt the proofs below using the structure theory of simple real Lie groups.

1.1 Previous results

We’ll say that a discrete group Γ is character rigid (or operator algebraic superrigid) ifthe consequences of Theorem A hold for Γ. That lattices in higher-rank simple Lie groups arecharacter rigid was conjectured by Connes in the early 80’s as a von Neumann algebraic analogueof Margulis’ superrigidity theorem (see [Jon00]). At the time no such lattes where known to becharacter rigid, the first examples being found by Bekka [Bek07] who showed that this holds forthe groups PSLn(Z) for n ≥ 3. Bekka’s techniques were generalized by the author and Thom in[PT13] where the same results were obtained for the groups SL2(A), where A = O is a ring ofintegers (or, more generally, A = OS−1 a localization) with infinitely many units. Unfortunatelythough, Bekka’s techniques do not appear to generalize to arbitrary lattices.

The first examples of higher-rank groups such that arbitrary irreducible lattices are characterrigid were found by the author and Creutz in [CP12, CP13], where this was shown to be the casefor irreducible lattices in certain product groups where one of the product factors is an algebraicgroup over a non-archimedian local field. These groups however are not simple real Lie groups andso are disjoint from the groups considered in Theorem A.

1.2 General strategy

Suppose G = PSLn(R) for n ≥ 3, Γ < G is a lattice, M is a finite factor and π : Γ → U(M) is arepresentation such that π(Γ)′′ = M . We let P < G be the minimal parabolic subgroup consistingof upper triangular matrices in PSLn(R).

The general strategy to prove Theorem D is to mimic Margulis [Mar79] (in the setting of groups)and Stuck-Zimmer [SZ94] (in the setting of discrete measured equivalence relations) by splittingthe result into a “property (T) half” and an “amenability half”. The strategy to prove each “half”is then roughly as follows:

Property (T) half:

1. (Kazhdan [Kaz67]) G and Γ have property (T).

2. (Connes-Jones [CJ85], Popa [Pop86]) M has property (T).

Amenability half:

1. (Zimmer [Zim77]) The von Neumann algebra (L∞(G/P ;B(L2M))Γ is amenable.

2. (Margulis [Mar79]) The action ΓyG/P satisfies a “Lebesgue density” type property.

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3. (P [Pet15a], Creutz-P [CP13]) If π does not extend to an isomorphism LΓ∼−→ M , then we

have M = (L∞(G/P ;B(L2M))Γ.

4. (Connes [Con76]) If M is amenable, then it satisfies a “Reiter type” property.

Once the results above are established the proof of Theorem D then follows from the easyobservation that if M is a finite factor with property (T) and satisfying a “Reiter type” propertythen M must be finite dimensional.

1.3 Background

We assume that the reader is familiar with basic concepts in Banach spaces/algebras, von Neumannalgebras, standard Borel/probability spaces, and locally compact groups. Familiarity with [Zhu93]and the first four chapters of [Fol95] will suffice for most of these notes. Familiarity with Chapters1, 2, 3, and 5 of [Pet15b] will also suffice. Another good resource is [Zim84], specifically Chapters2, 4, 7, and 8, which contain a detailed proof of Corollary C above.

2 The Howe-Moore property for SLn(R)

A continuous representation π : G→ U(H) is mixing if G is not compact and in the weak operatortopology we have limg→∞ π(g) = 0, i.e., for each ξ, η ∈ H, and ε > 0, there exists a compactset L ⊂ G so that |〈πxξ, η〉| < ε for all x 6∈ L. A locally compact group G has the Howe-Moore property if every continuous representation π : G → U(H) without non-trivial invariantvectors is mixing. In this section we’ll show that for n ≥ 2, SLn(R) has the Howe-Moore property[HM79]. We’ll present here the proof of Veech [Vee79] which works even for uniformly boundedrepresentations on Banach spaces.1

Lemma 2.1. Suppose G is a Polish group, X is a Banach space, and π : G → GL(X) is auniformly bounded continuous representation. If kn, kn, gn ∈ G such that kn → k, kn → k, andWOT-limn→∞ π(gn) = T , then WOT-limn→∞ π(kngnkn) = π(k)Tπ(k).

Proof. We first note that as G is separable, it is enough to consider the case when X is separable.Next, note that it is easy to see that WOT-limn→∞ π(gnkn) = Tπ(k). Thus, replacing gn withgnkn, and T with Tπ(k) we may assume that kn = k = e.

Since X is separable, B(X) with the weak operator topology is metrizable on bounded subsets.Moreover, the action of G on B(X) given by (x, T ) 7→ π(x)T , is separately continuous. Hence, byFort’s joint continuity theorem2 this action is jointly continuous on bounded subsets, and the resultthen follows.

Lemma 2.2 (Mautner [Mau57]). Suppose G is a Polish group, X is a Banach space, π : G →GL(X) is a uniformly bounded continuous representation, and x, y ∈ X. If b, an ∈ G, such that wehave the weak limit limn→∞ π(an)x = y, and such that a−1

n ban → e, then π(b)x = y.

1Veech’s interest was to show that every weakly almost periodic function on G has a limit at infinity, this followsfrom Theorem 2.3, together with Ryll-Nardzewski’s result that the space of weakly almost periodic functions has a(unique) G-invariant mean.

2See Theorem 8.51 in [Kec95] for a simple proof of Fort’s theorem.

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Proof. Taking limits in the weak topology we have

π(b)x = limn→∞

π(ban)x = limn→∞

π(an(a−1n ban))x = lim

n→∞π(an)x = y.

Fix n ≥ 2, and set G = SLn(R), and K = SO(n) < G. We let A+ denote the set of matricesg ∈ G such that g is a diagonal matrix whose diagonal entries are positive and non-increasing,and we let A be the subgroup generated by A+. Note that if g ∈ G, then if we consider thepolar decomposition of g we may write g = k0h, where h is positive-definite and k0 ∈ K. Sincepositive-definite matrices can be diagonalized there then exists k1 ∈ K so that h = k−1

2 ak2, fora ∈ A+. If we set k1 = k0k

−12 , then we have g = k1ak2, with k1, k2 ∈ K, and a ∈ A+. Hence, we

have established the Cartan decomposition G = KA+K.

Theorem 2.3 (Veech). Fix m ≥ 2 and set G = SLm(R). Suppose X is a Banach space, andπ : G→ GL(X) is a uniformly bounded continuous representation, then any weak operator topologycluster point of π(G) is a projection onto XG.

Proof. We first consider the case G = SL2(R). Suppose π : G → GL(X) is a uniformly boundedcontinuous representation on a Banach space X.

Let T be a WOT-cluster point of some sequence {π(gn)}n, where gn → ∞. Using the Cartandecomposition we may write gn = knank

′n where kn, k

′n ∈ K, and an ∈ A+, and taking a subsequence

we will assume that for k, k ∈ K we have kn → k, kn → k, and WOT-limn→∞ π(gn) = T . ByLemma 2.1 we then have S = WOT-limn→∞ π(an) = π(k)Tπ(k).

Write an =(rn 0

0 r−1n

), where rn → ∞, and consider the subgroup N ⊂ G consisting of upper

triangular matrices with entries 1 on the diagonal. Note that the conjugation action of A = 〈A+〉on N is given by (

r 00 r−1

)( 1 s

0 1 )(r−1 0

0 r

)=(

1 r2s0 1

),

thus, for x ∈ N we have a−1n xan → e ∈ G. Hence, by Mautner’s lemma it follows that π(x)S = S,

for all x ∈ N .Since N is non-compact we may again use the Cartan decomposition to conclude that there

is a sequence bn ∈ A+, hn, hn ∈ K, and h, h ∈ K, such that bn → ∞, hn → h, hn → h, andπ(hnbnhn)S = S for all n ∈ N. Therefore we have SOT-limn→∞ π(bn)π(h)S = π(h−1)S.

Now fix x ∈ X in the range of π(h)S = π(hk)Tπ(k). Since bn → ∞, and π(bn)x convergesit follows that for each ε > 0, {b ∈ A+ | ‖π(b)x − x‖ < ε} is non-compact. Thus, there exists asequence cn ∈ A such that cn →∞, and limn→∞ π(cn)x = x. Since the representation is uniformlybounded we then have also that limn→∞ π(c−1

n )x = x, and hence taking a subsequence and replacingcn with c−1

n if necessary we may assume that cn ∈ A+. Since limn→∞ π(cn)x = x we then havefrom Mautner’s lemma that π(g)x = x for all g ∈ N . Since we also have limn→∞ π(c−1

n )x = x,Mautner’s lemma also shows that π(g)x = x for all g ∈ NT .

Since 〈N,NT 〉 = G we then conclude that x is G-invariant. Since x was an arbitrary vector inthe range of π(kh)Tπ(k), and since π(kh), π(k) ∈ Isom(X) it then follows that every vector in therange of T is G-invariant. Thus, T is a projection onto the space of G-invariant vectors, finishingthe proof for G = SL2(R).

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We now suppose that G = SLm(R), with m > 2. We first note that the same argument asabove shows that if {gn}n is any sequence in G converging to infinity, and if T ∈ B(X) is a weakoperator topology cluster point of {π(gn)}n, then we may find S ∈ B(X), k, k, h ∈ K, such thatfor any x in the range of π(hk)Tπ(k), there is a sequence {cn}n ⊂ A, with cn →∞, such that S =WOT-limn→∞ π(an) = π(k)Tπ(k), and limn→∞ π(cn)x = x.

For i 6= j, let Ni,j ⊂ SLm(R) denote the subgroup consisting of matrices with diagonal entriesequal to 1, and all other entries zero except possibly the (i, j)-th entry. For I ⊂ {1, 2, . . . ,m} welet AI denote the subgroup of A consisting of all diagonal matrices whose diagonal entries are 1except possibly those diagonal entries corresponding to indices in I.

As cn →∞, taking a subsequence if necessary, there must be 1 ≤ i, j ≤ m, i 6= j so that the ithdiagonal entry of cn converges to infinity, while the jth diagonal entry of cn converges to 0. It thenfollows from Mautner’s lemma that x is fixed by the subgroup Ni,j , and applying Mautner’s lemmato the inverse sequence shows that x is also fixed by Nj,i, and hence is fixed by A{i,j} ⊂ 〈Ni,j , Nj,i〉.

We let Y denote the set of A{i,j}-invariant vectors, then to finish the proof it is enough toshow that Y is G-invariant. Indeed, if this is the case then A{i,j} is contained in the kernel of the

representation restricted to Y , and since G is simple we then have Y = XG.To see that Y is G-invariant note that Ni′,j′ commutes with A{i,j} whenever {i, j}∩{i′, j′} = ∅,

in which case Ni′,j′ leaves A invariant. On the other hand, if {i′, j′} ∩ {i, j} 6= ∅ then Ai,j actson Ni′,j′ by conjugation, and the closure of any orbit contains the identity matrix. Thus, againby Mautner’s lemma we have that any vector which is fixed by Ai,j is also fixed by Ni′,j′ and, inparticular, we have that Ni′,j′ leaves Y invariant in this case as well.

Since G is generated by Ni′,j′ , for 1 ≤ i′, j′ ≤ m, i′ 6= j′, this then shows that Y is indeedG-invariant.

Note that it may be the case that π(G) has no weak operator topology cluster point, e.g.,consider the translation action of G on L1G.

If X is reflexive, e.g., if X is a Hilbert space, then it follows easily from the Banach-Alaoglu,that Isom(X) ⊂ B(X) is compact in the weak operator topology, thus a consequence of the previoustheorem is the Howe-Moore property.

Corollary 2.4 (Howe-Moore). SLn(R) has the Howe-Moore property for n ≥ 2.

Corollary 2.5 (Moore’s ergodicity theorem). Set G = SLn(R). Let Γ < G be a lattice and supposeH < G is a non-compact closed subgroup. Then the action HyG/Γ is ergodic, as is the actionΓyG/H.

Proof. Let µ be the G-invariant probability measure on G/Γ. Suppose a measurable subset E ⊂G/Γ is H-invariant and consider the Koopman representation π : G → U(L2(G/Γ, µ)) given byπ(g)(f)(x) = f(g−1x). Then we have that 1E is H-invariant and since H is non-compact it thenfollows from the Howe-Moore property that 1E is contained in the space of G-invariant functions. Asthe G action is transitive it is clearly ergodic and hence 1E is essentially constant, i.e., µ(E) ∈ {0, 1}.

We have canonical identifications

L∞(G/Γ)H ∼= L∞(G)Γ×H ∼= L∞(G/H)Γ,

and hence ergodicity of the action ΓyG/H follows as well.

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3 Property (T)

Let G be a locally compact group and π : G→ U(H) a continuous unitary representation. A vectorξ ∈ H is an invariant vector if πxξ = ξ for all x ∈ G. The representation has almost invariantvectors if there is a sequence of unit vectors ξn ∈ H such that ‖π(x)ξn − ξn‖ → 0 uniformly oncompact subsets of G.

Proposition 3.1. Let G be a group, and π : G → U(H) a unitary representation. If there existsξ ∈ H and c > 0 such that Re(〈πxξ, ξ〉) ≥ c‖ξ‖2 for all x ∈ G, then π contains an invariant vectorξ0 such that Re(〈ξ0, ξ〉) ≥ c‖ξ‖2.

Proof. Let K be the closed convex hull of the orbit π(G)ξ. Then K is G-invariant and Re(〈η, ξ〉) ≥c‖ξ‖2 for every η ∈ K. Let ξ0 ∈ K be the unique element of minimal norm, then since G actsisometrically we have that for each x ∈ G, πxξ0 is the unique element of minimal norm for πxK = K,and hence πxξ0 = ξ0 for each x ∈ G. Since ξ0 ∈ K we have that Re(〈ξ0, ξ〉) ≥ c‖ξ‖2.

Corollary 3.2. Let G be a group, and π : G → U(H) a unitary representation. If there existsξ ∈ H and c <

√2 such that ‖πxξ − ξ‖ ≤ c‖ξ‖ for all x ∈ G, then π contains a non-zero invariant

vector.

Proof. For each x ∈ G we have

2Re(〈πxξ, ξ〉) = 2‖ξ‖2 − ‖πxξ − ξ‖2 ≥ (2− c2)‖ξ‖2.

Hence, we may apply Proposition 3.1.

Let G be a locally compact group, and H < G a closed subgroup. The pair (G,H) has relativeproperty (T) if every representation of G which has almost invariant vectors, has a non-zeroH-invariant vector. G has property (T) if the pair (G,G) has relative property (T).

Proposition 3.3. Let G be a locally compact group and H CG a closed normal subgroup, then thefollowing are equivalent:

1. The pair (G,H) has relative property (T).

2. For any sequence of positive definite functions ϕn : G → C such that ϕn → 1 uniformly oncompact subsets we have that supx∈H |1− ϕn(x)| → 0.

Proof. First suppose that the pair (G,H) has relative property (T), and ϕn : G→ C is a sequenceof positive definite functions such that ϕn → 1 uniformly on compact subsets. Normalizing ϕnwe assume that ϕn(e) = 1. By the GNS-construction there exists a sequence of representationsπn : G → U(Hn) and unit vectors ξn ∈ Hn so that ϕn(x) = 〈πn(x)ξn, ξn〉. We let H0

n denote thespace of H-invariant vectors in Hn and note that since H is normal we have that G preserves H0

n

and hence also preserves its orthogonal compliment H0n⊥

. We let ηn denote the projection of ξnto H0

n⊥

and ζn denote the projection of ξn to H0n, so that ζn is H-invariant. Then ⊕nH0

n⊥

is arepresentation without H-invariant vectors, and ηn are almost invariant. Since (G,H) has relative

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property (T) we then conclude that ‖ηn‖ → 0, and hence ξn − ζn → 0. Thus by Cauchy-Schwarzwe have

supx∈H|1− ϕn(x)| = sup

x∈H|〈ξn − πn(x)ξn, ξn〉| ≤ 2‖ξn − ζn‖ → 0.

Conversely, if (G,H) does not have relative property (T) then there exists a representation π :G→ U(H) without H-invariant vectors, but having almost invariant unit vectors ξn ∈ H. If we setϕn(x) = 〈π(x)ξn, ξn〉, then ϕn is a sequence of positive definite functions which converges uniformlyon compact subsets to 1. However ϕn cannot converge uniformly on H by Proposition 3.1.

We remark that the previous lemma also holds in the case when H is not normal, although theproof is quite a bit more complicated [Jol05].

Suppose that G and A are locally compact groups such that A is abelian. Suppose α : G →Aut(A) is a continuous homomorphism, and let α : G → Aut(A) denote the dual homomorphismgiven by αx(χ) = χ ◦ αx−1 , for x ∈ G, and χ ∈ A. Note that for f ∈ L1H, and x ∈ G we have

f ◦ αx(χ) =∫f ◦ αx(y)χ(y) dy =

∫f(y)αx(χ)(y)

dαx−1y

dy dy =(f

dαx−1y

dy

)◦ αx(χ).

Lemma 3.4. Let G and A be as above. Suppose {ϕi}i∈I ⊂ P1(G n A) is a net of positive defi-nite functions, and let νi ∈ Prob(A) denote the net of probability measures which corresponds byBochner’s theorem to the functions ϕi restricted to A. If ϕi → 1 uniformly on compact subsets ofG then ‖αxνi − νi‖ → 0 uniformly on compact subsets of G.

Proof. We let πi : GnA→ U(Hi) be the cyclic representations associated to ϕi with cyclic vectorξi ∈ Hi. For f ∈ L1H we have∫

f dαxνi =

∫f ◦ αx dνi

=

∫f ◦ αx(y)〈πi(y)ξi, ξi〉 dαx(y)

=

∫f(y)〈πi(y)π(x−1)ξi, π(x−1)ξi〉 dy

= 〈πi(f)πi(x−1)ξi, π(x−1)ξi〉.

Thus, we have |∫f dαxνi −

∫f dνi| ≤ 2‖πi(f)‖‖ξi − πi(x)ξi‖ = 2‖f‖∞‖ξi − π(x)ξi‖.

Since F(L1H) is dense in C0A it then follows that ‖αxνi− νi‖ ≤ 2‖ξi− πi(x)ξi‖ → 0 uniformlyon compact subsets of G.

Lemma 3.5. Let G and A be second countable locally compact groups such that A is abelian.Suppose α : G → Aut(A) is a continuous homomorphism. Then the following conditions areequivalent:

(i) (GnA,A) does not have relative property (T).

(ii) There exists a net {νi}i∈I ⊂ Prob(A), such that νi({e}) = 0, νi → δ{e} weak∗, and ‖αyνi −νi‖ → 0 uniformly on compact subsets of G.

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Proof. For (i) =⇒ (ii), suppose that (GnA,A) does not have relative property (T). Thus, thereexists a continuous representation π : G n A → U(H) without A-invariant vectors, and a net ofunit vectors {ξi}i ⊂ H such that ‖π(x)ξi − ξi‖ → 0 uniformly on compact subsets of GnH.

We let ϕi : G n A → C denote the function of positive definite given by ϕi(x) = 〈π(x)ξi, ξi〉,and we let νi ∈ Prob(A) denote the probability measure corresponding to ϕi restricted to A, givenby Bochner’s theorem. Since π does not have A-invariant vectors we have that νi({e}) = 0. Asϕi → 1 uniformly on compact sets of A we have νi → δ{e} weak∗, and since ϕi → 1 uniformly oncompact subsets of G by the previous lemma we have that ‖αxνi − νi‖ → 0 uniformly on compactsubsets of G.

Conversely, for (ii) =⇒ (i), let {νi}i∈I ⊂ Prob(A) be the net given by (ii). Fix µ0 ∈ Prob(G)in the same measure class as Haar measure, and set νi = α(µ0)∗νi =

∫αxνi dµ0(x). Then we again

have that νi → δ{e} weak∗, and ‖αxνi − νi‖ → 0 uniformly on compact subsets, and moreover we

have that νi is quasi-invariant for the G action on A.We define πi : GnH → U(L2(A, νi)) by πi(xh) = Uxh where U is the Koopman representation

corresponding to the action of G on (A, νi). Then πi gives a unitary representation, and as νi → δ{e}weak∗ we see that {ξi}i∈I forms a net of almost invariant vectors for H. Moreover, for x ∈ G wehave

|〈πi(x)ξi − ξi, ξi〉| ≤∫ ∣∣∣∣∣(

dαxνidνi

)1/2

− 1

∣∣∣∣∣ dνi

≤∥∥∥∥dαxνi

dνi− 1

∥∥∥∥1/2

1

≤ ‖αxνi − νi‖1/2.

Hence, {ξi}i∈I also forms a net of almost invariant vectors for G. We let Ki ⊂ L2(A, νi) denote thespace of A-invariant vectors. Then as ACGnA is normal it follows that Ki is ACG-invariant andhence so is K⊥i . If (GnA,A) had relative property (T) then since {P⊥Kiξi}i∈I is almost invariant for

Gn A in ⊕i∈Iπi it then follows that we must have P⊥Kiξi → 0. Hence, it follows that νi({e})→ 1.However, νi({e}) = νi({e})− νi({e})→ 0 and we would then have a contradiction.

Corollary 3.6. Let G and A be second countable locally compact groups such that A is abelian.Suppose α : G → Aut(A) is a continuous homomorphism. If (G n A,A) does not have relativeproperty (T) then there exists a state ϕ ∈ B∞(A)∗ such that ϕ(1{e}) = 0, ϕ(1O) = 1 for every

neighborhood O of e, and ϕ(f ◦ αx) = ϕ(f) for all f ∈ B∞(A), and x ∈ G.

Proof. Suppose (GnA,A) does not have relative property (T), and let {νi}i∈I ⊂ Prob(A) be as inthe previous lemma. Then each νi gives a state on B∞(A). If we let ϕ be a weak∗-cluster point of{νi}i∈I then we have ϕ(1{e}) = 0 since νi({e}) = 0 for all i ∈ I. We also have ϕ(1O) = 1 for every

neighborhood O of e since νi → δ{e} weak∗. And we have ϕ(f ◦ αx) = ϕ(f) for all f ∈ B∞(A), andx ∈ G, since ‖αxνi − νi‖ → 0 for all x ∈ G.

We consider the natural action of SL2(R) on R2 given by matrix multiplication, and let SL2(R)nR2 be the semi-direct product.

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Theorem 3.7. The pair (SL2(R)nR2,R2) has relative property (T).

Proof. Under the identification R2 = R2 given by the pairing 〈a, ξ〉 = eπia·ξ we have that the dualaction of SL2(R) is given by matrix multiplication with the inverse transpose.

Suppose that ϕ ∈ B∞(R2)∗ is a SL2(R)-invariant state such that ϕ(1O) = 1 for any neighbor-hood O of (0, 0).

We setA = {(x, y) ∈ R2 | x > 0,−x < y ≤ x};B = {(x, y) ∈ R2 | y > 0,−y ≤ x < y};C = {(x, y) ∈ R2 | x < 0, x ≤ y < −x};D = {(x, y) ∈ R2 | y < 0, y < x ≤ −y}.

A simple calculation shows that for k ≥ 0 the sets Ak =(

1 02k 1

)A are pairwise disjoint. Thus, we

must have that ϕ(1A) = 0. A similar argument also shows that ϕ(1B) = ϕ(1C) = ϕ(1D) = 0.Hence we conclude that ϕ(1{(0,0)}) = 1 − ϕ(1A∪B∪C∪D) = 1. By the previous corollary it thenfollows that (SL2(R)nR2,R2) has relative property (T).

Theorem 3.8 ([Kaz67]). SLm(R) has property (T) for m ≥ 3.

Proof. We consider the group SL2(R) < SLm(R) embedded as matrices in the upper left corner.We also consider the group R2 < SLm(R) embedded as those matrices with 1’s on the diagonal, andall other entries zero except possibly the (1, n)th, and (2, n)th entries. Note that the embedding ofSL2(R) normalizes the embedding of R2, and these groups generate a copy of SL2(R)nR2.

If π : SLm(R) is a representation which has almost invariant vectors, then by Theorem 3.7 wehave that the copy of R2 has a non-zero invariant vector. By the Howe-Moore property it thenfollows that π has an SLm(R)-invariant vector.

Suppose that G is a locally compact group and Γ < G is a lattice. If π : Γ → U(H) isa unitary representation, we denote by L2(G,H)Γ the set of measurable function f : G → Hwhich satisfy π(γ−1)f(gγ) = f(g) for all g ∈ G, and γ ∈ Γ, and

∫G/Γ ‖f(g)‖2dg < ∞, where we

identify two functions if they agree almost everywhere. We define an inner-product on L2(G,H)Γ

by 〈f1, f2〉 =∫G/Γ f1(g)f2(g) dg. With this inner-product it is not hard to see that L2(G,H)Γ forms

a Hilbert space.The induced representation π : G → U(L2(G,H)Γ) is given by (π(x)f)(y) = f(x−1y). It is

easy to see that π gives a continuous unitary representation of G.

Theorem 3.9. [Kaz67] Let G be a second countable locally compact group, and Γ < G a lattice, ifG has property (T) then Γ has property (T).

Proof. We fix a Borel fundamental domain Σ for Γ so that the map Σ × Γ 3 (σ, γ) → σγ ∈ Gis a Borel isomorphism, and we choose a Haar measure on G so that Σ has measure 1. We letα : G× Σ→ Γ be defined so that α(g, σ) is the unique element in Γ which satisfies gσα(g, σ) ∈ Σ.

If π : Γ → U(H) is a representation with almost invariant unit vectors {ξn}n∈N. We considerthe vectors ξn ∈ L2(G,H)Γ given by ξn(σγ) = π(γ)ξn, for σ ∈ Σ, γ ∈ Γ. Then for g ∈ G we have

‖π(g)ξn − ξn‖2 =

∫G/Γ‖ξn(g−1x)− ξn(x)‖2 dx =

∫Σ‖ξn − π(α(g−1, σ))ξn‖2 dσ → 0.

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Thus, {ξn}n∈N forms a sequence of almost invariant vectors for π. Since G has property (T), itfollows that there exists a non-zero invariant vector ξ0 ∈ L2(G,H)Γ, i.e., ξ0(gx) = ξ0(x) for almostall g, x ∈ G. It then follows that ξ0 is essentially constant. We let ξ0 6= 0 denote the essential rangeof ξ0. Since ξ0 ∈ L2(G,H)Γ we have that π(γ)ξ0 = ξ0 for all γ ∈ Γ. Thus, ξ0 ∈ H is a non-zeroinvariant vector and hence Γ has property (T).

We remark that the converse of the previous theorem is also true.

4 Amenability

A (left) invariant mean m on a locally compact group G is a finitely additive Borel probabilitymeasure on G, which is absolutely continuous with respect to Haar measure, and which is invariantunder the action of left multiplication, i.e., m : Borel(G) → [0, 1] such that m(G) = 1, m(E) = 0if λ(E) = 0, where λ is the Haar measure on G, and if A1, . . . , An ∈ Borel(G) are disjoint thenm(∪nj=1An) =

∑nj=1m(An), and if A ∈ Borel(G), then m(xA) = m(A) for all x ∈ G. If G possesses

an invariant mean then G is amenable. We can similarly define right invariant means, and infact if m is a left invariant mean then m∗(A) = m(A−1) defines a right invariant mean. Amenablegroups were first introduced by von Neumann in his investigations of the Banach-Tarski paradox.

Given a right invariant mean m on G it is possible to define an integral over G just as in thecase if m were a countably additive measure. We therefore obtain a state φm ∈ (L∞G)∗ by theformula φm(f) =

∫f dm, and this state is left invariant, i.e., φm(Lx(f)) = φm(f) for all x ∈ G,

f ∈ L∞G. Conversely, if φ ∈ (L∞G)∗ is a left invariant state, then restricting φ to characteristicfunctions defines a right invariant mean.

Example 4.1. Let F2 be the free group on two generators a, and b. Let A+ be the set of allelements in F2 whose leftmost entry in reduced form is a, let A− be the set of all elements in F2

whose leftmost entry in reduced form is a−1, let B+, and B− be defined analogously, and considerC = {e, b, b2, . . .}. Then we have that

F2 = A+ tA− t (B+ \ C) t (B− ∪ C)

= A+ t aA−

= b−1(B+ \ C) t (B− ∪ C).

If m were a left-invariant mean on F2 then we would have

m(F2) = m(A+) +m(A−) +m(B+ \ C) +m(B− ∪ C)

= m(A+) +m(aA−) +m(b−1(B+ \ C)) +m(B− ∪ C)

= m(A+ t aA−) +m(b−1(B+ \ C) t (B− ∪ C)) = 2m(F2).

Hence, F2 is non-amenable.

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An approximately invariant mean on G is a net fi ∈ L1(G)+ such that∫fi = 1, and

‖Lx(fi)− fi‖1 → 0, uniformly on compact subsets of G.A Følner net is a net of non-null finite measure Borel subsets Fi ⊂ G such that λ(Fi∆xFi)/λ(Fi)→

0, uniformly on compact subsets of G. Note that we do not require that Γ = ∪iFi, nor do we requirethat Fi are increasing, however, if G is not compact then it is easy to see that any Følner net {Fi}imust satisfy λ(Fi)→∞.

Theorem 4.2. Let G be a locally compact group, then the following conditions are equivalent.

(i) G is amenable.

(ii) CbG admits a left invariant state.

(iii) C lub (G) admits a left invariant state.

(iv) L∞G has an L1G-invariant state.

(v) G has an approximate invariant mean.

(vi) The left regular representation λ : G→ U(L2G) has almost invariant vectors.

(vii) The representation λ : G → U(L2G) has almost invariant vectors when G is viewed as adiscrete group.

(viii) The (discontinuous) action of G on its Stone-Cech compactification βG which is induced byleft-multiplication admits an invariant Radon probability measure.

(ix) Any continuous action GyK on a compact metric space K admits an invariant Radon prob-ability measure.

Proof. First note that (i) =⇒ (ii), and (ii) =⇒ (iii) are obvious.To see (iii) =⇒ (iv) suppose m is a left invariant state on C lu

b (G). Note that since Gacts continuously on C lu

b (G) it follows that for all f ∈ L1G, and g ∈ C lub (G) we have that the

integral f ∗ g =∫f(y)δy ∗ g dλ(y) converges in norm. Hence we have m(f ∗ g) = m(g), for all

f ∈ L1(G)1,+ = {f ∈ L1G | f ≥ 0;∫f = 1} and g ∈ C lu

b (G).Fix f0 ∈ L1(G)1,+ and define a state on L∞G by m(g) = m(f0 ∗ g) (recall that L1G ∗ L∞G ⊂

C lub (G)). If {fn}n∈N ⊂ L1(G)1,+ is an approximate identity then m(f0∗g) = limn→∞m(f0∗fn∗g) =

limn→∞m(fn ∗ g), and thus m is independent of f0. Thus, for f ∈ L1(G)1,+ and g ∈ L∞G we havem(f ∗ g) = m(f0 ∗ f ∗ g) = m(g).

We show (iv) =⇒ (v) using the method of Day: Since L∞G = (L1G)∗, the unit ball in L1G isweak∗-dense in the unit ball of (L∞G)∗ = (L1G)∗∗. It follows that L1(G)1,+ is weak∗-dense in thestate space of L∞G.

Let S ⊂ L1G+,1, be finite and let K ⊂∏f∈S L

1G be the weak-closure of the set {S 3 g 7→(g ∗ f − f) | f ∈ L1(G)1,+}. Since G has an L1G-invariant state on L∞G, and since L1(G)1,+ isweak∗-dense in the state space of L∞G, we have that 0 ∈ K. However, K is convex and so by theHahn-Banach separation theorem the weak-closure coincides with the norm closure. Thus, thereexists a net {fi} ⊂ L1(G)1,+ such that for all g ∈ L1G+,1 we have

‖g ∗ fi − fi‖1 → 0.

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If S ⊂ L1G+,1 is compact then the above convergence may be taken uniformly for g ∈ S.Indeed, if ε > 0 then let S0 ⊂ S be finite such that infg0∈S0 ‖g− g0‖1 < ε, for all g ∈ S. Then thereexists fi such that ‖g0 ∗ fi − fi‖1 < ε for all g0 ∈ S0, and we then have ‖g ∗ fi − fi‖1 < 2ε for allg ∈ S.

If K ⊂ G is compact and g0 ∈ L1G+,1, then since the action of G on L1G is continuous itfollows that {Lk(g0) | k ∈ K} ⊂ L1G+,1 is compact, and hence we have

limi→∞

supk∈K‖Lk(g0 ∗ fi)− (g0 ∗ fi)‖1 ≤ lim

i→∞supk∈K‖Lk(g0) ∗ fi − fi‖1 + ‖fi − g0 ∗ fi‖1 = 0.

For (v) =⇒ (vi) just notice that if {fi}i ⊂ L1(G)! is an approximately invariant mean, then{√fi}i ⊂ L2G is a net of almost invariant vectors.(vi) =⇒ (vii) is obvious. For (vii) =⇒ (i) let ξi ∈ L2G be a net of almost invariant vectors for

G as a discrete group. We define states ϕi on B(L2G) by ϕi(T ) = 〈Tξi, ξi〉. By weak∗ compactnessof the state space, we may take a subnet and assume that this converges in the weak∗ topology toϕ ∈ B(L2G)∗. We then have that for all T ∈ B(L2G) and x ∈ G,

|ϕ(λ(x)T − Tλ(x))| = limi|〈(λ(x)T − Tλ(x))ξi, ξi〉|

= limi|〈Tξi, λ(x−1)ξi〉 − 〈Tλ(x)ξi, ξi〉|

≤ limi‖T‖(‖λ(x−1)ξi − ξi‖+ ‖λ(x)ξi − ξi‖) = 0.

We consider the usual embedding M : L∞G→ B(L2G) given by point-wise multiplication. Forf ∈ L∞G and x ∈ G we have λ(x)Mfλ(x−1) = MLx(f). Thus, restricting ϕ to L∞G gives a stateon L∞G which is G-invariant.

(ii) ⇐⇒ (viii), follows from the G-equivariant identification CbG ∼= C(βG), together with theRiesz representation theorem.

For (viii) =⇒ (ix), suppose G acts continuously on a compact Hausdorff space K, and fix apoint x0 ∈ K. Then the map f(g) = gx0 on G extends uniquely to a continuous map βf : βG→ K,moreover since f is G-equivariant, so is βf . If µ is an invariant Radon probability measure for theaction on βG then we obtain the invariant Radon probability measure f∗µ on K.

For (ix) =⇒ (iii), fix F ⊂ C lub G any finite subset. Then since G acts continuously on C lu

b Git follows that the G-invariant unital C∗-subalgebra A ⊂ C lu

b G generated by F is separable, andhence σ(A) is a compact metrizable space and the natural action of G on σ(A) is continuous. Byhypothesis there then exists a G-invariant probabiliy measure on σ(A) which corresponds to a G-invariant state ϕF on A. By the Hahn-Banach theorem we may extend ϕF to a (possibly no longerG-invariant) state ϕF on C lu

b G. Considering the finite subsets of C lub G as a partially ordered set

by inclusion we then have a net of states {ϕF } on C lub G, and we may let ϕ∞ be a cluster point of

this set. It is then easy to see that ϕ∞ is G-invariant.

The previous theorem is the combined work of many mathematicians, including von Neumann,Følner, Day, Namioka, Hulanicki, Reiter, and Kesten.

Example 4.3. Any compact group is amenable, and any group which is locally amenable (eachcompactly generated subgroup is amenable) is also amenable. The group Zn is amenable (consider

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the Følner sequence Fk = {1, . . . , k}n for example). From this it then follows easily that all discreteabelian groups are amenable. Moreover, from part (ix) we see that if a locally compact group isamenable as a discrete group then it is also amenable as a locally compact group, thus all abelianlocally compact groups are amenable.

Closed subgroups of amenable groups are also amenable (hence any locally compact groupcontaining F2 as a closed subgroup is non-amenable). This follows from the fact that if H < G is aclosed subgroup, then there exists a Borel set Σ ⊂ G such that the map H ×Σ 3 (h, σ) 7→ hσ ∈ Ggives a Borel bijection. This then gives an H-equivariant homomorphism θ from L∞H → L∞G,given by θ(f)(hσ) = f(h). Restricting a G-invariant state to the image of L∞H then gives anH-invariant state on L∞H.

If G is amenable and H CG is a closed subgroup then G/H is again amenable. Indeed, we mayview L∞(G/H) as the space of right H-invariant functions in L∞G, and hence we may restrict aG-invariant mean to L∞(G/H).

From part (ix) in Theorem 4.2 it follows that if H C G is closed, such that H and G/H areamenable then G is also amenable. Indeed, if GyK is a continuous action on a compact Hausdorffspace, then if we consider Prob(K)H the set of H-invariant probability measures, then Prob(K)H

is a non-empty compact set on which G/H acts continuously. Thus there is a G/H-invariantprobability measure µ ∈ Prob(Prob(K)H) and if we consider the barycenter µ =

∫ν dµ(ν), then µ

is a G invariant probability measure on K.It then follows that all solvable groups amenable, e.g., the group of upper triangular real in-

vertible matrices.

5 Finite von Neumann algebras

Fix a separable Hilbert space H. A von Neumann algebra M ⊂ B(H) is finite if there exists alinear functional τ : M → C (called the trace) which satisfies the following properties:

1. τ is a state, i.e., τ(1) = 1 and τ(x∗x) ≥ 0 for all x ∈M ;

2. τ is faithful, i.e., τ(x∗x) = 0 if and only if x = 0;

3. τ is normal, i.e., τ is continuous with respect to the strong operator topology (or equivalently,τ is continuous with respect to the weak operator topology).

4. τ is tracial, i.e., τ(xy) = τ(yx) for all x, y ∈M .

By a tracial von Neumann algebra we mean a pair (M, τ), where M is a finite von Neumann algebraand τ is a trace on M . We remark that by a result of Murray and von Neumann, if M is a finitefactor then the trace is unique.

Given a finite von Neumann algebra M with a normal faithful tracial state τ , we obtain aninner-product on M by

〈x, y〉 = τ(y∗x).

We denote by ‖ · ‖2 the resulting norm from this inner-product, and we denote by L2(M, τ) theresulting Hilbert space completion. We then have a normal representation of M acting on L2(M, τ)given by left multiplication. (This is the GNS-construction). Thus, we may view M both as an

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algebra of bounded operators on B(L2(M, τ)) and as a dense subspace of the Hilbert space L2(M, τ).When we wish to emphasize the latter perspective we will write x for an element x ∈ M when itis viewed as an element in L2(M, τ). We will also write Px to denote the rank 1 projection ontoCx ⊂ L2(M, τ). When M is a finite factor, the trace is unique, and so, in this case, we use thenotation L2M , for L2(M, τ).

The tracial property of τ entails that the conjugation map M 3 x 7→ x∗ extends to a conjugatelinear isometry J : L2(M, τ) → L2(M, τ). We than have JyJx = xy∗, and from this we see thatJMJ ⊂M ′, and hence M ⊂ JMJ ′.

Lemma 5.1. Suppose (M, τ) is a tracial von Neumann algebra. Then on bounded sets the strongoperator topology is equivalent to the topology induced from 〈·, ·〉, and the weak operator topology isequivalent to the weak topology induced from 〈·, ·〉.

Proof. If {xi}i is a uniformly bounded net such that xi → x in the strong operator topology, thenwe have ‖xi − x‖2 = ‖(xi − x)1‖2 → 0. Conversely, if we have ‖xi − x‖2 → 0, then for any y ∈ Mwe have ‖(xi − x)y‖ = ‖Jy∗J(xi − x)1‖ ≤ ‖y‖‖xi − x‖2 → 0. Since M is dense in L2(M, τ) andsince {xi}i is uniformly bounded it then follows that xi → x in the strong operator topology.

Similarly, if {xi}i is a uniformly bounded net such that xi → x weakly, then for all ξ ∈ L2(M, τ),we have 〈(xi − x)1, ξ〉 → 0. Conversely, if 〈(xi − x)1, ξ〉 → 0 for every ξ ∈ L2(M, τ), then for eachy ∈ M and ξ ∈ L2(M, τ) we have 〈(xi − x)y, ξ〉 = 〈(xi − x)1, (JyJ)ξ〉 → 0. Just as above it thenfollows that xi → x in the weak operator topology.

Corollary 5.2. The unit ball (M)1 is a closed subset of L2(M, τ).

Proof. Any limit point in (M)1 ⊂ L2(M, τ) must also be a weak limit point. Since (M)1 is compactin the weak operator topology it then follows from the previous lemma that any weak limit pointin (M)1 ⊂ L2(M, τ) must be contained in (M)1.

Proposition 5.3. If (M, τ) is a tracial von Neumann algebra then we have the equality JMJ = M ′

and M = JM ′J .

Proof. If x, y ∈ M then 〈Jx, Jy〉 = τ(yx∗) = 〈y, x〉. Thus, for all ξ, η ∈ L2(M, τ) we have〈Jξ, Jη〉 = 〈η, ξ〉.

If a ∈M ′ and z ∈M we have

〈JaJ 1, z〉 = 〈Jz, aJ 1〉 = 〈a∗z∗1, 1〉 = 〈a∗1, z〉

Since M is dense in L2(M, τ) we then have JaJ 1 = a∗1.If we also have b ∈M ′ then

〈bJaJ 1, z〉 = 〈a∗z∗, b∗1〉 = 〈z∗, aJbJ 1〉 = 〈JaJb1, z〉,

Since z ∈ M is arbitrary we then have bJaJ 1 = JaJb1. If a, b, c ∈ M ′ then applying this lastequation with either ac or c in place of a gives

(bJaJ)Jc1 = JacJb1 = (JaJb)Jc1.

Since JM ′1 is dense in L2(M, τ) it then follows that b(JaJ) = (JaJ)b. Thus, JaJ ∈M ′′ = M andso JM ′J ⊂M ⊂ JM ′J .

Therefore JM ′J = M and taking commutants give JMJ = M ′.

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If Γ is a countable group, the group von Neumann algebra is the von Neumann algebra LΓ ⊂B(`2Γ) generated by the left-regular representation of Γ. This is a finite von Neumann algebra asLΓ 3 x 7→ 〈xδe, δe〉 is easily seen to give a normal faithful tracial state. Since this gives a trace onLΓ to each operator x ∈ LΓ we may associate the vector xδe ∈ `2Γ. If xδe =

∑γ∈Γ αγδγ then we

call the coefficients αγ the Fourier coefficients of x and we sometimes write x =∑

γ∈Γ αγuγ fornotational convenience.

We also denote by RΓ the von Neumann algebra generated by the right-regular representation.Note that LΓ is standardly represented on B(`2Γ) and we have RΓ = JLΓJ . In particular, itfollows from Proposition 5.3 that we have RΓ = LΓ′.

Exercise 5.4. Let Γ be a countable group. Given vectors ξ =∑

γ∈Γ αγδγ , η =∑

γ∈Γ βγδγ ∈ `2Γthe convolution of ξ with η is given by

ξ ∗ η =∑γ∈Γ

(∑x∈Γ

αxβx−1γ

)δγ ∈ `1Γ.

A vector ξ ∈ `2Γ is a left-convolver if ξ ∗ η ∈ `2Γ for all η ∈ `2Γ. By the closed graph theorem toeach left-convolver ξ we obtain a bounded operator Lξ given by Lξη = ξ ∗ η.

Prove that LΓ ={Lξ | ξ ∈ `2Γ is a left− convolver.

}.

Example 5.5. If Γ is abelian then we may consider the dual group Γ = Hom(Γ,T) which is acompact group when endowed with the topology of pointwise convergence. We consider this groupendowed with a Haar measure µ normalized so that µ(Γ) = 1. The Fourier transform F : `2Γ→ L2Γis defined as (Fξ)(χ) =

∑g∈Γ ξ(g)〈χ, g〉. The Fourier transform implements a unitary between `2Γ

and `2Γ.If ξ ∈ `2Γ is a (left) convolver, then we have Lξ = F−1MF(ξ)F , and hence we obtain a canonical

isomorphism LΓ ∼= L∞Γ. Moreover, we have τ(Lξ) =∫F(ξ) dµ, for each Lξ ∈ LΓ.

A discrete group Γ is i.c.c. if every non-trival conjugacy class is infinite.

Theorem 5.6 (Murray-von Neumann [MvN43]). Let Γ be a discrete group. Then LΓ is a factorif and only if Γ is i.c.c.

Proof. First suppose that h ∈ Γ \ {e}, such that hΓ = {ghg−1 | g ∈ G} is finite. Let x =∑

k∈hΓ uk.Then x 6= C, and for all g ∈ G we have ugxu

∗g =

∑k∈hΓ ugkg−1 = x, hence x ∈ {ug}′g∈Γ ∩ LΓ =

Z(LΓ).Conversely, suppose that Γ is i.c.c. and x =

∑g∈Γ αgug ∈ Z(LΓ) \C, then for all h ∈ Γ we have

x = uhxu∗h =

∑g∈Γ αguhgh−1 =

∑g∈Γ αh−1ghug. Thus the Fourier coefficients for x are constant on

conjugacy classes, and since x ∈ LΓ ⊂ `2Γ we have αg = 0 for all g 6= e, hence x = τ(x) ∈ C.

5.1 Group characters and tracial von Neumann algebras

Let Γ be a discrete group. Recall that a function ϕ : Γ → C is a character if it is of positivedefinite, is constant on conjugacy classes, and is normalized so that ϕ(e) = 1. Characters arisefrom representations into finite von Neumann algebras. Indeed, if M is a finite von Neumann

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algebra with a normal faithful trace τ , and if π : Γ → U(M) ⊂ U(L2(M, τ)) is a representationthen ϕ(g) = τ(π(g)) = 〈π(g)1τ , 1τ 〉 defines a character on Γ. Conversely, if ϕ : Γ → C is acharacter then the cyclic vector ξ in the corresponding GNS-representation π : Γ→ U(H) satisfies〈π(gh)ξ, ξ〉 = ϕ(gh) = ϕ(hg) = 〈π(hg)ξ, ξ〉 for all g, h ∈ Γ. Since the linear functional T 7→ 〈Tξ, ξ〉is normal we may then extend this to a normal faithful trace τ : π(Γ)′′ → C by the formulaτ(x) = 〈xξ, ξ〉. In particular this shows that π(Γ)′′ is finite since it has a normal faithful trace.

If (Mi, τi) are finite von Neumann algebras with normal faithful traces τi, i ∈ {1, 2}, andπi : Γ → U(Mi) then we will consider π1 and π2 to be equivalent if there is a trace preservingautomorphism α : M1 → M2, such that α(π1(g)) = π2(g) for all g ∈ Γ. Clearly, this is equivalentto requiring that there exist a unitary U : L2(M1, τ1) → L2(M2, τ2) such that U1τ1 = 1τ2 , andUπ1(g) = π2(g)U for all g ∈ G.

Note that the space of characters is a convex set, which is closed in the topology of pointwiseconvergence.

Theorem 5.7 (Thoma). Let Γ be a discrete group. There is a one to one correspondence between:

1. Equivalence classes of embeddings π : Γ → U(M) where M is a finite von Neumann algebrawith a given normal faithful trace τ , and such that π(Γ)′′ = M , and

2. Characters ϕ : Γ→ C,

which is given by ϕ(g) = τ(π(g)). Moreover, M is a factor if and only if ϕ is an extreme point inthe space of characters.

Proof. The one to one correspondence follows from the discussion preceding the theorem, thus weonly need to show the correspondence between factors and extreme points. If p ∈ P(Z(M)), is anon-trivial projection then we obtain characters ϕ1, and ϕ2 by the formulas ϕ1(g) = 1

τ(p)τ(π(g)p),

and ϕ2(g) = 1τ(1−p)τ(π(g)(1 − p)), and we have ϕ = τ(p)ϕ1 + τ(1 − p)ϕ2. Since p ∈ M = π(Γ)′′,

there exists a sequence xn ∈ CΓ such that 1τ(p)τ(π(xn)p) → 1, and 1

τ(1−p)τ(π(xn)(1 − p)) → 0, itthen follows that ϕ1 6= ϕ2 and hence ϕ is not an extreme point.

Conversely, if ϕ = 12(ϕ1+ϕ2) with ϕ1 6= ϕ2 then if we consider the corresponding representations

πi : Γ → U(Ni), we obtain a trace preserving embedding α : N → N1 ⊕ N2, which satisfiesα(π(g)) = π1(g) ⊕ π2(g). If we denote by p the projection 1 ⊕ 0 then it need not be the casethat p ∈ α(N), however by considering p we may then define a new trace τ ′ on N by τ ′(x) =14τ1(α(x)p) + 3

4τ2(α(x)(1 − p)). Since ϕ1 6= ϕ2 we must have that τ ′(π(g)) 6= τ(π(g)) for someg ∈ Γ. Thus, N does not have unique trace and so is not a factor by a result of Murray and vonNeumann [MvN37].

6 Completely positive maps

Given a C∗-algebra A we denote by Mn(A) the space of n× n matrices over A.If φ : A → B is a linear map between C∗-algeras, then we denote by φ(n) : Mn(A) → Mn(B)

the map defined by φ(n)([ai,j ]) = [φ(ai,j)]. We say that φ is positive if φ(a) ≥ 0, whenever a ≥ 0.If φ(n) is positive then we say that φ is n-positive and if φ is n-positive for every n ∈ N then wesay that φ is completely positive. If A and B are unital and φ : A→ B such that φ(1) = 1 thenwe say that φ is unital.

17

Example 6.1. Let A be a C∗-algebra and K a compact Hausdorff space. If π : K → S(A) is acontinuous map, then we obtain a unital (completely) positive map φ : A→ C(K) given by

φ(a)(x) = π(x)(a), a ∈ A, x ∈ K.

Conversely, this formula also defines a continuous map π : K → S(A) whenever φ : A → C(K) isunital (completely) positive.

Similarly, if (X,µ) is σ-finite measure space and π : X → S(A) is a measurable map, then weobtain a unital (completely) positive map φ : A → L∞(X,µ) given by the same formula. Lessobvious is that if A is separable then every unital (completely) positive map from A into L∞(X,µ)arrises in this way.

If π : A → B(K) is a representation of a C∗-algebra A and V ∈ B(H,K), then the operatorφ : A→ B(H) given by φ(x) = V ∗π(x)V is completely positive. Indeed, if we consider the operatorV (n) ∈ B(H⊕n,K⊕n) given by V (n)((ξi)i) = (V ξi)i then for all x ∈Mn(A) we have

φ(n)(x∗x) = V (n)∗π(n)(x∗x)V (n)

= (π(n)(x)V (n))∗(π(n)(x)V (n)) ≥ 0.

Generalizing the GNS construction Stinespring showed that every completely positive map from Ato B(H) arises in this way.

Theorem 6.2 (Stinespring’s dilation theorem). Let A be a unital C∗-algebra, and suppose φ : A→B(H), then φ is completely positive if and only if there exists a representation π : A → B(K) anda bounded operator V ∈ B(H,K) such that φ(x) = V ∗π(x)V . We also have ‖φ‖ = ‖V ‖2, and ifφ is unital then V is an isometry. Moreover, if A is a von Neumann algebra and φ is a normalcompletely positive map, then π is a normal representation.

Proof. Consider the sesquilinear form on A⊗H given by 〈a⊗ ξ, b⊗ η〉φ = 〈φ(b∗a)ξ, η〉, for a, b ∈ A,ξ, η ∈ H. If (ai)i ∈ A⊕n, and (ξi)i ∈ H⊕n, then we have

〈∑i

ai ⊗ ξi,∑j

aj ⊗ ξj〉φ =∑i,j

〈φ(a∗jai)ξi, ξj〉

= 〈φ((ai)∗i (ai)i)(ξi)i, (ξi)i〉 ≥ 0.

Thus, this form is non-negative definite and we can consider Nφ the kernel of this form so that〈·, ·〉φ is positive definite on K0 = (A ⊗ H)/Nφ. Hence, we can take the Hilbert space completionK = K0.

As in the case of the GNS construction, we define a representation π : A→ B(K) by first settingπ0(x)(a ⊗ ξ) = (xa) ⊗ ξ for a ⊗ ξ ∈ A ⊗ H. Note that since φ is positive we have φ(a∗x∗xa) ≤‖x‖2φ(a∗a), applying this to φ(n) we see that ‖π0(x)

∑i ai ⊗ ξi‖2φ ≤ ‖x‖2‖

∑i ai ⊗ ξi‖2φ. Thus,

π0(x) descends to a well defined bounded map on K0 and then extends to a bounded operatorπ(x) ∈ B(K).

If we define V0 : H → K0 by V0(ξ) = 1⊗ ξ, then we see that V0 is bounded by ‖φ(1)‖ and henceextends to a bounded operator V ∈ B(H,K). For any x ∈ A, ξ, η ∈ H we then check that

〈V ∗π(x)V ξ, η〉 = 〈π(x)(1⊗ ξ), 1⊗ η〉φ= 〈x⊗ ξ, 1⊗ η〉φ = 〈φ(x)ξ, η〉.

18

Thus, φ(x) = V ∗π(x)V as claimed.

Corollary 6.3 (Kadison’s inequality). If A and B are unital C∗-algebras, and φ : A→ B is unitalcompleley positive then for all x ∈ A we have φ(x)∗φ(x) ≤ φ(x∗x)

Proof. We assume that B ⊂ B(H). If we consider the Stinespring dilation φ(x) = V ∗π(x)V , thensince φ is unital we have that V is an isometry. Hence 1− V V ∗ ≥ 0 and so we have

φ(x∗x)− φ(x)∗φ(x) = V ∗π(x∗x)V − V ∗π(x)∗V V ∗π(x)V

= V ∗π(x∗)(1− V V ∗)π(x)V ≥ 0.

Lemma 6.4. Let A be a C∗-algebra, if

(0 x∗

x y

)∈M2(A) is positive, then x = 0, and y ≥ 0.

Proof. We may assume A is a C∗-subalgebra of B(H), hence if ξ, η ∈ H we have

2Re(〈x∗η, ξ〉) + 〈yη, η〉 =

⟨(0 x∗

x y

)(ξη

),

(ξη

)⟩≥ 0.

The result then follows easily.

Theorem 6.5 (Choi). If φ : A→ B is a unital 2-positive map between C∗-algebras, then for a ∈ Awe have φ(a∗a) = φ(a∗)φ(a) if and only if φ(xa) = φ(x)φ(a), and φ(a∗x) = φ(a∗)φ(x), for allx ∈ A.

Proof. Applying Kadison’s inequality to φ(2) it follows that for all x ∈ A we have(φ(a∗a) φ(a∗x)φ(x∗a) φ(aa∗ + x∗x)

)= φ(2)

(∣∣∣∣( 0 a∗

a x

)∣∣∣∣2)≥∣∣∣∣φ(2)

((0 a∗

a x

))∣∣∣∣2=

(φ(a∗)φ(a) φ(a∗)φ(x)φ(x∗)φ(a) φ(a)φ(a∗) + φ(x∗)φ(x)

).

Since φ(a∗a) = φ(a)∗φ(a) it follows from the previous lemma that φ(x∗a) = φ(x∗)φ(a), andφ(a∗x) = φ(a)∗φ(x).

6.1 Conditional expectations

If A is a unital C∗-algebra, and B ⊂ A is a unital C∗-subalgebra, then a conditional expectationfrom A to B is a unital completely positive E : A → B such that E|B = id. Note that by Choi’stheorem we have E(axb) = aE(x)b for all a, b ∈ B, x ∈ A.

Theorem 6.6 (Umegaki). Let M be a finite von Neumann algebra with normal faithful trace τ , andlet N ⊂M be a von Neumann subalgebra, then there exists a unique normal conditional expectationE : M → N such that τ ◦ E = τ .

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Proof. Let eN ∈ B(L2(M, τ)) be the projection onto L2(N, τ) ⊂ L2(M, τ), and let J be the con-jugation operator on L2(M, τ) which we also view as the conjugation operator on L2(N, τ). Notethat N ′ ∩ B(L2(N, τ)) = JNJ by Proposition 5.3. Since L2(N, τ) is invariant under JNJ we haveeN (JyJ) = (JyJ)eN for all y ∈ N .

If x ∈M , y ∈ N , then

eNxeNJyJ = eNxJyJ = eNJyJxeN

= JyJeNxeN .

Thus, eNxeN ∈ (JNJ)′ = N and we denote this operator by E(x). Clearly, E : M → N is normalunital completely positive, and E|N = id, thus E is a normal conditional expectation. Also, forx ∈M we have

τ(E(x)) = 〈eNxeN1τ , 1τ 〉 = 〈x1τ , 1τ 〉 = τ(x).

If E were another trace preserving conditional expectation, then for x ∈ M , and y ∈ N wewould have

τ(E(x)y) = τ(E(xy)) = τ(xy)

= τ(E(xy)) = τ(E(x)y),

from which it follows that E = E.

7 Amenable von Neumann algebras

A von Neumann algebra M ⊂ B(H) is amenable if there exists a (not in general normal) condi-tional expectation E : B(H)→M .

If X and Y are Banach spaces and x ∈ X, and y ∈ Y then we define the linear map x ⊗ y :B(X,Y ∗) → C by (x ⊗ y)(L) = L(x)(y). Note that |(x ⊗ y)(L)| ≤ ‖L‖‖x‖‖y‖ and hence x ⊗ y isbounded and indeed ‖x ⊗ y‖ ≤ ‖x‖‖y‖. Let Z be the norm closed linear span in B(X,Y ∗)∗ of allx⊗ y.

Lemma 7.1. The pairing 〈L, x ⊗ y〉 = (x ⊗ y)(L) extends to an isometric identification betweenZ∗ and B(X,Y ∗).

Proof. It is easy to see that this pairing gives an isometric embedding of B(X,Y ∗) into Z∗. To seethat this is onto consider ϕ ∈ Z∗, and for each x ∈ X define Lx : Y → C by Lx(y) = ϕ(x ⊗ y).Then we have |Lx(y)| ≤ ‖ϕ‖‖x‖‖y‖ and hence Lx ∈ Y ∗. The mapping x 7→ Lx is easily seen to belinear and thus we have L ∈ B(X,Y ∗), and clearly L is mapped to ϕ under this pairing.

Lemma 7.2. If X and Y are Banach spaces and Li is a bounded net in B(X,Y ∗) then Li → L inthe weak*-topology described above if and only if Li(x)→ L(x) in the weak*-topology for all x ∈ X.

Proof. If Li converges to L in the weak*-topology then for all x ∈ X and y ∈ Y we have Li(x)(y) =(x ⊗ y)(Li) → (x ⊗ y)(L) = L(x)(y) showing that Li(x) → L(x) is the weak*-topology for allx ∈ X. Conversely, if Li(x)→ L(x) in the weak*-topology for all x ∈ X then in particular we have(x⊗ y)(Li)→ (x⊗ y)(L) for each x ∈ X and y ∈ Y , thus this also holds for the linear span of allx⊗ y and since the net is bounded we then have convergence on the closed linear span.

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Corollary 7.3. Let A be a unital C∗-algebra, then the set of contractive completely positive mapsfrom E to B(H) is compact in the topology of point-wise weak operator topology convergence.

Proof. First note that it is easy to see that the space of contractive completely positive mapsfrom E to B(H) is closed in this topology. Since B(H) is a dual space, and on bounded sets theweak operator topology is the same as the weak*-topology, the result then follows from Alaoglu’stheorem, together with the previous two lemmas.

Theorem 7.4 (Schwartz [Sch63]). Let G be a locally compact amenable group, M a von Neumannalgebra, and suppose that α : G→ Aut(M) is a continuous action of G on M . Then there exists aconditional expectation E : M →MG.

Proof. First suppose thatG is amenable and let {µi}i ⊂ Prob(G) be a net such that ‖µi−g∗µi‖1 → 0for each g ∈ G. We then consider the completely positive maps φi : B(`2Γ) → B(`2Γ) given byφi(T ) =

∫αg(T ) dµi.

By the previous corollary we may take a subsequence and assume that {φi}i converges to acompletely positive map E in the topology of pointwise weak operator topology convergence. Notethat for x ∈MG we have φi(x) = x and hence it follows that E(x) = x for each x ∈MG. Moreover,if T ∈M then for g ∈ G we have ‖αg(φi(T ))−φi(T )ρ(γ)‖ ≤ ‖T‖‖µi−g∗µi‖1 → 0. Hence it followsthat E(T ) ∈MG and thus E : M →MG is a conditional expectation.

Corollary 7.5. Let Γ be a discrete group, then Γ is amenable if and only if LΓ is amenable.

Proof. If Γ is amenable then we may consider the action of Γ on B(`2Γ) given by αγ(T ) =ρ(γ)Tρ(γ−1). From the previous theorem there then exists a conditional expectation from B(`2Γ)onto ρ(Γ)′ = LΓ and hence LΓ is amenable.

Conversely, if LΓ is amenable and E : B(`2Γ)→ LΓ is a conditional expectation, then we mayconsider the state ϕ = τ ◦E|`∞Γ and a simple calculation shows that ϕ is Γ-invariant showing thatΓ is amenable.

7.1 Amenable finite von Neumann algebras

Proposition 7.6 (Powers-Stormer inequality). If S, T ∈ B(H) are positive Hilbert-Schmidt opera-tors, then

‖S − T‖2HS ≤ ‖S2 − T 2‖TC ≤ ‖S + T‖HS‖S − T‖HS.

Proof. Note that if x, y ∈ B(H) positive, at least one of which is a trace class operator, thenTr(xy) ≥ 0. We consider the spectral projection p = χ[0,∞)(S − T ), so that p(S − T ) ≥ 0 and

p⊥(T − S) ≥ 0. Then

Tr((S − T )2) = Tr(p(S − T )2 + p⊥(T − S)2)

≤ Tr(p(S + T )(S − T ) + p⊥(T − S)(T + S))

= Tr(p(S2 − T 2) + p⊥(T 2 − S2))

≤ Tr(p|S2 − T 2|+ p⊥|T 2 − S2|) = Tr(|S2 − T 2|),

where the third line follows from the fact that Tr(p(S + T )(S − T )) = Tr((S + T )(S − T )p).

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The second inequality follows form the formula S2−T 2 = 12((S+T )(S−T ) + (S−T )(S+T )),

the Cauchy-Schwarz inequality, and the fact that ‖x‖HS = ‖|x|‖HS for any Hilbert-Schmidt operatorx.

The following theorem is the von Neumann algebra analogue of Theorem 4.2.

Theorem 7.7 (Connes [Con76]). Let (M, τ) be a tracial von Neumann algebra, then the followingconditions are equivalent.

1. M is amenable.

2. There exists a state ϕ on B(L2(M, τ)) such that ϕ|M = τ , and ϕ(xT ) = ϕ(Tx) for all x ∈Mand T ∈ B(L2(M, τ)).

3. There exists a sequence of positive trace class operators {Ti}i ⊂ B(L2(M, τ)) so that τ(x) =Tr(xTi) for all x ∈M and i ∈ N, and such that ‖xTi − Tix‖Tr → 0 for all x ∈M .

4. There exists a sequence of Hilbert-Schmidt operators ξi ∈ B(L2(M, τ)) such that 〈xξi, ξi〉 =〈ξix, ξi〉 = τ(x) for every x ∈M and i ∈ N, and such that ‖xξi−ξix‖HS → 0 for every x ∈M .

Proof. (1) =⇒ (2): Suppose that E : B(L2(M, τ)) → M is a conditional expectation and setϕ(T ) = τ(E(T )). We then have that ϕ|M = τ and for T ∈ B(L2(M, τ)) and x ∈M we have

ϕ(xT ) = τ(xE(T )) = τ(E(T )x) = ϕ(Tx).

(2) =⇒ (1): Suppose now that ϕ is a state on B(L2(M, τ)) such that ϕ|M = τ , and ϕ(xT ) = ϕ(Tx)for all x ∈ M and T ∈ B(L2(M, τ)). Performing the GNS-construction gives a representationπ : B(L2(M, τ)) → B(K) and a unit vector ξ0 ∈ K so that for all T ∈ B(L2(M, τ)) we haveϕ(T ) = 〈π(T )ξ0, ξ0〉.

Since ϕ|M = τ we see that the map x 7→ π(x)ξ0 extends to an isometry V : L2(M, τ)→ K. It’seasy to see that for x ∈M we have π(x)V = V x.

We then obtain a unital completely positive map E : B(L2(M, τ)) → B(L2(M, τ)) by E(T ) =V ∗π(T )V . Since π(x)V = V x we have that E(x) = x for all x ∈M .

Also, if T ∈ B(L2(M, τ)), and x, a, b ∈M we have

〈JxJE(T )a, b〉 = 〈V ∗π(T )V a, Jx∗Jb〉 = 〈π(T )π(a)ξ0, π(bx)ξ0〉= ϕ(x∗b∗Ta) = ϕ(b∗Tax∗)

= 〈V ∗π(T )V ˆax∗, b∗〉 = 〈E(T )JxJa, b〉.

We therefore conclude that E(T ) ∈ JMJ ′ = M and it follows that E : B(L2(M, τ)) → M is aconditional expectation.

(2) =⇒ (3): We let X denote the space of trace class operators T such that Tr(xT ) =τ(x)Tr(T ). Then X is a closed subspace of the trace class operators and the annihilator of X inB(L2(M, τ)) is {x ∈M | τ(x) = 0}. It then follows that ϕ ∈ X∗∗.

Fix a sequence {xn}n ⊂M which is dense in the strong operator topology. For each n ∈ N welet Cn denote the convex set {⊕ni=1(xiT − Txi) | T ∈ X,T ≥ 0,Tr(T ) = 1} ⊂ X ⊂ X∗∗. Sinceϕ ∈ X∗∗ it follows that 0 is in the weak∗ closure of Cn in X∗∗, and since Cn is convex we have that

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0 must also be in the strong closure of Cn. Therefore there exist positive trace class operators Tnsuch that Tr(xTn) = τ(x) for all x ∈M , and ‖xiTn − Tnxi‖TC < 1/n.

Since {xn}n is dense in the strong operator topology we then have that ‖xTn−Tnx‖TC → 0 foreach x ∈M .

(3) =⇒ (4): If we set Sn = T1/2n then the Powers-Stormer inequality gives ‖xSn−Snx‖HS → 0,

and for x ∈M we have 〈xSn, Sn〉 = 〈Snx, Sn〉 = Tr(xTn) = τ(x).(4) =⇒ (2): We consider the states on B(L2(M, τ)) given by ϕn(T ) = 〈TSn, Sn〉. We let ϕ be

a weak∗ cluster point of {ϕn}n∈N. Then ϕ|M = τ and if x ∈M and T ∈ B(L2(M, τ)) we have thatϕ(xT − Tx) is a cluster point of ϕn(xT − Tx). However, from the Cauchy-Schwarz inequality wehave

|ϕn(xT − Tx)| = |〈(xT − Tx)Sn, Sn〉|≤ |〈xTSn − TSnx, Sn〉|+ ‖T‖‖xSn − Snx‖HS

= |〈xTSn − TSn, Snx∗〉|+ ‖T‖‖xSn − Snx‖HS

≤ 2‖T‖‖xSn − Snx‖HS → 0.

While we will not use this below, we also mention the following result of Connes:

Theorem 7.8 (Connes [Con76]). There is a unique (up to isomorphism) amenable separable II1

factor R.

8 Property (T) for finite von Neumann algebras

A tracial von Neumann algebra (M, τ) has property (T) if for any normal M -bimodule H, andany sequence of vectors ξi ∈ H satisfying ‖xξi‖2, ‖ξix‖2 ≤ ‖x‖22 for all x ∈ M , and i ∈ N, and‖xξi− ξix‖ → 0 for all x ∈M , there exists a sequence of vectors ηi ∈ H, such that xηi = ηix for allx ∈M and i ∈ N, and such that ‖ηi − ξi‖ → 0.

Theorem 8.1 (Connes-Jones [CJ85]). Suppose Γ is a discrete group with property (T), (M, τ) isa tracial von Neumann algebra, and π : Γ → U(M) is a representation so that M = π(Γ)′′. Then(M, τ) has property (T).

Proof. Let H be a normal Hilbert M -bimodule, and {ξi}i ⊂ H a sequence of vectors satisfying‖xξi‖2, ‖ξix‖2 ≤ ‖x‖22 for all x ∈M , and i ∈ N, and ‖xξi − ξix‖ → 0 for all x ∈M .

Consider the representation of Γ on H given by conjugation ξ 7→ π(γ)ξπ(γ−1). Then we have‖ξi − π(γ)ξiπ(γ−1)‖ → 0 for all γ ∈ Γ and since Γ has property (T) there then exists a sequence ofΓ-invariant vectors ηi ∈ H so that ‖ξi − ηi‖ → 0.

Since ηi are Γ-invariant we have that π(γ)ηi = ηiπ(γ) for all γ ∈ Γ. Taking linear combinationswe then have that xηi = ηix for any x in the linear span of π(Γ). Since this linear span is weakoperator topology dense in M it then follows that xηi = ηix for all x ∈M .

In the case of the group-von Neumann algebra the converse of the previous result also holds,although we will not need that here.

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Theorem 8.2 (Connes-Jones [CJ85]). Let Γ be a discrete group. then Γ has property (T) only ifLΓ has property (T).

Just as the case for groups, amenability and property (T) intersect only in very elementarysituations.

Proposition 8.3. Let M be a finite factor, if M is amenable and has property (T) then M is finitedimensional.

Proof. Suppose M is a finite factor which is amenable and has property (T). By Theorem 7.7 therethen exists a sequence of Hilbert-Schmidt operators ξn ∈ H(L2(M)) so that 〈xξi, ξi〉 = 〈ξix, ξi〉 =τ(x) for each x ∈M and ‖xξi − ξix‖HS → 0 for each x ∈M .

Since left and right multiplication on the space of Hilbert-Schmidt operators gives a normalbimodule it then follows from property (T) that there exists a sequence of Hilbert-Schmidt operatorsηi ∈ B(L2(M)) so that xηi = ηix for all x ∈ M and ‖ξi − ηi‖HS → 0. In particular, there mustexist some non-zero Hilbert-Schmidt operator η which commutes with M . We then have thatJηJ ∈ JMJ ′ = M and is Hilbert-Schmidt. Taking spectral projections then shows that M containsa projection onto a finite dimensional subspace and hence must be a type I factor. As M is a finitetype I factor it must then be finite dimensional.

9 Induced actions and ergodicity

Let G be a locally compact group, and Γ < G a closed subgroup. Suppose M is a von Neumannalgebra and θ : Γ→ Aut(M) is a continuous action of Γ on M . We let α : G×G/Γ→ Γ be a cocyclecorresponding to the identity map Γ → Γ, e.g., α(g, x) = ψ(gx)−1gψ(x), where ψ : G/Γ → G is aBorel section. We may then induce the action of Γ on M to an action of G on L∞(G/Γ)⊗M asθg(f)(x) = θα(g,g−1x)(f(g−1x)).

We consider the action L : G → Aut(L∞G) (resp. R : G → Aut(L∞G)) induced by left(resp. right) multiplication. An alternate way to view the induced action of G on L∞(G/Γ)⊗Mis to consider the action of ΓyR⊗θL∞(G)⊗M . Fixing a Borel section ψ : G/Γ → G then givesan isomorphism Ψ : L∞(G/Γ)⊗M → (L∞(G)⊗M)(R⊗θ)(Γ) by Ψ(f)(g) = θψ(Γ)α(g−1,gΓ)(f(gΓ)).Under this isomorphism, the induced action of G on L∞(G/Γ)⊗M then translates to the actionGyL⊗id(L∞(G)⊗M)(R⊗θ)(Γ).

Theorem 9.1 (Zimmer [Zim77]). Let G be a locally compact group, H < G a closed amenablesubgroup, and Γ < G a discrete subgroup. For any unitary representation π : Γ → U(H), we mayconsider the corresponding action of Γ on B(H) given by θγ(T ) = πγTπ

∗γ. Then the von Neumann

algebra (L∞(G/H)⊗B(H))Γ is amenable.

Proof. As described above, we may identify the von Neumann algebras (L∞(G/H)⊗B(H))Γ and(L∞(G)⊗B(H))H×Γ. We similarly have an identification between this von Neumann algebra and(L∞(G/Γ)⊗B(H))H (where the action of H here is the one induced from the action of Γ). SinceH and L∞(G/Γ)⊗B(H) are amenable the result then follows from Theorem 7.4.

An action of a group on a von Neumann algebra is ergodic if the fixed point subalgebra is C.Note that from the isomorphism given by Ψ above, we have that the action ΓyθM is ergodic ifand only if the induced action GyθL∞(G/Γ)⊗M is ergodic.

24

Suppose now that Γ < G is a lattice and M is a finite von Neumann algebra with a normalfaithful trace τ , and assume also that θγ preserves the trace on M for each γ ∈ Γ. We then havethat θ preserves the trace on L∞(G/Γ)⊗M given by

∫⊗ τ (where the integral corresponds to the

unique G-invariant probability measure on G/Γ).Suppose H < G is a closed subgroup such that ΓH is dense in G. For O ⊂ G a non-empty

open subset we set ΓO = Γ ∩ OH, which is non-empty since ΓH is dense in G. For x ∈ M we setKx(O) = co{θγ(x) | γ ∈ ΓO}, and Kx(O) = co{θγ−1(x) | γ ∈ ΓO}. For each g ∈ G we let κx(g)

(resp. κx(g)) be the element of minimal ‖ · ‖2 in ∩O∈N (g)Kx(O) (resp. ∩O∈N (g)Kx(O)), where N (g)denotes the set of open neighborhoods of g. Note that for each g ∈ G, κx(g), κx(g) ∈M ⊂ L2(M, τ)by Corollary 5.2.

Proposition 9.2. Using the notation above, if the induced action HyL∞(G/Γ)⊗M is ergodic,then κx(g) = τ(x) for all g ∈ G.

Proof. Using the isomorphism (L∞(G/Γ)⊗M)H ∼= (L∞(G)⊗M)H×Γ as described above, ergodicityof the H action is equivalent to ergodicity of the H × Γ action on L∞(G)⊗M , where the action ofH is given by h 7→ Lh ⊗ id, and the action of Γ is given by γ 7→ Rγ ⊗ θγ .

Note that κx : G→ M is a bounded Borel map. Indeed, we have ‖κx(g)‖ ≤ ‖x‖ for all g ∈ G,hence κx is bounded. If {gn}n∈N is a countable dense subset of G, then for each O ∈ N (e) wemay consider the simple function κx,O : G→M given by setting κx,O(g) to be the unique elementof minimal norm in Kx(gjO), where j is the smallest natural number such that g ∈ gjO. Takinga sequence On ∈ N (e) such that ∩On = {e} we then have that κx,On converges pointwise in thestrong operator topology to κx, hence κx is Borel.

For g ∈ G, h ∈ H, γ ∈ Γ, and O ∈ N (g), we have θγ(Kx(O)) = Kx(γOh), hence it follows thatθγ(κx(g)) = κx(γgh).

Thus, κx ∈ (L∞(G)⊗M)H×Γ = C, and so κx(g) = τ(x) for almost every g ∈ G. As {g ∈ G |κx(g) = τ(x)} is closed, it then follows that κx(g) = τ(x) for all g ∈ G.

In the sequel we will need to consider convex combinations of the form θγ−1(x) for γ ∈ ΓO. Inthe case when H < G is normal we have κx(g) = κx(g), and the above proposition suffices. For thegeneral case we have the following argument:

Proposition 9.3. Using the notation above, if the induced action HyL∞(G/Γ)⊗M is ergodic,then κx(g) = τ(x) for all g ∈ G.

Proof. Let ε > 0 be given. We claim that there exists U ∈ N (e) such that for all γ ∈ ΓU we have‖θγ(κx(e))− κx(e)‖2 = ‖κx(e)− θγ−1(κx(e))‖2 < ε.

Indeed, from the definition of κx(e) there exists O ∈ N (e) such that if y is the element ofminimal norm in Kx(O) then ‖κx(e) − y‖2 < ε/2. We may write y as a convex combinationy =

∑i αiθγ−1

i(x) where γi = gihi with gi ∈ O, and hi ∈ H. If no such U existed, then there would

exist a sequence γk = gkhk with hk ∈ H, and gk → e such that ‖y − θγ−1k

(y)‖2 ≥ ε/4.

However, θγ−1k

(y) =∑

i αiθγ−1k γ−1

i(x), and since gk → e, for each i there exists large enough k so

that γiγk = gi(higkh−1i )hihk ∈ OH. It then follows from uniqueness of y that ‖y − θγ−1

k(y)‖2 → 0,

proving the claim.

25

From the claim it then follows that κx(e) is the element of minimal ‖ · ‖2 in ∩O∈N (e)Kκx(e)(O),and it then follows from Proposition 9.2 that κx(e) = τ(κx(e)) = τ(x).

For γ ∈ Γ, h ∈ H, and O ⊂ G a non-empty open set, we have Kx(γOh) = Kθγ−1 (x)(O), hence

it follows that κx(γh) = κθγ−1 (x)(e) = τ(θγ−1(x)) = τ(x). Since ΓH is dense in G and since

{g ∈ G | κx(g) = τ(x)} is closed, it then follows that κx(g) = τ(x) for all g ∈ G.

10 Contracting automorphisms

Let H be a locally compact, compactly generated group. An automorphism α ∈ Aut(H) is con-tracting if for any neighborhood U of e and any compact set K ⊂ H, there is an integer N sothat αn(K) ⊂ U for all n ≥ N .

Theorem 10.1 (Lebesgue’s density theorem). Suppose X is a locally compact seperable Hausdorffspace and µ is a σ-finite measure on X which is positive on open sets and finite on compact sets.Suppose that for each x ∈ X we have a decreasing sequence Bx,n of precompact open neighborhoodsof x, forming a basis for the open sets at x, and such that

1. µ(Bx,n−1)/µ(Bx,n) is constant independent of x, n.

2. If n ≤ p and Bx,n ∩By,p 6= ∅, then By,p ⊂ Bx,n−2.

If E ⊂ X is a measurable set, then for almost all x ∈ E,

limn→∞

µ(Bx,n ∩ E)/µ(Bx,n) = 1.

Proposition 10.2. Suppose that H is a compactly generated locally compact group and α ∈ Aut(H)is a contracting automorphism of H. Suppose that U ⊂ H is a precompact open neighborhood ofthe identity with U−1 = U and such that α(U2) ⊂ U . For x ∈ H let Bx,n = xαn(U). Then thesesets satisfy the hypothesis of Theorem 10.1, where µ is the left Haar measure.

Proof. First note that since µ and α∗µ are both left Haar measures it follows that there is someconstant c > 0 so that α∗µ = cµ. Thus, for x ∈ H and n > 1 we have

µ(Bx,n−1)

µ(Bx,n)=µ(xαn−1(U))

µ(xαn(U))= c.

To verify (2) suppose there exists z ∈ yαp(U) ∩ xαn(U), with n ≤ p. Let w ∈ yαp(U). Then

z−1w ∈ αp(U)y−1yαp(U) = αp(U2) ⊂ αp−1(U) ⊂ αn−1(U).

Thus w ∈ zαn−1(U) and since z ∈ xαn(U) we have

w ∈ xαn(U)αn−1(U) ⊂ xαn−1(U2) ⊂ xαn−2(U).

Since w ∈ yαp(U) was arbitrary we then have yαp(U) ⊂ xαn−2(U).

26

Proposition 10.3 (Margulis). Suppose that H is a compactly generated locally compact group andα ∈ Aut(H) so that α−1 is contracting. Then replacing α by some power of α if necessary, it followsthat for any measurable set E ⊂ H and almost every h ∈ E we have that αn(h−1E) converges inmeasure to H.

Proof. Fix a precompact open neighborhood U of the identity such that U−1 = U , and ∪n≥1Un =

H. Since α−1 is contracting we may replace α with a fixed power and assume that α−1(U2) ⊂ U .Note that we then have α−1((Uk)2) ⊂ Uk for all k > 1. We again have α∗µ = cµ, for some c > 0.

Let E ⊂ H be a measurable set and fix k ≥ 1. Then by the previous proposition and Lebesgue’sdensity theorem we have that for almost every h ∈ E

1 = limn→∞

µ(hα−n(Uk) ∩ E)/µ(hα−n(Uk)) = limn→∞

cn

cnµ(Uk ∩ αn(h−1E))/µ(Uk).

Hence limn→∞ µ(Uk ∩ αn(h−1E)) = µ(Uk). Since H = ∪k≥1Uk it then follows that αn(h−1E)

converges to H in measure.

Notation 10.4. For the remaining of this section, and through all of the next section we use thefollowing notation:

Fix G = SLn(R) for n ≥ 3. We denote by P the subgroup of upper triangular matrices, andV the subgroup of upper triangular matrices whose diagonal entries are 1. Suppose also that1 ≤ j1 < j2 < · · · < jk = n, with k > 1. We denote by P0 < G the subgroup of block triangularmatrices of determinant 1 of the form

A11 A12 · · · A1k

0 A22 · · · A2k

· · · · · · · · · · · ·0 0 · · · Akk

,

where Ahh is a square matrix of order jh−jh−1 (here we assume j0 = 0). We also denote by V0 < Vthe subgroup block triangular matrices of the form

E1 A12 · · · A1k

0 E2 · · · A2k

· · · · · · · · · · · ·0 0 · · · Ek

,

where Eh denotes the jh − jh−1 identity matrix, and R0 consists of the block diagonal matrices ofthe form

A11 0 · · · 00 A22 · · · 0· · · · · · · · · · · ·0 0 · · · Akk

.

We let L0 be the subgroup of R0 whose block matrices Ahh are upper triangular with diagonalentries equal to 1.

We also denote by P , V , etc. the corresponding transpose subgroups.

27

Example 10.5. Choose real numbers λ1 < λ2 < · · · < λk so that∏kj=1 λj = 1 and let s be the

block diagonal matrix

s =

A11 0 · · · 00 A22 · · · 0· · · · · · · · · · · ·0 0 · · · Akk

,

where Aii is the diagonal matrix with diagonal entries equal to λi. Then s ∈ Z(R0) and Int(s−1)|V0

and Int(s)|V0are contracting.

Lemma 10.6. The mapping V → G/P given by v 7→ vP is an injective Borel map onto a conullsubset, hence gives rise to a measure space isomorphism.

Proof. If A ∈ SLn(R), we let Aj be the upper left-hand corner j × j submatrix. The lemma caneasily be reduced to the following statement in linear algebra which is readily verified by inductionon n: For A ∈ SLn(R) we can write A = BC where B ∈ V and C ∈ P if and only if for all1 ≤ j ≤ n, det(Aj) 6= 0, and if this is the case then the representation of A is unique.

Under the above isomorphism, the action of V on G/P transforms to left multiplication on V ,while the action of R on G/P transforms to the action induced by conjugation on V . As V = V0oL0

we then have a measure space isomorphism between G/P and V0×L0, and hence we obtain a corre-sponding isomorphism between von Neumann algebras L∞(G/P ) ∼= L∞(V0)⊗L∞(L0). Moreover,the natural projectiion map G/P to G/P0 gives rise to an embedding L∞(G/P0) ⊂ L∞(G/P ) whichcorresponds under the above isomorphism to the natural embedding L∞(V0) ⊂ L∞(V0)⊗L∞(L0).Note that the functions in L∞(L0) are fixed by any element in V0 o Z(R0).

Lemma 10.7. Suppose X is a second countable locally compact Hausdorff space equipped with aprobability µ ∈ Prob(X) such that µ gives positive measure to any non-empty open set. Supposealso that ZyX by homeomorphisms, preserving the measure µ, such that the action Zy(X,µ) isergodic. Then for almost every x ∈ X the set N · x is dense.

Proof. If W ⊂ X is a non-empty open set, then E = (−N) ·W ⊂ X is a positive measure subset ofX such that (−1) ·E ⊂ E. Since the action is measure-preserving it then follows that E is invariantand hence must be conull in (X,µ). Thus, if {Wi}i is a countable basis of non-empty open setsthen we have that F = ∩i(∪j≤−1j ·Wi) is conull in (X,µ), and for each x ∈ F we have that N · xis dense since it intersects each basis set Wi.

Lemma 10.8. If s ∈ Z(R0) is as in Example 10.5, then for almost every v ∈ V0 the set {Γvs−j |j ∈ N} is dense in G.

Proof. Set W = {g ∈ G | {Γgs−j | j ∈ N} is dense in G}. By Moore’s ergodicity theorem {s−n |n ∈ Z} acts ergodically on G/Γ and hence by Lemma 10.7 W is conull.

Recall from Lemma 10.6 that the multiplication map (V0 o L0) × P → G is an injective maponto a conull subset of G. For each v ∈ V0 we set Yv = {(g, h) | g ∈ L0, h ∈ P, vgh ∈ W}, andset U = {v ∈ V0 | Yv is conull in L0 × P}. Then by Fubini, U is conull in V0. Thus, to prove thelemma it suffices to show that {Γvs−j | j ∈ N} is dense in G for each v ∈ U .

28

Fix v ∈ U , and choose g ∈ L0 and h ∈ P so that (g, h) ∈ Yv, so that {Γvghs−j | j ∈ N} is densein G. Since gh ∈ P0 = R0nV0, we may write gh = ur where u ∈ V0, and r ∈ R0. Since [s, r] = e itthen follows that {Γvus−j | j ∈ N} is dense in G. Note also that since Γ is discrete we must thenhave that {Γvus−j | j ≥ N} is dense for any N ≥ 1.

Suppose now that O ⊂ G is a non-empty open set. We let O1 ⊂ O be a non-empty precompactopen set with O1 ⊂ O, and we let C ⊂ G be an open neighborhood of the identity so thatO1C ⊂ O. Since Int(s)|V0

is contracting there exists N > 1 so that sju−1s−j ∈ C for all j ≥ N .Since {Γvus−j | j ≥ N} is dense, there exists j ≥ N and γ ∈ Γ so that γvus−j ∈ O1. We then have

γvs−j = γvus−j(sju−1s−j) ∈ (γvus−j)C ⊂ O1C ⊂ O.

Since O ⊂ G was arbitrary it follows that {Γvs−j | j ∈ N} is dense in G.

Theorem 10.9 (Margulis). Let ν ∈ Prob(V0) be in the same measure class as Haar measure. IfE ⊂ V0 has positive measure and g ∈ G, then there exist sequences {γj} ⊂ Γ and {hj} ⊂ V0oZ(R0)such that γjh

−1j → g, and ν(γjE)→ 1.

Proof. From Example 10.5 there exists s ∈ Z(R0) such that Ints−1|V0

and Ints|V0are contracting,

and by Lemma 10.8, for almost every v ∈ V0 we have that {Γvs−j | j ∈ N} is dense in G. Wedenote by F1 the set of such v ∈ V0.

As Ints−1|V0

is contracting, Proposition 10.3 shows that for almost every v ∈ V0 we have

ν(sjv−1E)→ 1. We denote by F2 the set of such v ∈ V0.Fix v0 ∈ F1 ∩ F2. Since v0 ∈ F1 there exists a sequence {γi} ⊂ Γ, and a subsequence {s−ji}

such that γiv0s−ji → g. Hence

limi→∞

ν(γiE) = limi→∞

ν(gsjiv−10 E) = 1.

Since hj = v0s−ji ∈ V0 o Z(R0), this finishes the proof.

Margulis’ normal subgroup theorem (Corollary C) follows from the following ergodicity result.

Theorem 10.10 (Margulis). Using the notation above with G = SLn(R) and Γ < G a lattice, letµ ∈ Prob(G/P ) be a quasi-invariant probability measure. If N C Γ is a normal subgroup such thatN 6< Z(G), then the action Ny(G/P, µ) is ergodic.

Proof. Using the notation in 10.4 we measurably identify G/P with V = V0oL0 as above. Supposethat E ⊂ V is invariant under the action of N . Write E = ∪h∈V0

{h} × Eh. We consider the

measure algebra M(L0) consisting of measurable subsets of L0 where we identify two subsets iftheir symmetric difference is null. ThenM(L0) is a Polish space when equipped with the topologyof convergence in measure and the map ψ : V0 →M(L0) is a measurable map. We fix a probabilitymeasures ζ ∈ Prob(V0), and ν ∈ Prob(L0) in the class of Haar measure.

We fix g ∈ G and let E0 ⊂M(L0) be in the essential range of ψ, and we fix a countable densecollection {Fi}i ⊂M(L0). Then for any j > 1 and, there exists a positive measure set Aj ⊂ L0 sothat |ν(Eh∆Fi)−ν(E0∆Fi)| < 1/j for all 1 ≤ i ≤ j, and all h ∈ Aj . By the previous theorem therethen exists sequences {γj}j ⊂ Γ and {hj}j ⊂ V0 o Z(R0) so that ζ(γjAj)→ 1, and γjh

−1j → g.

29

Since the action of V0 o Z(R0) on L0 is trivial we then have that γjE converges in measure togE0 ∈ M(L0), and since N C Γ is normal we have that each γjE is N -invariant and hence so isgE0. Equivalently, we have shown that E0 is invariant under the action of the group g−1Ng. Sincethis was true for an arbitrary element g ∈ G we then have that E0 is invariant under the action ofthe closed normal subgroup in G which is generated by N . However, as N 6< Z(G) it then followsthat E0 is invariant under all of G and so must be either null or conull.

Since E0 was an arbitrary set in the essential range of ψ it follows that there exists A ⊂L0 measurable so that E = A × L0 up to measure zero, i.e., the characteristic function 1E ∈L∞(G/P0) ⊂ L∞(G/P ). Since P < P0 � G was arbitrary and since G is generated by suchsubgroups (here is where we use n ≥ 3) it then follows that 1E ∈ C, i.e., E is either null or conulland hence the action Ny(G/P, µ) is ergodic.

11 Character rigidity for lattices

We continue to use the notation from 10.4.Suppose that M is a finite factor with normal faithful trace τ , and π : Γ → U(M) is a homo-

morphism such that π(Γ)′′ = M . We let Γ act on B(L2(M)) by conjugation by Jπ(γ)J and welet

B = (L∞(G/P )⊗B(L2M))Γ,

denote the von Neumann algebra of essentially bounded Γ-equivariant functions from G/P toB(L2M). We denote by σ : Γ → Aut(L∞(G/P )) the action given by σγ(f)(x) = f(γ−1x). Wealso fix a quasi-invariant probability measure µ ∈ Prob(G/P ) and let σ0 : Γ→ L2(G/P, µ) denotethe Koopman representation σ0

γ(f)(x) = f(γ−1x)(dγ∗µdµ )1/2(x). Note that viewing L∞(G/P ) as

bounded operators on L2(G/P, µ) we have σγ(f) = σ0γfσ

0γ−1 .

We have the following operator algebraic consequence of Lemma 10.9:

Lemma 11.1. Using the notation from 10.4, suppose x = x∗ ∈ L∞(V0)⊗L∞(L0)⊗B(L2M) andsuppose x0 ∈ L∞(L0)⊗B(L2M) is in the strong operator topology V0-essential range of x, then thereexists y = y∗ ∈ B such that yP1 ∈ L∞(L0)⊗B(L2M), and P1yP1 = P1x0P1.

Proof. For each j ∈ N we let Ej ⊂ V0 be a positive measure subset such that x(vj)−x0 → 0 stronglywhenever vj ∈ Ej . From Lemma 10.9 there exists sequences {γj} ⊂ Γ and {hj} ⊂ V0oZ(R0) suchthat γjh

−1j → e, and ν(γjEj)→ 1.

Hence, for all ξ ∈ L2(V0 × L0) and η ∈ L2M we have

‖1γjEj (σγj (x)− x0)ξ ⊗ η‖ ≤‖1γjEj (σγj (x− x0))ξ ⊗ η‖+ ‖1γjEj (σγjh−1j

(x0)− x0)ξ ⊗ η‖

=

(∫γjEj

‖(σγj (x− x0))η‖22|ξ|2)1/2

+ ‖1γjEj (σγjh−1j

(x0)− x0)ξ ⊗ η‖ → 0.

Therefore, 1γjEj (σγj (x) − x0) → 0 strongly. Since ν(γjEj) → 1, we then have that 1γjEj → 1strongly, and hence σγj (x)→ x0 strongly.

30

We let y ∈ B be a weak operator topology cluster point of the set {π(γj)xπ(γ−1j )}. It then

follows that yP1 is a weak operator topology cluster point of

{π(γj)xπ(γ−1j )P1} = {(π(γj)(Jπ(γj)J))(Jπ(γ−1

j )J)x(Jπ(γj)J)P1}= {(π(γj)(Jπ(γj)J))σγj (x)P1}.

Since σγj (x) → x0 strongly, we then have that yP1 is a weak operator topology cluster point of

{(π(γj)(Jπ(γj)J))x0P1} ⊂ L∞(L0)⊗B(L2M), thus y ∈ B, and yP1 ∈ L∞(L0)⊗B(L2M).Similarly, P1yP1 is a weak operator topology cluster point of

{P1π(γj)xπ(γ−1j )P1} = {P1σγj (x)P1}

and hence P1yP1 = P1x0P1.

We say the representation π : Γ → U(M) is induced from a character on Z(Γ) if τ(π(γ)) = 0for all γ 6∈ Z(Γ).

Theorem 11.2. Using the notation above, either π is induced from a character on Z(Γ), or elseM = B.

Proof. Suppose that x = x∗ ∈ B. Using the notation above consider x ∈ L∞(V0)⊗L∞(L0)⊗B(L2M).Let x0 ∈ L∞(L0)⊗B(L2M) be in the V0-essential range of x. By Lemma 11.1 there exists y = y∗ ∈ Bsuch that yP1 ∈ L∞(L0)⊗B(L2M), and P1yP1 = P1x0P1.

Since the action GyL∞(G/Γ)⊗M is ergodic Moore’s ergodicity theorem shows that the actionrestricted to V0 o Z(R0) is ergodic (this group is non-compact since P0 6= G). By Proposition 9.3,for every open neighborhood O ⊂ G of e, setting ΓO = Γ ∩ O(V0 o Z(R0)) we have τ(x) ∈co{π(γ−1)xπ(γ) | γ ∈ ΓO}.

If π is not induced from a character on Z(Γ) there exists γ0 ∈ Γ\Z(Γ) such that α0 = τ(π(γ0)) 6=0. Thus, for each open neighborhood O ⊂ G of e there exists 0 ≤ αi ≤ 1,

∑ji=1 αi = 1, and γi ∈ ΓO

such that

[σ0γ0⊗ α0, P1x0P1] = [σ0

γ0⊗ α0, P1yP1]

∼j∑i=1

αiP1[σ0γ0⊗ Jπ(γ−1

i γ0γi)J, y]P1,

where the approximation above is in the strong operator topology. We may write γi = gihi wheregi ∈ O, and hi ∈ V0oZ(R0), and since yP1, P1y ∈ L∞(L0)⊗B(L2M), we have that σhi(yP1) = yP1,and σhi(P1y) = P1y. Taking O to be a small enough neighborhood, we then have the strong operatortopology approximations σγi(yP1) ∼ yP1, and σγi(P1y) ∼ P1y.

It follows that we then have the weak operator topology approximation

[σ0γ0⊗ α0, P1x0P1] ∼

j∑i=1

αiP1[σ0γ0⊗ Jπ(γ−1

i γ0γi)J, (σ0γi ⊗ 1)y(σ0

γ−1i⊗ 1)]P1

=

j∑i=1

αiP1(σ0γi ⊗ 1)[σ0

γ−1i γ0γi

⊗ Jπ(γ−1i γ0γi)J, y](σ0

γ−1i⊗ 1)P1 = 0.

31

Hence, we conclude that [σ0γ0⊗ α0, P1x0P1] = 0, and since α0 6= 0 we then have that [σ0

γ0⊗

1, P1x0P1] = 0.Hence σγ0(P1x0P1) = P1x0P1, and since τ(π(γγ0γ

−1)) = τ(π(γ0)) the same argument showsthat σγ(P1x0P1) = P1x0P1 for all γ ∈ 〈〈γ0〉〉, the normal closure of γ0. Since γ0 6∈ Z(Γ),Theorem 10.10 then shows that 〈〈γ0〉〉 acts ergodically on L∞(G/P ). Thus, we have P1x0P1 ∈P1B(L2M)P1 = CP1.

As x0 was an arbitrary element in the essential range of x, we conclude that P1xP1 ∈ L∞(V0)⊗CP1,and, as x was an arbitrary self-adjoint element, we then have P1BP1 ⊂ L∞(V0)⊗CP1. If a, b ∈M and z ∈ B, we then have 〈za, b〉 = 〈(b∗za)1, 1〉 ∈ L∞(V0), and hence it follows that B ⊂L∞(V0)⊗B(L2M) = L∞(G/P0)⊗B(L2M).

However, P0 was an arbitrary subgroup such that P < P0 � G. Since G is generated by allsuch subgroups (note that this is were we need n ≥ 3) we therefore have B ⊂ B(L2M). Thus,B = B(L2M) ∩ {Jπ(γ)J | γ ∈ Γ}′ = M .

Proof of Theorem D. From Theorem 3.8 we see that G has property (T), and hence so does Γ byTheorem 3.9. Thus, M has property (T) by Theorem 8.1.

Since Z(G) = {e}, we have that Z(Γ) = {e}. Thus, if π does not extend to an isomorphismLΓ

∼−→ M , then by Theorem 11.2 we have that M = B and hence is amenable by Theorem 9.1.Proposition 8.3 would then show that M is finite dimensional.

12 Appendix: Background material

In this section we collect the basic results we need along with references to proofs.

12.1 Bounded linear operators

If H is a Hilbert space then B(H), the algebra of all bounded linear operators, is a C∗-algebra withnorm

‖x‖ = supξ∈H,‖ξ‖≤1

‖xξ‖ = supξ,η∈H,‖ξ‖,‖η‖≤1

|〈xξ, η〉|,

and involution given by the adjoint, i.e., x∗ is the unique bounded linear operator such that

〈ξ, x∗η〉 = 〈xξ, η〉,

for all ξ, η ∈ H. Note that ‖x∗‖ = ‖x‖ since

‖x‖ = supξ,η∈H,‖ξ‖,‖η‖≤1

|〈xξ, η〉| = supξ,η∈H,‖ξ‖,‖η‖≤1

|〈ξ, x∗η〉| = ‖x∗‖.

To see the C∗-identity note that we clearly have ‖x∗x‖ ≤ ‖x∗‖‖x‖, and for the reverse inequalitywe have

‖x‖2 = supξ∈H,‖ξ‖≤1

〈xξ, xξ〉

≤ supξ,η∈H,‖ξ‖,‖η‖≤1

|〈xξ, xη〉|

= supξ,η∈H,‖ξ‖,‖η‖≤1

|〈x∗xξ, η〉| = ‖x∗x‖.

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Proposition 12.1 (Polar decomposition). Let H be a Hilbert space, and x ∈ B(H), then thereexists a partial isometry v such that x = v|x|, and ker(v) = ker(|x|) = ker(x). Moreover, thisdecomposition is unique, in that if x = wy where y ≥ 0, and w is a partial isometry with ker(w) =ker(y) then y = |x|, and w = v.

Proof. We define a linear operator v0 : R(|x|)→ R(x) by v0(|x|ξ) = xξ, for ξ ∈ H. Since ‖|x|ξ‖ =‖xξ‖, for all ξ ∈ H it follows that v0 is well defined and extends to a partial isometry v from R(|x|)to R(x), and we have v|x| = x. We also have ker(v) = R(|x|)⊥ = ker(|x|) = ker(x).

To see the uniqueness of this decomposition suppose x = wy where y ≥ 0, and w is a partialisometry with ker(w) = ker(y). Then |x|2 = x∗x = yw∗wy = y2, and hence |x| = (|x|2)1/2 =

(y2)1/2 = y. We then have ker(w) = R(|x|)⊥, and ‖w|x|ξ‖ = ‖xξ‖, for all ξ ∈ H, hence w = v.

12.2 Locally convex topologies on the space of operators

Let H be a Hilbert space. On B(H) we define the following locally convex topologies:

• The weak operator topology (WOT) is defined by the family of semi-norms T 7→ |〈Tξ, η〉|,for ξ, η ∈ H.

• The strong operator topology (SOT) is defined by the family of semi-norms T 7→ ‖Tξ‖,for ξ ∈ H.

Note that the from coarsest to finest topologies we have

WOT ≺ SOT ≺ Uniform.

Also note that since an operator T is normal if and only if ‖Tξ‖ = ‖T ∗ξ‖ for all ξ ∈ H, itfollows that the adjoint is SOT continuous on the set of normal operators.

Lemma 12.2. Let ϕ : B(H)→ C be a linear functional, then the following are equivalent:

(i) There exists ξ1, . . . , ξn, η1, . . . , ηn ∈ H such that ϕ(T ) =∑n

i=1〈Tξi, ηi〉, for all T ∈ B(H).

(ii) ϕ is WOT continuous.

(iii) ϕ is SOT continuous.

Proof. The implications (i) =⇒ (ii) and (ii) =⇒ (iii) are clear and so we will only show (iii)=⇒ (i). Suppose ϕ is SOT continuous. Thus, the inverse image of the open ball in C is open inthe SOT and hence by considering the semi-norms which define the topology we have that thereexists a constant K > 0, and ξ1, . . . , ξn ∈ H such that

|ϕ(T )|2 ≤ Kn∑i=1

‖Tξi‖2.

If we then consider {⊕ni=1Tξi | T ∈ B(H)} ⊂ H⊕n, and let H0 be its closure, we have that

⊕ni=1 Tξi 7→ ϕ(T )

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extends to a well defined, continuous linear functional on H0 and hence by the Riesz representationtheorem there exists η1, . . . , ηn ∈ H such that

ϕ(T ) =n∑i=1

〈Tξi, ηi〉,

for all T ∈ B(H).

Corollary 12.3. Let K ⊂ B(H) be a convex set, then the WOT, SOT, and closures of K coincide.

Proof. By Lemma 12.2 the three topologies above give rise to the same dual space, hence thisfollows from the the Hahn-Banach separation theorem.

If H is a Hilbert space then the map id⊗ 1 : B(H)→ B(H⊗`2N) defined by (id⊗ 1)(x) = x⊗ 1need not be continuous in either of the locally convex topologies defined above even though it is anisometric C∗-homomorphism with respect to the uniform topology. Thus, on B(H) we define thefollowing additional locally convex topologies:

• The σ-weak operator topology (σ-WOT) is defined by pulling back the WOT of B(H⊗`2N)under the map id⊗1.

• The σ-strong operator topology (σ-SOT) is defined by pulling back the SOT of B(H⊗`2N)under the map id⊗1.

Note that the σ-weak operator topology can alternately be defined by the family of semi-normsT 7→ |Tr(Ta)|, for a ∈ L1(B(H)). Hence, under the identification B(H) = L1(B(H))∗, we have thatthe weak∗-topology on B(H) agrees with the σ-WOT.

Lemma 12.4. Let ϕ : B(H)→ C be a linear functional, then the following are equivalent:

(i) There exists a trace class operator a ∈ L1(B(H)) such that ϕ(x) = Tr(xa) for all x ∈ B(H)

(ii) ϕ is σ-WOT continuous.

(iii) ϕ is σ-SOT continuous.

Proof. Again, we need only show the implication (iii) =⇒ (i), so suppose ϕ is σ-SOT continuous.Then by the Hahn-Banach theorem, considering B(H) as a subspace of B(H ⊗ `2N) through themap id ⊗ 1, we may extend ϕ to a SOT continuous linear functional on B(H ⊗ `2N). Hence byLemma 12.2 there exists ξ1, . . . , ξn, η1, . . . , ηn ∈ H⊗`2N such that for all x ∈ B(H) we have

ϕ(x) =n∑i=1

〈(id⊗ 1)(x)ξi, ηi〉.

For each 1 ≤ i ≤ n we may define ai, bi ∈ HS(H, `2N) as the operators corresponding to ξi, ηi inthe Hilbert space isomorphism H⊗ `2N ∼= HS(H, `2N). By considering a =

∑ni=1 b

∗i ai ∈ L1(B(H)),

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it then follows that for all x ∈ B(H) we have

Tr(xa) =n∑i=1

〈aix, bi〉2

=n∑i=1

〈(id⊗ 1)(x)ξi, ηi〉 = ϕ(x).

Corollary 12.5. The unit ball in B(H) is compact in the σ-WOT.

Corollary 12.6. The WOT and the σ-WOT agree on bounded sets.

Proof. The identity map is clearly continuous from the σ-WOT to the WOT. Since both spaces areHausdorff it follows that this is a homeomorphism from the σ-WOT compact unit ball in B(H).By scaling we therefore have that this is a homeomorphism on any bounded set.

Exercise 12.7. Show that the adjoint T 7→ T ∗ is continuous in the WOT, and when restricted tothe space of normal operators is continuous in the SOT, but is not continuous in the SOT on thespace of all bounded operators.

Exercise 12.8. Show that operator composition is jointly continuous in the SOT on boundedsubsets.

Exercise 12.9. Show that the SOT agrees with the σ-SOT on bounded subsets of B(H).

Exercise 12.10. Show that pairing 〈x, a〉 = Tr(a∗x) gives an identification between K(H)∗ and(L1(B(H)), ‖ · ‖1).

12.3 Von Neumann algebras and the double commutant theorem

A von Neumann algebra (over a Hilbert space H) is a ∗-subalgebra of B(H) which contains 1and is closed in the weak operator topology.

Note that since subalgebras are of course convex, it follows from Corollary 12.3 that von Neu-mann algebras are also closed in the strong operator topology.

If A ⊂ B(H) then we denote by W ∗(A) the von Neumann subalgebra which is generated by A,i.e., W ∗(A) is the smallest von Neumann subalgebra of B(H) which contains A.

Lemma 12.11. Let A ⊂ B(H) be a von Neumann algebra. Then (A)1 is compact in the WOT.

Proof. This follows directly from Corollary 12.5.

Corollary 12.12. Let A ⊂ B(H) be a von Neumann algebra, then (A)1 and As.a. are closed in theweak and strong operator topologies.

Proof. Since taking adjoints is continuous in the weak operator topology it follows that As.a. isclosed in the weak operator topology, and by the previous result this is also the case for (A)1.

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If B ⊂ B(H), the commutant of B is

B′ = {T ∈ B(H) | TS = ST, for all S ∈ B}.

We also use the notation B′′ = (B′)′ for the double commutant.

Theorem 12.13. Let A ⊂ B(H) be a self-adjoint set, then A′ is a von Neumann algebra.

Proof. It is easy to see that A′ is a self-adjoint algebra containing 1. To see that it is closed in theweak operator topology just notice that if xα ∈ A′ is a net such that xα → x ∈ B(H) then for anya ∈ A, and ξ, η ∈ H, we have

〈[x, a]ξ, η〉 = 〈xaξ, η〉 − 〈xξ, a∗η〉

= limα→∞

〈xαaξ, η〉 − 〈xαξ, a∗η〉 = limα→∞

〈[xα, a]ξ, η〉 = 0.

Corollary 12.14. A self-adjoint maximal abelian subalgebra A ⊂ B(H) is a von Neumann algebra.

Proof. Since A is maximal abelian we have A = A′.

Lemma 12.15. Suppose A ⊂ B(H) is a self-adjoint algebra containing 1. Then for all ξ ∈ H, andx ∈ A′′ there exists xα ∈ A such that limα→∞ ‖(x− xα)ξ‖ = 0.

Proof. Consider the closed subspace K = Aξ ⊂ H, and denote by p the projection onto thissubspace. Since for all a ∈ A we have aK ⊂ K, it follows that ap = pap. But since A is self-adjointit then also follows that for all a ∈ A we have pa = (a∗p)∗ = (pa∗p)∗ = pap = ap, and hence p ∈ A′.

We therefore have that xp = xp2 = pxp and hence xK ⊂ K. Since 1 ∈ A it follows that ξ ∈ Kand hence also xξ ∈ Aξ.

Theorem 12.16 (Von Neumann’s double commutant theorem). Suppose A ⊂ B(H) is a self-adjoint algebra containing 1. Then A′′ is equal to the weak operator topology closure of A.

Proof. By Theorem 12.13 we have that A′′ is closed in the weak operator topology, and we clearlyhave A ⊂ A′′, so we just need to show that A ⊂ A′′ is dense in the weak operator topology. Forthis we use the previous lemma together with a matrix trick.

Let ξ1, . . . , ξn ∈ H, x ∈ A′′ and consider the subalgebra A of B(Hn) ∼= Mn(B(H)) consistingof diagonal matrices with constant diagonal coefficients contained in A. Then the diagonal matrixwhose diagonal entries are all x is easily seen to be contained in A′′, hence the previous lemmaapplies and so there exists a net aα ∈ A such that limα→∞ ‖(x − aα)ξk‖ = 0, for all 1 ≤ k ≤ n.This shows that A ⊂ A′′ is dense in the strong operator topology.

We also have the following formulation which is easily seen to be equivalent.

Corollary 12.17. Let A ⊂ B(H) be a self-adjoint algebra. Then A is a von Neumann algebra ifand only if A = A′′.

Corollary 12.18. Let A ⊂ B(H) be a von Neumann algebra, x ∈ A, and consider the polardecomposition x = v|x|. Then v ∈ A.

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Proof. Note that ker(v) = ker(|x|), and if a ∈ A′ then we have a ker(|x|) ⊂ ker(|x|). Also, we have

‖(av − va)|x|ξ‖ = ‖axξ − xaξ‖ = 0,

for all ξ ∈ H. Hence av and va agree on ker(|x|) +R(|x|) = H, and so v ∈ A′′ = A.

Proposition 12.19. Let (X,µ) be a σ-finite measure space. Consider the Hilbert space L2(X,µ),and the map M : L∞(X,µ) → B(L2(X,µ)) defined by (Mgξ)(x) = g(x)ξ(x), for all ξ ∈ L2(X,µ).Then M is an isometric ∗-isomorphism from L∞(X,µ) onto a maximal abelian von Neumannsubalgebra of B(L2(X,µ)).

Proof. The fact that M is a ∗-isomorphism onto its image is clear. If g ∈ L∞(X,µ) then bydefinition of ‖g‖∞ we can find a sequence En of measurable subsets of X such that 0 < µ(En) <∞,and |g||En ≥ ‖g‖∞ − 1/n, for all n ∈ N. We then have

‖Mg‖ ≥ ‖Mg1En‖2/‖1En‖2 ≥ ‖g‖∞ − 1/n.

The inequality ‖g‖∞ ≤ ‖Mg‖ is also clear and hence M is isometric.To see that M(L∞(X,µ)) is maximal abelian let’s suppose T ∈ B(L2(X,µ)) commutes with

Mf for all f ∈ L∞(X,µ). We take En ⊂ X measurable sets such that 0 < µ(En) <∞, En ⊂ En+1,and X = ∪n∈NEn. Define fn ∈ L2(X,µ) by fn = T (1En).

For each g, h ∈ L∞(X,µ) ∩ L2(X,µ), we have

|∫fnghdµ| = |〈MgT (1En), h〉| = |〈T (g|En), h〉| ≤ ‖T‖‖g‖2‖h‖2.

Since L∞(X,µ) ∩ L2(X,µ) is dense in L2(X,µ), it then follows from Holder’s inequality thatfn ∈ L∞(X,µ) with ‖fn‖∞ ≤ ‖T‖, and that M1EnT = Mfn . Note that for m ≥ n, 1Emfn =1EmT (1En) = T (1En) = fn. Hence, {fn} converges almost every where to a measurable function f .Since ‖fn‖∞ ≤ ‖T‖ for each n, we have ‖f‖∞ ≤ ‖T‖. Moreover, if g, h ∈ L2(X,µ) then we have∫

fghdµ = limn→∞

∫fnghdµ = lim

n→∞〈1EnT (g), h〉 = 〈T (g), h〉.

Thus, T = Mf .

Because of the previous result we will often identify L∞(X,µ) with the subalgebra of B(L2(X,µ))as described above. Conversely, every abelian von Neumann algebra A is isomorphic to L∞(X,µ)for some measure space (X,µ), if A is separable in the strong operator topology then (X,µ) maybe taken be taken to be a standard probability space.

Exercise 12.20. Let X be an uncountable set, B1 the set of all subsets of X, B2 ⊂ B1 the setconsisting of all sets which are either countable or have countable complement, and µ the countingmeasure on X. Show that the identity map implements a unitary operator id : L2(X,B1, µ) →L2(X,B2, µ), and we have L∞(X,B2, µ) ( L∞(X,B2, µ)′′ = idL∞(X,B1, µ) id∗.

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12.4 The spectral theorem and Borel functional calculus

Lemma 12.21. Let xα ∈ B(H) be an increasing net of positive operators such that supα ‖xα‖ <∞,then there exists a bounded operator x ∈ B(H) such that xα → x in the SOT.

Proof. We may define a quadratic form on H by ξ 7→ limα ‖√xαξ‖2. Since supα ‖xα‖ < ∞ we

have that this quadratic form is bounded and hence there exists a bounded positive operatorx ∈ B(H) such that ‖

√xξ‖2 = limα ‖

√xαξ‖2, for all ξ ∈ H. Note that xα ≤ x for all α, and

supα ‖(x− xα)1/2‖ <∞. Thus for each ξ ∈ H we have

‖(x− xα)ξ‖2 ≤ ‖(x− xα)1/2‖2‖(x− xα)1/2ξ‖2

= ‖(x− xα)1/2‖2(‖√xξ‖2 − ‖

√xαξ‖2)→ 0.

Hence, xα → x in the SOT.

Corollary 12.22. Let A ⊂ B(H) be a von Neumann algebra. If {pι}ι∈I ⊂ A is a collection ofpairwise orthogonal projections then p =

∑ι∈I pι ∈ A is well defined as a SOT limit of finite sums.

Let K be a locally compact Hausdorff space and let H be a Hilbert space. A spectral measureE on K relative to H is a mapping from the Borel subsets of K to the set of projections in B(H)such that

(i) E(∅) = 0, E(K) = 1.

(ii) E(B1 ∪B2) = E(B1) + E(B2) for all disjoint Borel sets B1 and B2.

(iii) For all ξ, η ∈ H the functionB 7→ Eξ,η(B) = 〈E(B)ξ, η〉

is a finite Radon measure on K.

Example 12.23. If K is a locally compact Hausdorff space and µ is a σ-finite Radon measure onK, then the map E(B) = 1B ∈ L∞(K,µ) ⊂ B(L2(K,µ)) defines a spectral measure on K relativeto L2(K,µ).

We denote by B∞(K) the space of all bounded Borel functions onK. This is clearly a C∗-algebrawith the sup norm.

For each f ∈ B∞(K) it follows that the map

(ξ, η) 7→∫f dEξ,η

gives a continuous sesqui-linear form on H and hence it follows that there exists a bounded operatorT such that 〈Tξ, η〉 =

∫f dEξ,η. We denote this operator T by

∫f dE so that we have the formula

〈(∫f dE)ξ, η〉 =

∫f dEξ,η, for each ξ, η ∈ H.

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Theorem 12.24. Let K be a locally compact Hausdorff space, let H be a Hilbert space, and supposethat E is a spectral measure on K relative to H. Then the association

f 7→∫f dE

defines a continuous unital ∗-homomorphism from B∞(K) to B(H). Moreover, the image of B∞(K)is contained in the von Neumann algebra generated by the image of C(K), and if fn ∈ B∞(K) is anincreasing sequence of non-negative functions such that f = supn fn ∈ B∞, then

∫fndE →

∫fdE

in the SOT.

Proof. It is easy to see that this map defines a linear contraction which preserves the adjointoperation. If A,B ⊂ K are Borel subsets, and ξ, η ∈ H, then denoting x =

∫1AdE, y =

∫1BdE,

and z =∫

1A∩BdE we have

〈xyξ, η〉 = 〈E(A)yξ, η〉 = 〈E(B)ξ, E(A)η〉= 〈E(B ∩A)ξ, η〉 = 〈zξ, η〉.

Hence xy = z, and by linearity we have that (∫f dE)(

∫g dE) =

∫fg dE for all simple functions

f, g ∈ B∞(K). Since every function in B∞(K) can be approximated uniformly by simple functionsthis shows that this is indeed a ∗-homomorphism.

To see that the image of B∞(K) is contained in the von Neumann algebra generated by theimage of C(K), note that if a commutes with all operators of the form

∫f dE for f ∈ C(K) then

for all ξ, η ∈ H we have

0 = 〈(a(

∫f dE)− (

∫f dE)a)ξ, η〉 =

∫f dEξ,a∗η −

∫f dEaξ,η.

Thus Eξ,a∗η = Eaξ,η and hence we have that a also commutes with operators of the form∫g dE

for any g ∈ B∞(K). Therefore by Theorem 12.16∫g dE is contained in the von Neumann algebra

generated by the image of C(K).Now suppose fn ∈ B∞(K) is an increasing sequence of non-negative functions such that f =

supn fn ∈ B∞(K). For each ξ, η ∈ H we have∫fn dEξ,η →

∫f dEξ,η,

hence∫fn dE converges in the WOT to

∫f dE. However, since

∫fn dE is an increasing sequence

of bounded operators with ‖∫fn dE‖ ≤ ‖f‖∞, Lemma 12.21 shows that

∫fn dE converges in the

SOT to some operator x ∈ B(H) and we must then have x =∫f dE.

The previous theorem shows, in particular, that if A is an abelian C∗-algebra, and E is aspectral measure on σ(A) relative to H, then we obtain a unital ∗-representation π : A→ B(H) bythe formula

π(x) =

∫Γ(x)dE.

We next show that in fact every unital ∗-representation arises in this way.

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Theorem 12.25 (The spectral theorem). Let A be an abelian C∗-algebra, H a Hilbert space andπ : A → B(H) a ∗-representation, which is non-degenerate in the sense that ξ = 0 if and only ifπ(x)ξ = 0 for all x ∈ A. Then there is a unique spectral measure E on σ(A) relative to H suchthat for all x ∈ A we have

π(x) =

∫Γ(x)dE.

Proof. For each ξ, η ∈ H we have that f 7→ 〈π(Γ−1(f))ξ, η〉 defines a bounded linear functional onσ(A) and hence by the Riesz representation therorem there exists a Radon measure Eξ,η such thatfor all f ∈ C(σ(A)) we have

〈π(Γ−1(f))ξ, η〉 =

∫fdEξ,η.

Since the Gelfand transform is a ∗-homomorphism we verify easily that fdEξ,η = dEπ(Γ−1(f))ξ,η =dEξ,π(Γ−1(f))η.

Thus for each Borel set B ⊂ σ(A) we can consider the sesquilinear form (ξ, η) 7→∫

1BdEξ,η. Wehave |

∫fdEξ,η| ≤ ‖f‖∞‖ξ‖‖η‖, for all f ∈ C(σ(A)) and hence this sesquilinear form is bounded

and there exists a bounded operator E(B) such that 〈E(B)ξ, η〉 =∫

1BdEξ,η, for all ξ, η ∈ H. Forall f ∈ C(σ(A)) we have

〈π(Γ−1(f))E(B)ξ, η〉 =

∫1BdEξ,π(Γ−1(f))η =

∫1BfdEξ,η.

Thus it follows that E(B)∗ = E(B), and E(B′)E(B) = E(B′ ∩B), for any Borel set B′ ⊂ σ(A). Inparticular, E(B) is a projection and since π is non-degenerate it follows easily that E(σ(A)) = 1,thus E gives a spectral measure on σ(A) relative to H. The fact that for x ∈ A we have π(x) =∫

Γ(x)dE follows easily from the way we constructed E.

If H is a Hilbert space and x ∈ B(H) is a normal operator, then by applying the previoustheorem to the C∗-subalgebra A generated by x and 1, and using the identification σ(A) = σ(x)we obtain a homomorphism from B∞(σ(x)) to B(H) and hence for f ∈ B∞(σ(x)) we may define

f(x) =

∫fdE.

Note that it is straight forward to check that considering the function f(z) = z we have

x =

∫zdE(z).

We now summarize some of the properties of this functional calculus which follow easily fromthe previous results.

Theorem 12.26 (Borel functional calculus). Let A ⊂ B(H) be a von Neumann algebra and supposex ∈ A is a normal operator, then the Borel functional calculus defined by f 7→ f(x) satisfies thefollowing properties:

(i) f 7→ f(x) is a continuous unital ∗-homomorphism from B∞(σ(x)) into A.

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(ii) If f ∈ B∞(σ(x)) then σ(f(x)) ⊂ f(σ(x)).

(iii) If f ∈ C(σ(x)) then f(x) agrees with the definition given by continuous functional calculus.

Exercise 12.27. Suppose that K is a compact Hausdorff space and E is a spectral measure forK relative to a Hilbert space H, show that if {fn}n ⊂ B∞(K) is a uniformly bounded sequence,and f ∈ B∞(K) such that fn(k) → f(k) for every k ∈ K, then

∫fn dE →

∫f dE in the strong

operator topology.

12.5 Locally compact groups

If G is a locally compact group there exists a Haar measure λ, which is a non-zero Radon measureon G satisfying λ(gE) = λ(E) for all g ∈ G, and Borel subsets E ⊂ G. When a Haar measure λ onG is fixed, we’ll often use the notation

∫f(x) dx for the integral

∫f(x) dλ(x).

Any two Haar measures differ by a constant. If h ∈ G, and λ is a Haar measure then E 7→ λ(Eh)is again a Haar measure and hence there is a scalar ∆(h) ∈ (0,∞) such that λ(Eh) = ∆(h)λ(E)for all Borel subsets E ⊂ G. The map ∆ : G → (0,∞) is the modular function, and is easilyseen to be a continuous homomorphism. The group G is unimodular if ∆(g) = 1 for all g ∈ G,or equivalently if a Haar measure for G is right invariant. For example, all abelian groups and allcompact groups are unimodular.

Example 12.28. (i) Lebesgue measure is a Haar measure for Rn, and similarly for Tn.

(ii) Counting measure is a Haar measure for any discrete group.

(iii) Consider the group GLn(R) of invertible matrices. As the determinant of a matrix is givenby a polynomial function of the coefficients of the matrix it follows that GLn(R) is an opendense subset of Mn(R) ∼= Rn2

. If x ∈ GLn(R) then the transformation induced on Mn(R) byleft multiplication has Jacobian given by det(x)n, and hence if λ denotes Lebesgue measureon Mn(R) ∼= Rn2

, then we obtain a Haar measure for GLn(R) by

µ(B) =

∫B

1

|det(X)|ndX.

Since the linear transformation induced by right multiplication has the same Jacobian we seethat this also gives a right Haar measure, and hence GLn(R) is unimodular.

(iv) If G ∼= G1 ×G2, and λi are Haar measures for Gi, then λ1 × λ2 is a Haar measure for G. Forexample, consider the group SLn(R) of matrices with determinant 1. Then SLn(R)CGLn(R),and we also have an embedding of R∗ in GLn(R) as constant diagonal matrices. The subgroupsSLn(R), and R∗ are both closed and normal, and their intersection is {e}, hence we have anisomorphism GLn(R) ∼= R∗×SLn(R). Thus, for suitably chosen Haar measures µ on GLn(R),and λ on SLn(R) we have dµ(tx) = 1

|t|dt dλ(x). Note that SLn(R) is again unimodular.

(v) Consider the group N of upper triangular n× n matrices with real coefficients and diagonalentries equal to 1. We may identify N with Rn(n−1)/2 by means of the homeomorphism

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N 3 n 7→ (nij)1≤i<j≤n ∈ Rn(n−1)/2. Under this identification, Lebesgue measure on Rn(n−1)/2

is a Haar measure on N . Indeed, if n, x ∈ N , then for i < j we have

(nx)ij = nij + xij +∑i<k<j

nikxkj .

If we endow the set of pairs (i, j), 1 ≤ i < j ≤ n with the lexicographical order, then it isclear that the Jacobi matrix corresponding to the transformation x 7→ nx is upper triangularwith diagonal entries equal to 1, and hence the Jacobian of this transformation is 1.

The same argument shows that the transformation n 7→ nx also has Jacobian equal to 1 andhence Lebesgue measure is also right invariant, i.e., N is unimodular.

(vi) Suppose K, and H are locally compact groups with Haar measures dk, and dh respectively.Suppose also that α : K → Aut(H) is a homomorphism which is continuous in the sensethat the map K ×H 3 (k, h) 7→ αk(h) ∈ H is jointly continuous. Then for each k ∈ K, thepush-forward of dh under the transformation αk is again a Haar measure and hence must beof the form δ(k)−1dh, where δ : K → R>0 is a continuous homomorphism.

The semi-direct product of K with H is denoted by K nH. Topologically, it is equal tothe direct product K ×H, however the group law is given by

(k1, h1)(k2, h2) = (k1k2, α−1k2

(h1)h2); k1, k2 ∈ K, h1, h2 ∈ H.

A Haar measure for K nH is given by the product measure dkdh. Indeed, if f ∈ Cc(GnH),and k′ ∈ K, h′ ∈ H then we have∫

f((k′, h′)(k, h)) dkdh =

∫f(k′k, α−1

k (h′)h) dhdk

=

∫f(k′k, h) dkdh =

∫f(k, h) dkdh.

The modular function for K n H is given by ∆KnH(k, h) = δ(k)∆K(k)∆H(h). Indeed, iff ∈ Cc(K nH) then∫

f((k, h)−1)dkdh =

∫f(k−1, αk(h

−1))dhdk

=

∫δ(k)f(k−1, h−1)dhdk

=

∫δ(k)∆K(k)∆H(h)f(k, h)dkdh.

A specific case to consider is when G = A is the group of diagonal matrices in Mn(R) withpositive diagonal coefficients, and H = N is the group of upper triangular matrices in Mn(R)with diagonal entries equal to 1. In this case A acts on N by conjugation and the resultingsemi-direct product can be realized as the group B of upper triangular matrices with positivediagonal coefficients.

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Both N and A are unimodular, and for each a = diag(a1, a2, . . . , an) ∈ A we see from theprevious example that conjugating N by a multiplies the Haar measure on N by a factor ofδ(a) =

∏1≤i<j≤n

aiaj

. Hence, the modular operator for B is given by ∆B(g) =∏

1≤i<j≤ngiigjj

.

In the case when n = 2 we have the group G = {( x y

0 x−1

)| x ∈ R∗+, y ∈ R}, so that G = R∗+nR,

δ : R∗+ → R∗+ is given by δ(x) = x2, and a left Haar measure for G is given by 1xdx dy, while

a right Haar measure is given by 1x3 dx dy.

Theorem 12.29. Let G be a locally compact group and H < G a closed subgroup. Then thereexists a unique measure class on G/H which is quasi-invariant. Moreover, G/H has a σ-finiteinvariant measure if and only if the restriction of ∆G to H agrees with ∆H .

Proof. See Theorem 2.49 in [Fol95].

12.6 Positive definite functions

Let G be a locally compact group with a Haar measure λ. A function ϕ ∈ L∞G is of positivedefinite if for all f ∈ L1G we have

∫ϕ(x)(f∗ ∗ f)(x) dλ(x) ≥ 0. Note that this condition is

weak∗-closed in L∞G so that the space of positive definite functions is weak∗-closed.

Proposition 12.30. Suppose π : G→ U(H) is a continuous representation of G, and ξ0 ∈ H, thenthe function defined by ϕ(x) = 〈π(x)ξ0, ξ0〉 is of positive definite.

Proof. If we consider the associated representation of L1G, then for f ∈ L1G we have∫〈π(x)ξ0, ξ0〉(f∗ ∗ f)(x) dλ(x) = 〈π(f∗ ∗ f)ξ0, ξ0〉

= ‖π(f)ξ0‖2 ≥ 0.

Thus, ϕ is positive definite.

We also have the following converse to Proposition 12.30.

Theorem 12.31 (The GNS-construction). If ϕ ∈ L∞G is positive definite there exists a continuousrepresentation π : G→ U(H), and a vector ξ0 ∈ H, such that ϕ(g) = 〈π(g)ξ0, ξ0〉, for locally almostevery g ∈ G. In particular, every function positive definite agrees with a continuous function locallyalmost everywhere.

Proof. See Theorem 3.20 in [Fol95].

Corollary 12.32. Every positive definite function on G agrees locally almost everywhere with acontinuous positive definite function.

Corollary 12.33. If ϕ ∈ CbG is positive definite then ϕ is uniformly continuous, and the followingstatements hold:

• ‖ϕ‖∞ = f(e);

• ϕ(x−1) = ϕ(x), for all x ∈ G;

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• |ϕ(y−1x)− ϕ(x)|2 ≤ 2ϕ(e)Re(ϕ(e)− ϕ(y)), for all x, y ∈ G.

Proof. Suppose ϕ(x) = 〈π(x)ξ0, ξ0〉 as in the GNS-construction. For the first statement we haveϕ(e) ≤ ‖ϕ‖∞ = supx∈G |〈π(x)ξ0, ξ0〉| ≤ ‖ξ0‖2 = ϕ(e).

For the second statement we have ϕ(x−1) = 〈π(x−1)ξ0, ξ0〉 = 〈ξ0, π(x)ξ0〉 = ϕ(x).For the third statement we have

|ϕ(y−1x)− ϕ(x)|2 = |〈π(x)ξ0, π(y)ξ0 − ξ0〉|2 ≤ ‖ξ0‖2‖π(y)ξ0 − ξ0‖2

= 2‖ξ0‖2(‖ξ0‖2 − Re(〈π(y)ξ0, ξ0〉))= 2ϕ(e)Re(ϕ(e)− ϕ(y)).

Since ϕ is continuous at e, this also shows that ϕ is left uniformly continuous, and right uniformcontinuity then follows since ϕ(x−1) = ϕ(x), for all x ∈ G.

We let P(G) ⊂ L∞G denote the convex cone of positive definite functions, P1(G) = {ϕ ∈P(G) | ϕ(e) = 1}, and P≤1(G) = {ϕ ∈ P(G) | ϕ(e) ≤ 1}. We let E(P1(G)), and E(P≤1(G)) denote,respectively, the extreme points of these last two convex sets.

Theorem 12.34 (Raikov). The weak*-topology on P1(G) agrees with the topology of uniform con-vergence on compact sets.

Proof. See Theorem 3.31 in [Fol95].

12.7 Duality and Bochner’s theorem

Let G be a locally compact abelian group. A (unitary) character on G is a continuous homomor-phism χ : G → T. We denote the set of all characters by G, which by Corollary ?? agrees withE(P1(G)), which is canonically identified with σ(C∗G), and we endow G with the weak*-topology,which by Theorem 12.34 is the same as the topology of uniform convergence on compact sets. Theset of all characters is also a group under pointwise operations, and these operations are clearlycontinuous and hence G is an abelian locally compact group, which is the Pontryagin dual groupof G. If χ ∈ G, and x ∈ G, then we’ll also use the notation 〈x, χ〉 = χ(x).

We leave it to the reader to verify the following.

Example 12.35. • R ∼= R with the pairing 〈x, ξ〉 = e2πixξ.

• Z ∼= T with the pairing 〈n, λ〉 = λn.

• T ∼= Z with pairing 〈λ, n〉 = λn.

• Z/nZ ∼= Z/nZ with the pairing 〈j, k〉 = e2πjki/n.

• If G1, G2, are locally compact abelian groups, then we have G1 ×G2∼= G1 × G2 under the

pairing 〈(x, y), (χ, ω)〉 = 〈x, χ〉〈y, ω〉.

• If G is a finite abelian group then G ∼= G.

44

The Gelfand transform gives an isomorphism C∗G ∼= C0(G), which on the dense subspaceL1G ⊂ C∗G is given by

Γ(f)(χ) =

∫f(x)χ(x) dλ(x).

We introduce the closely related Fourier transform F : L1G → C0(G) given by F(f)(χ) =f(χ) =

∫f(x)χ(x) dλ(x). Note that since χ 7→ χ is a homeomorphism on G it follows that F also

extends continuously to an isomorphism from C∗G onto C0(G).

Theorem 12.36 (Bochner’s Theorem). Let G be a locally compact abelian group. If ϕ ∈ P(G)then there exists a unique positive measure µ ∈M(G) such that ϕ(x) =

∫χ(x)dµ(χ), for all x ∈ G.

Proof. See Theorem 4.18 in [Fol95].

Theorem 12.37 (The Plancherel Theorem). The Fourier transform on L1G∩L2G extends uniquelyto a unitary isomorphism from L2G to L2G.

Proof. See Theorem 4.25 in [Fol95].

Theorem 12.38 (Pontrjagin Duality). The map Φ : G 7→ ˆG given by Φ(x)(χ) = χ(x) gives an

isomorphism of topological groups.

Proof. See Theorem 4.31 in [Fol95].

12.8 Lattices

Let G be a locally compact group. A lattice in G is a discrete subgroup Γ < G such that G/Γ hasa G-invariant probability measure. Note that by Theorem 12.29 only unimodular groups may havelattices.

Fix n ∈ N, and set G = SLn(R). We also set K = SO(n) < G, A the abelian subgroup ofG consisting of diagonal matrices in G with positive diagonal coefficients, and N the subgroup ofupper triangular matrices in G with diagonal entries equal to one. We also set Γ = SLn(Z). Wedenote by {ei}1≤i≤n, the standard basis vectors for Rn. For 1 ≤ i, j ≤ n, i 6= j, we denote byEij ∈ Γ the elementary matrix which has diagonal entries and the ijth entry equal to 1, and allother entries equal to 0.

Proposition 12.39. For every v ∈ Zn \ {0}, there exists γ ∈ SLn(Z) such that γv ∈ Ne1.

Proof. The proposition is trivially true for n = 1. For n = 2, suppose v0 ∈ Z2 \ {0}, multiplyingby −I if necessary we may obtain a vector v1 with at least 1 positive entry, multiplying by anappropriate power of E12, or E21 we obtain a vector v2 = (α1, α2) with both entries non-negative.We now try to minimize these entries as follows: If 0 < α1 ≤ α2 we multiply by E−1

21 , and if0 < α2 < α1 we multiply by E−1

12 . Repeating this procedure we eventually obtain a vector v3 withone positive entry and the other entry equal to 0. Thus, either v3 ∈ Ne1, or v3 ∈ Ne2. In the lattercase, multiplying by

(0 1−1 0

)gives a vector v4 ∈ Ne1.

If n > 2 then for each 1 ≤ i < j ≤ n, we may realize SL2(Z) as the subgroup of SLn(Z)which fixes ek, for k 6= i, j. By considering the embeddings corresponding to (1, j) as j decreasesfrom n to 2, then for v ∈ Zn \ {0} we may inductibely find elements γj ∈ SLn(Z) such thatγ2γ3 · · · γnv ∈ Ne1.

45

Theorem 12.40 (Iwasawa decomposition for SLn(R)). The product map K × A × N → G is ahomeomorphism.

Proof. Fix g ∈ G, and suppose that g has column vectors x1, . . . , xn. Using the Gram-Schmidt pro-cess we inductively construct unit orthogonal vectors y1, . . . , yn by setting yi = xi−

∑1≤j<i〈yj , xj〉yj ,

and then yi = yi/‖yi‖, for 1 ≤ i ≤ n.We may then consider the orthogonal transformation k ∈ O(n), such that kei = yi. It is then

easy to check that k−1g is an upper triangular matrix with diagonal entries equal to ‖yi‖ > 0. Notethat we have k ∈ SO(n), since det(k−1) = det(k−1g) =

∏ni=1 ‖yi‖ 6= −1.

If a = diag(‖y1‖, ‖y2‖, . . . , ‖yn‖) then we have g = kan where n ∈ N . This association is clearlycontinuous and it is easy to see that it is an inverse to the product map K ×A×N → G. Hence,this must be a homeomorphism.

Given the decomposition above, it is then natural to see how Haar measures for K, A, and Nrelate to Haar measures for G.

Theorem 12.41. Suppose dk, da, and dn, are Haar measure for K, A, and N respectively. Then,with respect to the Iwasawa decomposition, a Haar measure for G is given by dg = δ(a)dk da dn,where δ(a) =

∏1≤i<j≤n

aiaj

, for a = diag(a1, a2, . . . , an) ∈ A.

Proof. From Example 12.28, we see that a right Haar measure for B = AN is given by δ(a) dadn.Hence a right Haar measure for K ×B is given by δ(a) dkdadn.

We denote by dx a measure on K × B which is obtained from a Haar measure on G throughthe homeomorphism K × B 3 (k, b) 7→ kb ∈ G. Since G is unimodular it follows that dx is rightinvariant under the actions of B, and left invariant under the action of K. Hence ψ∗dx is a rightinvariant measure on K × B, where ψ : K × B → K × B is given by ψ(k, b) = (k−1, b), andhence must be a scalar multiple of δ(a) dkdadn. But K is unimodular, and hence it follows thatdx = ψ∗dx is a scalar multiple of δ(a) dkdadn.

For t > 0 we let At be the subset of A given by diagonal matrices a such that aii/a(i+1)(i+1) ≤ t,for all 1 ≤ i ≤ n − 1. For u > 0 we let Nu be the subset of N consisting of those matrixes (nij)such that |nij | ≤ u, for all 1 ≤ i < j ≤ n. Note that Nu is compact. A Segal set in SLnR is a setof the form Σt,u = KAtNu.

Lemma 12.42. We have N = N1/2(N ∩ Γ).

Proof. We will prove this by induction on n. Note that for n ∈ {1, 2} this is easy. If n > 2, andu ∈ N , then we have u =

(1 ∗0 u0

), where u0 ∈ SLn−1(R) is upper triangular with diagonal entries

equal to one. Thus, by induction there exists γ0 ∈ SLn−1(Z) upper triangular with diagonal entriesequal to one, such that the non-diagonal entries in u0γ0 have magnitude at most 1/2. We may thenwrite u

(1 00 γ0

)=(

1 x0 u0γ0

).

If we take y ∈ Zn−1, such that y + x has entires with magnitude at most 1/2, then we haveu(

1 00 γ0

) (1 y0 I

)∈ N1/2.

Lemma 12.43. If g ∈ AN such that ‖ge1‖ ≤ ‖gv‖, for all v ∈ Zn \ {0}, then g11/g22 ≤ 2/√

3.

46

Proof. Suppose g ∈ AN is as above. Note that since N stabilizes e1, if γ ∈ Γ ∩N , then gγ againsatisfies ‖gγe1‖ ≤ ‖gγv‖, for all v ∈ Zn \ {0}. Also, g and gγ have the same diagonal entries ifγ ∈ Γ ∩N . Thus, from the previous lemma it is enough to consider the case g = an, with a ∈ A,and n ∈ N1/2.

In this case we have ge1 = a11e1, and ge2 = a11n12e1 + a22e2, with |n12| ≤ 1/2. Hence,

a211 = ‖ge1‖2 ≤ ‖ge2‖2 = a2

11|n12|2 + a222 ≤ a2

11/4 + a222.

Hence, 34g

211 = 3

4a211 ≤ a2

22 = g222.

Theorem 12.44. For t ≥ 2/√

3, and u ≥ 1/2 we have G = Σt,uΓ.

Proof. By Lemma 12.42 it is enough to show G = KA2/√

3NΓ, which we will do by induction, withthe case n = 1 being trivial.

Assume therefore that n > 1, and this holds for n − 1. We fix g ∈ G. Since g(Zn \ {0}) isdiscrete, there exists v0 ∈ Zn \ {0} which achieves the minimum of {‖gv‖ | v ∈ Zn \ {0}}. Notethat we cannot have v0 = αv, for some v ∈ Zn \ {0}, and α ∈ Z, unless α = ±1. Hence, byProposition 12.39 there exists γ ∈ Γ such that γe1 = v0.

We consider the Iwasawa decomposition gγ = kan, and write an =(λ ∗0 λ−1h0

), where h0 ∈

SLn−1(R). By the induction hypothesis there then exists k0 ∈ SO(n − 1), and γ0 ∈ SLn−1(Z)such that k−1

0 h0γ0 is upper triangular and whose positive diagonal entries {ai,i}i=1,n−1 satisfyai,i/ai+1,i+1 ≤ 2/

√3, for 1 ≤ i ≤ n− 2.

Thus, if we consider g =(

1 00 k−1

0

)k−1gγ

(1 00 γ0

), then we see that g ∈ AN , and the diagonal

entries of g satisfy gi,i/gi+1,i+1 ≤ 2/√

3 for all 2 ≤ i ≤ n − 1. Thus, to finish the proof it sufficesto show that we also have g1,1/g2,2 ≤ 2/

√3. However, this follows from Lemma 12.43 since for all

v ∈ Zn \ {0} we have

‖ge1‖ = ‖gγ(

1 00 γ0

)e1‖ = ‖gγe1‖ = ‖gv0‖

≤ ‖gγ(

1 00 γ0

)v‖ = ‖gv‖.

Theorem 12.45. SLn(Z) is a lattice in SLn(R).

Proof. By the previous theorem it suffices to show that Σt,u has finite Haar measure for t = 2/√

3,and u = 1/2. By Theorem 12.41 a Haar measure for G is given by dg = δ(a)dk da dn, whereδ(a) =

∏1≤i<j≤n

aiaj

for a = diag(a1, . . . , an). Hence,∫

Σt,udg = (

∫K dk)(

∫Atδ(a) da)(

∫Nu

du).

Note that since K and Nu are compact, this integral is finite if and only if the integral∫Atδ(a) da

is finite.Consider the isomorphism Rn−1 → A given by

(x1, x2, . . . , xn−1) 7→ λdiag(ex1+x2+···+xn−1 , ex2+···+xn−1 , . . . , exn−1 , 1),

where λn =∏n−1i=1 e

−ixi . We then have an explicit Haar measure for A given by the push forwardof Lebesgue measure on Rn−1. Moreover the preimage of At under this map is given by E =(−∞, log(2/

√3)]n−1. Thus, we may compute directly∫At

δ(a) da =

∫E

∏1≤i≤j≤n−1

exi+···+xj dx1 · · · dxn−1 =

n−1∏i=1

∫ log(2/√

3)

−∞ebix dx,

47

where bi are positive integers. Hence,∫Atδ(a) da <∞.

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