Proofs of the Main Results

IV. ON THE STRUCTURE OF II 1 FACTORS OF NEGATIVELY CURVED GROUPS

IV.4 Proofs of the Main Results

t→0limkϕ_i(u)ξ_tϕ_i(u^∗)−ξ_tk

= lim

t→0k(1−e)(ϕ_i(u)ζ_tϕ_i(u^∗)−ζ_t)k

≤lim

t→0kϕ_i(u)J ϕ_i(u)J(ζ_t)−ζ_tk

≤lim

t→0kα_t(ϕ_i(u))α_t(J ϕ_i(u)J)(ζ_t)−ζ_tk+ 2 lim

t→0k(α_−t(ϕ_i(u))−ϕ_i(u))ˆx_tk

≤lim

t→0kV_t(ϕ_i(u)x_tϕ_i(u^∗)−x_t)k+ 2 lim

t→0k(α_t(ϕ_i(u))−ϕ_i(u))◦ek∞

≤lim

t→0kϕ_i(u)xtϕi(u^∗)−xtk₂

≤lim

t→0kux_tu^∗−xtk₂+ 2 lim

t→0kx_tk∞kϕ_i(u)−uk₂

≤2kϕ_i(u)−uk₂

(IV.4.2)

Given ε > 0, let us choose i ∈ I such that P

u∈Fkϕ_i(u)−uk₂ ≤ ₄^ε. It then follows from the calculation above that there exists t >0 such that

kX

u∈F

u⊗uk¯ _∞≥ kP

u∈Fuξtu^∗k kξ_tk

≥ kP

u∈Fϕ_i(u)ξ_tϕ_i(u^∗)k

kξ_tk ≥ |F| −ε Hence, by Haagerup’s criterion we have that N is amenable, a contradiction.

Proof of Theorem B. The proof follows the proof of Theorem B in [61]. Let P ⊂ LΓ = M be a diffuse amenable von Neumann subalgebra, N_M(P) = {u ∈ U(M) : uP u^∗ = P}, N = N_M(P)⁰⁰, and fix p∈N⁰∩M a projection. Since Γ is weakly amenable, by Theorem B in [58], we have that P is weakly compact inM. That is, there exists a net of unit vectors (ηn)n∈N inL²(M)⊗L²( ¯M) such that:

1. kη_n−(v⊗¯v)η_nk →0, for allv∈ U(P);

2. k[u⊗u, η¯ _n]k →0, for allu∈ N_M(P); and

3. h(x⊗1)ηn, ηni=τ(x) =h(1⊗x)η¯ n, ηni, for all x∈M.

Fix t > 0 and denote by ˜η_n,t = (V_t⊗1)(p⊗1)η_n ∈ H ⊗L²(M), ζ_n,t = (e⊗1)˜η_n,t = (e·V_t⊗ 1)(p⊗1)ηn, and ξn,t= ˜ηn,t−ζn,t. We will begin by showing that

Limn kξ_n,tk ≥ 5

12kpk₂. (IV.4.3)

Using the triangle inequality, we have that

k˜ηn,t−(e◦αt(v)⊗v)ζ¯ n,tk ≤ k˜ηn,t−(e◦αt(v)⊗v)˜¯ ηn,tk₂+kξ_n,tk

≤ kζ_n,t−(e◦α_t(v)⊗v)˜¯ η_n,tk+ 2kξ_n,tk

≤ kη˜n,t−(αt(v)⊗v)˜¯ ηn,tk+ 2kξ_n,tk

≤ kη_n−(v⊗v)η¯ _nk+ 2kξ_n,tk for all v∈ U(P) and all n.

Consequently, since by (3) we have k˜η_n,tk=kpk₂, then using the triangle inequality again we get

k(e◦αt(v)⊗v)ζ¯ n,tk ≥ kpk₂−2kξ_n,tk − kη_n−(v⊗v)η¯ nk. (IV.4.4) Since the operator e·Vt is compact on L²(M), one can find a finite set F ⊂ Γ such that k(P_F ⊗1)(e·V_t⊗1)−e·V_t⊗1k_∞ ≤ ^kpk₆²; here P_F denotes the orthogonal projection on the span of F. Hence using the formulae◦α_t=m_ttogether (IV.4.4) and triangle inequality we obtain

k(m_t(v)⊗1)(P_F ⊗1)ζ_n,tk ≥ 5

6kpk₂−2kξ_n,tk − kη_n−(v⊗¯v)η_nk, (IV.4.5) for all v∈ U(P) and all n∈N.

On the other hand, consider the decomposition (p⊗1)ηn=P

g,hµⁿ_g,hδg⊗δh withµⁿ_g,h scalars.

Applying the formula for the multiplierm_t together with Cauchy-Schwartz inequality, we obtain

k(m_t(v)⊗1)(P_F ⊗1)ζ_n,tk²

=X

h

kX

g∈F

e^−t2kq(g)k2µⁿ_g,hm_t(v)δgk²

≤X

h



 X

g∈F

e^{−2t2kq(g)k2}|µⁿ_g,h|²







 X

g∈F

km_t(v)δ_gk²





≤ k(p⊗1)η_nk²



 X

g∈F

km_t(v)δ_gk²



=kpk²₂X

g∈F

km_t(v)δ_gk²,

(IV.4.6)

for all v ∈ U(P) and all n ∈N. Thus, combining (IV.4.5) with (IV.4.6) and taking an arbitrary Banach limit Limn, by relation (1), we obtain that



kpk²₂X

g∈F

km_t(v)δ_gk²





1 2

≥ 5

6kpk₂−2 Lim

n kξ_n,tk, (IV.4.7)

for allv ∈ U(P). This shows that the limit Lim_nkξ_n,tk ≥ ₁₂⁵ kpk₂. Indeed, since P is diffuse there exists a sequence of unitariesv_k ∈ U(P) which converges weakly to 0. Then by Proposition IV.3.8 the infimum achieved by the left side of (IV.4.7) is 0, and we are done.

LetH=L²₀(X)⊗`²(Γ) which as we remarked before is weakly contained as anM-bimodule in the coarse bimodule. Following the same argument as in Theorem B of [61] we define a stateψt on N =B(H)∩ρ(M^op)⁰. Explicitly, ψ_t(x) = Lim_nkξ¹

n,tk²h(x⊗1)ξ_n,t, ξ_n,ti for every x∈ N. Next we prove the following technical result

Lemma IV.4.2. For everyε >0 and every finite set K ⊂C_λ^∗(Γ) withdist_k·k2(y,(N)₁)≤ε for all y∈K one can findt_ε>0 and a finite setL_K,ε ⊂ N_M(P) such that

|h((yx−xy)⊗1)ξn,t, ξn,ti| ≤4ε+ 2 X

v∈L_K,ε

k[v⊗¯v, ηn]k, (IV.4.8)

for all y∈K, kxk_∞≤1, t_ε> t >0, and n.

Proof. Fixε >0 andy∈K. SinceN =N_M(P)⁰⁰ by the Kaplansky density theorem there exists a

finite setF_y ={v_i} ⊂ N_M(P) and scalars µ_i such thatkP

iµ_iv_ik_∞≤1 and ky−X

i

µivik₂ ≤ε. (IV.4.9)

Also using Proposition IV.3.6 one can find a positive numbert_ε>0 such that for all t_ε > t >0

ky−α−t(y)k∞≤ε. (IV.4.10)

Next we will proceed in several steps to show inequality (IV.4.8). First we fixt_ε> t >0. Then, using the triangle inequality in combination with

kxk_∞≤1, (IV.4.10), and the M-bimodularity of 1−e=e^⊥, we see that

|h(x⊗1)ξn,t,(y^∗⊗1)ξn,ti − h(xy⊗1)ξn,t, ξn,ti|

≤ kα_−t(y^∗)−y^∗k_∞+|h(x⊗1)ξ_n,tε,(e^⊥V_tεy^∗p⊗1)η_ni − h(xy⊗1)ξ_n,t, ξ_n,ti|

≤ε+|h(x⊗1)ξn,t,(e^⊥Vty^∗p⊗1)ηni − h(xy⊗1)ξn,t, ξn,ti|

Furthermore, Cauchy-Schwartz inequality together with (3) and (IV.4.9) allow us to see that the last quantity above is smaller than

≤ε+k((y^∗p−X

i

¯

µiv^∗_i)p⊗1)ηnk+|X

i

µih(xy⊗1)ξn,t,(e^⊥Vtv^∗_ip⊗1)ηni − h(xy⊗1)ξn,t, ξn,ti|

≤2ε+|X

i

µ_ih(x⊗v¯_i^∗)ξ_n,t,(e^⊥V_tpv_i^∗⊗¯v^∗_i)η_ni − h(xy⊗1)ξ_n,t, ξ_n,ti|

Applying successively the triangle inequality and using (IV.4.9),vibeing a unitary, and (IV.4.10) we have that the previous quantity is smaller than

≤2ε+X

i

k[v_i^∗⊗¯v^∗_i, η_n]k+|X

i

µ_ih(x⊗¯v^∗_i)ξ_n,t, ξ_n,t(v^∗_i ⊗v¯_i^∗)i − h(xy⊗1)ξ_n,t, ξ_n,ti|

≤2ε+X

i

k[v_i^∗⊗¯v^∗_i, η_n]k+|X

i

µ_ih(x⊗¯v^∗_i)ξ_n,t(v_i⊗¯v_i), ξ_n,ti − h(xy⊗1)ξ_n,t, ξ_n,ti|

≤2ε+ 2X

i

k[v_i⊗v¯i, ηn]k+|h(xe^⊥Vt(X

i

µivi)p⊗1)ηn, ξn,ti − h(xy⊗1)ξn,t, ξn,ti|

≤2ε+k((up−X

i

µivi)p⊗1)ηnk+ 2X

i

k[v_i⊗¯vi, ηn]k+

+|h(xe^⊥V_typ⊗1)η_n, ξ_n,ti − h(xy⊗1)ξ_n,t, ξ_n,ti|

≤3ε+ 2X

i

k[v_i⊗v¯i, ηn]k+ky−a−t(y)k∞≤4ε+ 2X

i

k[v_i⊗v¯i, ηn]k

In conclusion, (IV.4.8) follows from the previous inequalities by taking L_K,ε =∪_y∈KF_y. Lemma IV.4.3. For every ε >0 and F0 ⊂ U(N) finite set there exist F0 ⊂F ⊂M finite set, a c.c.p. map ϕ_F,ε:span(F)→C_λ^∗(Γ), and t_ε>0 such that

|ψ_tε(ϕ_F,ε(up)^∗xϕ_F,ε(up))−ψtε(x)| ≤47ε, (IV.4.11)

for all u∈F₀ and kxk∞≤1.

Proof. Fix ε >0. Denote by F = {up, u^∗p} ∪F0∪F₀^∗ and E =span(F) and by local reflexivity, we may choose a c.c.p. mapϕF,ε:E →C^∗_λ(Γ) such that for allu∈F

kϕ_F,ε(up)−upk₂ ≤ε. (IV.4.12)

This shows in particular that distk·k₂(ϕF,ε(up),(N)1) ≤ ε for all u ∈ F. Therefore applying the previous lemma for K = {ϕ_F,ε(up) | u ∈ F} ⊂ C_λ^∗(Γ) there exists a t_ε > 0 and a finite set K⁰ ⊂ N_M(P) such that for all u∈F, all kxk_∞≤1, and alln we have

|h((ϕ_F,ε(up)^∗xϕF,ε(up)−xϕF,ε(up)ϕF,ε(up)^∗)⊗1)ξn,tε, ξn,tεi| ≤4ε+ 2 X

v∈K⁰

k[v⊗v, η¯ n]k (IV.4.13)

Also using Proposition IV.3.6, after shrinking t_ε if necessary, we can assume in addition that for all u∈F we have

kϕ_F,ε(up)−α−tε(ϕF,ε(up))k_∞≤ε. (IV.4.14) Hence, using triangle inequality together with (IV.4.13) and Cauchy-Schwartz inequality, we have that

|h(ϕ_F,ε(up)^∗xϕ_F,ε(up)⊗1)ξ_n,tε, ξ_n,tεi − h(x⊗1)ξ_n,tε, ξ_n,tεi|

≤4ε+ 2X

i

k[v⊗v, η¯ n]k+|h(x(ϕ_F,ε(up)ϕF,tε(up)^∗−1)⊗1)ξn,tε, ξn,tei

≤4ε+ 2X

v

k[v⊗v, η¯ n]k+kxk_∞k(ϕ_F,ε(up)ϕF,tε(up)^∗−1)⊗1)ξn,tεk

≤4ε+ 2X

v

k[v⊗v, η¯ _n]k+ 2kα_−tε(ϕ_F,ε(up))−ϕ_F,ε(up)k_∞ +k(V_tε(ϕF,ε(up)ϕF,ε(up)^∗−1)p⊗1)ηnk

Furthermore, using (IV.4.12) together with Cauchy-Schwartz inequality, (3) and (IV.4.14) we see that the last quantity above is smaller than

≤6ε+ 2X

v

k[v⊗v, η¯ n]k+k(ϕ_F,ε(up)ϕF,ε(up)^∗−1)p⊗1)ηnk

≤6ε+ 2X

v

k[v⊗v, η¯ _n]k+kϕ_F,ε(up)ϕ_F,ε(up)^∗−pk₂

≤6ε+ 2X

v

k[v⊗v, η¯ n]k+ 2kϕ_F,ε(up)−upk₂

≤8ε+ 2X

v

k[v⊗v, η¯ _n]k.

Altogether the above sequence of inequalities shows that

|h(ϕ_F,ε(up)^∗xϕF,ε(up)⊗1)ξn,tε, ξn,tεi − h(x⊗1)ξn,tε, ξn,tεi| ≤8ε+ 2 X

v∈K⁰

k[v⊗v, η¯ n]k,

and combining this with (2) and (IV.4.3) we obtain

|ψ_tε(ϕ_F,ε(up)^∗xϕ_F,ε(up))−ψ_tε(x)| ≤Lim

n

8ε+ 2P

vk[v⊗v, η¯ n]k kξ_n,tεk²

≤ 8ε

5 12

2 <47ε, which finishes the proof.

For the remaining part of the proof we mention that one can use Haagerup criterion to show that N is amenable. In fact the reasoning in Theorem B in [61] applies verbatim in our case and we leave the details to the reader.

Proof of Corollary C. In the case that Γ is hyperbolic, a result of Ozawa shows that Γ is weakly amenable [56]. In the case that Γ is a lattice in Sp(n,1), choose a co-compact lattice Λ<Sp(n,1).

We have that Λ is Gromov hyperbolic; hence, by [43] Λ belongs to the class QH_reg. A result of Shalom (Theorem 3.7 in [91]) shows that Γ<Sp(n,1) is integrable, thus`¹-measure equivalent to Λ. This implies that Γ ∈ QH_reg. The work of Cowling and Haagerup [20] shows that Sp(n,1) is weakly amenable, which implies, by an unpublished result of Haagerup, that any lattice in Sp(n,1) is also weakly amenable. Therefore, the hypotheses of Theorem B are satisfied.

Proof of Theorem D. Let M = LΓ = LΓ1⊗ · · · ⊗LΓ_n, Mi = LΓi, and for each Γi, 1 ≤ i≤ n, let qi : Γi → H_i be a proper array for some weakly-`<sup>2</sup> unitary representation (H<sub>i</sub>, πi). LetV<sub>t</sub><sup>i</sup>∈ U(H<sup>i</sup>), whereH<sup>i</sup> =L<sup>2</sup>(Xi)⊗`²(Γi), be the one-parameter family of unitaries as defined in§IV.3.3 with the attendant family of automorphismsαⁱ_tofB(Hⁱ). Let us denote by ˜Hⁱthe Hilbert spaceHⁱ⊗L²( ˆMi) with the natural M–M bimodular structure. We extendV_tⁱ to the unitary ˜V_tⁱ =V_tⁱ⊗1∈ U( ˜Hⁱ).

Define F to be the net of finite sets of unitaries in U(N). Note that by local reflexivity, for each ε > 0 and F ∈ F, there exists a c.c.p. map ϕ_F,ε : E → C^∗_λ(Γ), where E is the spanned by {1} ∪F∪F^∗, such thatP

u∈Fkϕ_F,ε(u)−uk₂≤ε.

To begin, suppose thatB ⊂LΓ is a II₁ subfactor whose relative commutant N =B⁰∩LΓ is a non-amenable factor. Let’s assume that by way of contradiction, for all 1≤i≤n,kV˜_tⁱ(ˆx)−xkˆ ₂ does not converge uniformly ast→0, wherex ranges in (B)1. By the proof of Theorem A, there would exist c > 0 such that, for every ε > 0, every finite subset F ⊂ U(N), and every 1≤i ≤n, there

would be a vectorξ_F,εⁱ ∈H˜ⁱ satisfying: (1)kξ_F,εⁱ k₂ ≥cand (2)P

u∈Fkϕ_F,ε(u)ξ_F,εⁱ ϕ_F,ε(u^∗)−ξ_F,εⁱ k₂≤ ε. SettingM_i =B( ˜Hⁱ)∩(M^o)⁰, we define a stateψ_F,εⁱ on M_i by

ψ_F,εⁱ (x) = 1

kξ_F,εⁱ k²₂hxξ_F,εⁱ , ξ_F,εⁱ i (IV.4.15) An easy computation shows that |ψ_F,εⁱ (x)−ψ_F,εⁱ (ϕF,ε(u)xϕF,ε(u^∗))| ≤c⁻¹εkxk∞, for all x ∈ M_i, u∈F. Sinceψⁱ_F,ε∈(M_i)^∗, it belongs to the closed convex hull of states of the formσ_η(x) =hxη, ηi, where η ∈ L²(M_i). Hence, by the generalized Powers–Størmer inequality (cf. §2.1 of [60]) we conclude that

ε→0limkX

u∈F

ϕ_F,ε(u) ⊗ ϕ_F,ε(u)k_L2(M_i),∞=|F| (IV.4.16)

Canonically identifying L²(M_i) with ˜Hⁱ⊗_M H˜ⁱ as in§2 of [92], we have that

kX

u∈F

u ⊗ uk¯ _˜

Hⁱ⊗_MH˜ⁱ,∞≥ lim

ε→0kX

u∈F

ϕF,ε(u) ⊗ ϕF,ε(u)k_L2(Mi),∞=|F|.

Hence, by the generalized Haagerup’s criterion for amenability ([92], Corollary 2.4), we have that H˜ⁱ is a left N-amenable Hilbert M–M bimodule. That is, there exists a net of unital vectors ξ_nⁱ ∈H˜ⁱ⊗_M H˜ⁱ∼= (Hⁱ⊗_M Hⁱ)⊗L²( ˆM_i) such that: (1) hxξⁱ_n, ξⁱ_ni=τ(x) =hξⁱ_nx, ξ_nⁱi, for allx∈M;

and (2)k[y, ξ_nⁱ]k →0, for ally∈N. Now, since ˜H¹⊗_MH˜¹⊗_M · · · ⊗_M H˜ⁿ⊗_M H˜ⁿ may be seen to be weakly contained in L²(M)⊗L²( ¯M), by the proof of Theorem 3.2 in [92] it follows that N is an amenable II1 factor, a contradiction.

Therefore, we must have that kV˜_tⁱ(ˆx)−xkˆ ₂ →0 uniformly on (B)1 for some 1≤i≤n. Since eM◦V˜_tⁱ is ˆMi– ˆMi bimodular and compact when restrictedL²(Mi), an examination of the proof on Theorem 6.2 in [70] shows thatB _M Mˆi. The result then follows by appealing to Proposition 12 of [59].