A path of automorphisms of the extended Roe algebra associated with the

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

IV.3 Deformations of the uniform Roe algebra

IV.3.3 A path of automorphisms of the extended Roe algebra associated with the

Following [65], any group representationπ: Γ→ H_π, gives rise to a measure-preserving action Γy^σ (X, µ) called the Gaussian action. We consider the associated extended Roe algebra C_u^∗(Γy^σ X) and below we indicate a procedure to construct a one-parameter family (αt)t∈Rof∗-automorphisms of C_u^∗(Γ y^σ X). Specifically, αt is obtained by exponentiating an array q : Γ→ H_π in a similar way to the construction of the malleable deformation ofLΓ from a cocyclebas carried out in§3 of [92]. Crucially, this family will be continuous with respect to the uniform norm as t→0 (Lemma IV.3.6).

Following the construction presented in §1.2 of [92], given an array q : Γ→ H_π, there exists a one-parameter family of maps υ_t : Γ → U(L^∞(X, µ)) defined by υ_t(γ)(x) = exp(itq(γ)(x)) where γ ∈Γ,x∈X. Using similar computations as in [65, 92] one can verify that we have the following properties:

Proposition IV.3.4.

Ifπ is weakly-`² then the Koopman representationπσ|L²

0(X,µ) (IV.3.4)

is also weakly-`²; Z

υ_t(γ)(x)υ_t(δ)^∗(x)dµ(x) =κ_t(γ, δ) for all γ, δ∈Γ. (IV.3.5)

These maps give rise naturally to a path of operators V_t∈B(L²(X)⊗`²(Γ)) by letting V_t(ξ⊗ δγ) =υt(γ)ξ⊗δγ for everyξ ∈L²(X) and γ ∈Γ. For further reference we summarize below some basic properties of Vt.

Proposition IV.3.5. For every t, s∈Rwe have the following properties:

1. VtVs =Vt+s,VtV_t^∗ =V_t^∗Vt= 1

2. J VtJ =Vt where J :L²(L^∞(X)oΓ)→L²(L^∞(X)oΓ) is Tomita’s conjugation.

Proof. The first part follows directly from the definitions, so we leave the details to the reader.

To get the second part it suffices to verify that the two operators coincide on vectors of the form ξ⊗δγ ∈ L²(X)⊗`²(Γ). Employing the formulas for J, Vt, υt and then using the fact that q is anti-symmetric we see that

J V_tJ(ξ⊗δ_γ) =J V_t((σ_γ⁻¹(ξ^∗))⊗δ_γ⁻¹)

=J(υ_t(γ⁻¹)σ_γ⁻¹(ξ^∗)⊗δ_γ⁻¹)

= (σγ(υ−t(γ⁻¹))ξ)⊗δγ

= (exp(−itπ_γ(q(γ⁻¹)))ξ)⊗δ_γ

= (exp(itq(γ))ξ)⊗δγ

=Vt(ξ⊗δγ), which finishes the proof.

SinceVtis a unitary onL²(X)⊗`²(Γ), we may consider an inner automorphismαtofB(L²(X)⊗

`<sup>2</sup>(Γ)) by lettingα<sub>t</sub>(x) = V<sub>t</sub>xV<sub>t</sub><sup>∗</sup> for all x∈B(L<sup>2</sup>(X)⊗`²(Γ)). Notice that this forms a family of inner automorphism of the extended Roe algebra. Moreover, when restricting to the uniform Roe algebra one can can recover from α_t the multipliers introduced above: E ◦α_t(x) = m_t(x) for all x∈C_u^∗(Γ).

However, one can see right away that these automorphisms do not move the group von Neumann algebra LΓ into itself. Hence, applying the deformation/rigidity arguments at the level of von Neumann algebra LΓ is rather inadequate. As we will see in the next section, this difficulty is overcome by working with the reduced C^∗-algebra C_λ^∗(Γ) rather than LΓ. The following result

underlines that the path α_t is a deformation at the C^∗-algebraic level i.e., with respect to the operatorial norm.

Lemma IV.3.6. Assuming the notations above, for every x∈C_λ^∗(Γ)we have

k(α_t(x)−x)·ek_∞→0 as t→0; (IV.3.6) k(α_t(J xJ)−J xJ)·ek_∞→0 as t→0, (IV.3.7)

where k · k∞ denotes the operatorial norm in B(L²(X)⊗`<sup>2</sup>(Γ)). Here e denotes the orthogonal projection on `²(Γ).

Proof. Since elements in C_λ^∗(Γ) can be approximated in the uniform norm by finitely supported elements, using the triangle inequality it suffices to show (IV.3.6) only forx=P

g∈Fxgug, a finite sum. Fix an arbitrary ξ =P

γξγ⊗δγ ∈C⊗`²(Γ). Using the formula for αt in combination with Cauchy-Schwartz inequality, we have

k(α_t(x)−x)ξk² =kX

g∈F

X

γ∈Γ

xg(VtugV−t−ug)(ξγ⊗δγ)k²

≤



 X

g∈F

|x_g|²







 X

g∈F

kX

γ∈Γ

(V_tu_gV−t−u_g)(ξ_γ⊗δ_γ)k²





=kxk²₂X

g∈F

kX

γ∈Γ

(VtugV−t−ug)(ξγ⊗δγ)k²

(IV.3.8)

Applying the definitions and the formula forVtwe see thatug(ξγ⊗δ_γ) =ξγ⊗δgγ andVtugV−t(ξγ⊗ δγ) =υt(gγ)σg(υ−t(γ))ξγ⊗δgγ. Therefore, continuing in (IV.3.8) we obtain

=kxk²₂X

g∈F

kX

γ∈Γ

ξγ(υt(gγ)σg(υ−t(γ))−1)⊗δgγk²

=kxk²₂X

g∈F

X

γ∈Γ

|ξ_γ|²kυ_t(gγ)σ_g(υ−t(γ))−1k²

= 2kxk²₂X

g∈F

X

γ∈Γ

|ξ_γ|²(1−τ(υ_t(gγ)σ_g(υ−t(γ)))).

(IV.3.9)

On the other hand, the same computations as in the proof of (IV.3.5) together with inequality

(IV.2.1) imply that, there existK ≥0 such that for everyg∈F and γ ∈Γ we have τ(υ_t(gγ)σ_g(υ−t(γ))) =

Z

X

exp (it(q(gγ)−π_g(q(γ)))(x))dµ(x)

= exp −t²kq(gγ)−π_g(q(γ))k²

≥exp −t²K .

(IV.3.10)

Thus, combining (IV.3.8), (IV.3.9) and (IV.3.10) we conclude that, for all ξ∈C⊗`²(Γ), we have

k(α_t(x)−x)ξk² ≤2kxk²₂kξk²|F| 1−exp −t²K ,

which further implies

k(α_t(x)−x)·ek_∞≤2kxk²₂|F| 1−exp(−t²K)

. (IV.3.11)

Since F is finite, then exp(−t²K)→1 as t→0, andIV.3.6 follows from (IV.3.11).

It remains to show (IV.3.7). Since [e, J] = 0, by Proposition IV.3.5 we see that (αt(J xJ)− J xJ)·e=J((αt(x)−x)·e)J. Therefore, (IV.3.7) follows from (IV.3.6) becauseJ is an anti-linear isometry.

Next we show that the path of unitaries Vtsatisfies a “transversality” property very similar to Lemma 2.1 in [77]. Our proof follows closely the proof of Lemma 3.1 in [98]: we include it here only for the sake of completeness.

Lemma IV.3.7. If V_t is the unitary defined above, then for all ξ ∈ C⊗`²(Γ) and all t ∈ R we have

2kV_t(ξ)−e·Vt(ξ)k² ≥ kξ−Vt(ξ)k². (IV.3.12) Proof. Fix ξ ∈ C⊗`²(Γ) and assume that it can be written as ξ = P

γξ_γ ⊗δ_γ with ξ_γ scalars.

Straightforward computations show thatV_t(x) =P

γξ_γυ_t(γ)⊗δ_hande·V_t(ξ) =P

γξ_γτ(υ_t(γ))1⊗δ_γ;

thus, the left side of (IV.3.12) is equal to

2kV_t(ξ)−e·V_t(ξ)k² = 2 kV_t(ξ)k²− ke·V_t(ξ)k²

= 2X

γ

|ξ_γ|²kυ_t(γ)k²− |ξ_h|²|τ(υt(γ))|²

= 2X

γ

|ξ_γ|² 1− |τ(υ_t(γ))|² .

(IV.3.13)

Applying the same formulas as above, we see that the right side of (IV.3.12) is equal to kξ−Vt(ξ)k² =kξk²+kV_t(ξ)k²−2RehV_t(ξ), ξi

= 2 kξk²−RehV_t(ξ), ξi

= 2X

γ

|ξ_γ|²(1−τ(υt(γ))).

(IV.3.14)

Since we haveτ(υ_t(γ)) = exp(−t²kq(γ)k²)≥exp(−2t²kq(γ)k²) =|τ(υ_t(γ))|², the conclusion follows from (IV.3.13) and (IV.3.14).

The multipliers m_t arising from a proper quasi-cocycle behave in some sense as compact opera- tors on LΓ i.e.,m_t is continuous from the weak operator topology to the strong operator topology on bounded sets. This result will be used crucially in the next section.

Proposition IV.3.8. Let m_t be the Schur multiplier associated to some proper quasi-cocycle on Γ. Ifv_k∈LΓ is a bounded net of elements such thatv_kconverges to vweakly, then for everyt >0 and everyξ ∈L²(Γ) we have that

km_t(vk)ξ−m_t(v)ξk →0. (IV.3.15)

Proof. For simplicity we may assume that ξ =δ_γ with γ ∈ Γ and v = 0. Let v_k =P

hµ^k_hu_h with µ^k_h scalars. Then applying the formula for m_t we have that

km_t(v_k)ξk² =keV_tv_kV−tδ_γk² =kX

h

µ^k_heV_tu_hV−tδ_γk²

=kX

h

µ^k_hτ(υt(hγ)σ_h(υ−t(γ)))1⊗δ_hγk²

=X

h

|µ^k_h|²|τ(υt(hγ)σh(υ−t(γ)))|²

=X

h

|µ^k_h|²exp −t²kq(hγ)−π_h(q(γ))k²

≤X

h

|µ^k_h|²exp

−t²

2kq(h)k²+t²D(q)

(IV.3.16)

Fixε >0. Sinceqis proper there exists a finite setF_ε ⊂Γ such that _t²2ln

2 exp(^t2D(q))

ε

≤ kq(h)k² for all h∈Γ\Fε. This obviously implies that

exp

−t²

2kq(h)k²+t²D(q)

≤ ε

2 (IV.3.17)

for allh∈Γ\Fε. Also, since v_kis weakly convergent to 0 andFεis finite, there existskε such that for all k≥kε and all h∈Fε we have

|µ^k_h| ≤





ε 2|F_ε|maxh∈Fεexp

−^t₂²kq(h)k²+t²∆(q)





1 2

(IV.3.18)

Using (IV.3.16), (IV.3.17), and (IV.3.18) together with P

h|µ^k_h|² = 1, for allk≥kε we have km_t(vk)ξk² ≤X

h

|µ^k_h|²exp

−t²

2kq(h)k²+t²D(q)

≤



 X

h∈Fε

|µ^k_h|²



max

h∈Fε

exp

−t²

2kq(h)k²+t²D(q)

+ε 2



 X

h∈Γ\Fε

|µ^k_h|²





≤ ε 2+ ε

2 =ε, giving the desired conclusion.