Twisted Roe algebras and twisted localization algebras

g∈C_L^∗(X). There exists a local index map [Y97]

indL: K∗(X)→K∗(C_L^∗(X)).

Theorem 27 ([Y97]). For every finite dimensional simplicial complex X endowed with the spherical metric, the local index map ind_L: K∗(X)→K∗(C_L^∗(X))is an isomorphism.

Consequently, if Γ is a discrete metric space with bounded geometry, we have the following commuting diagram:

d→∞lim K∗(C_L^∗(P_d(Γ)))

e∗

d→∞lim K∗(P_d(Γ))

indL

∼=

55j

jj jj jj jj jj jj jj

ind// lim

d→∞K∗(C^∗(P_d(Γ))).

exponential map at any point x∈M

exp_x : TxM −→M

is a homeomorphism, which induces a∗-isomorphism:

A∼=C0(R²ⁿ)⊗M2ⁿ(C),

where byMk(C) we denote the algebra of k×kcomplex matrices.

Let Γ be a discrete metric space with bounded geometry. Let f : Γ→M be a coarse embedding.

For eachd >0, we shall extend the mapf to the Rips complex P_d(Γ) in the following way. Note thatf is a coarse map, i.e., there exists R >0 such that for all γ1, γ2 ∈Γ,

d(γ₁, γ₂)≤d =⇒ d_M(f(γ₁), f(γ₂))≤R.

For any pointx=P

γ∈Γcγγ ∈P_d(Γ), wherecγ ≥0 andP

γ∈Γcγ= 1, we choose a point fx∈M such that

d(fx, f(γ))≤R

for all γ ∈Γ withc_γ 6= 0. The correspondencex7→f_x gives a coarse embedding P_d(Γ)→M, also denoted byf.

Choose a countable dense subset Γ_dof P_d(Γ) for eachd >0 in such a way that Γ_d⊂Γ_d⁰ when d < d⁰. LetK be the algebra of all compact operators on a separable Hilbert space H0.

LetC_alg^∗ (Pd(Γ),A) be the set of all functions

T : Γ_d×Γ_d→A⊗K

such that

1. there existsC >0 such thatkT(x, y)k ≤C for all x, y∈Γ_d;

2. there existsR >0 such thatT(x, y) = 0 if d(x, y)> R;

3. there existsL >0 such that for everyz∈Pd(Γ), the number of elements in the following set

{(x, y)∈Γ_d×Γ_d: d(x, z)≤3R, d(y, z)≤3R, T(x, y)6= 0}

is less than L;

4. there existsr >0 such that

Supp(T(x, y))⊂B(f(x), r)

for all x, y∈Γ_d, where B(f(x), r) ={p∈M : d(p, f(x))< r} and, for allx, y∈Γ_d, the entryT(x, y)∈A⊗K is a function on M withT(x, y)(p)∈Cliff(TpM)⊗K for each p∈M so that the supportof T(x, y) is defined by

Supp(T(x, y)) :={p∈M : T(x, y)(p)6= 0}.

Remark. For any T ∈C_alg^∗ (P_d(Γ),A), there is r >0 such that

Supp(T(x, y))⊂B(f(x), r)

for all x, y∈Γ_d. Since f : P_d(Γ)→M is a coarse embedding, there existsS >0 such that d(f(x), f(y))< S whenever d(x, y)< R. It follows that

Supp(T(x, y))⊂B(f(y), S+r)

for all x, y∈Γ_d. Hence, the adjointT^∗ ofT defined by

T^∗(x, y) = (T(y, x))^∗ ∈A⊗K (∀x, y∈Γd)

is also an element ofC_alg^∗ (P_d(Γ),A). Therefore, C_alg^∗ (P_d(Γ),A) is a ∗-algebra.

A product structure onC_alg^∗ (P_d(Γ),A) can be defined by

(T1T2)(x, y) = X

z∈Γ_d

T1(x, z)T2(z, y).

Let

E =





 X

x∈Γ_d

ax[x] : ax ∈A⊗K, X

x∈Γ_d

a^∗_xax converges in norm





 .

Then E is a Hilbert module overA⊗K:

* X

x∈Γd

a_x[x], X

x∈Γd

b_x[x]

+

= X

x∈Γd

a^∗_xb_x,



 X

x∈Γ_d

ax[x]



a= X

x∈Γ_d

axa[x]

for all a∈A⊗K. The ∗-algebra C_alg^∗ (P_d(Γ),A) acts on E by

T



 X

x∈Γ_d

a_x[x]



= X

y∈Γ_d



 X

x∈Γ_d

T(y, x)a_x



[y],

whereT ∈C_alg^∗ (P_d(Γ),A) andP

x∈Γ_da_x[x]∈E. Note that T is a module homomorphism which has an adjoint module homomorphism.

Definition 49. The twisted Roe algebra C^∗(P_d(Γ),A) is defined to be the operator norm closure of C_alg^∗ (P_d(Γ),A) in B(E), the C^∗-algebra of all module homomorphisms from E to E for which there is an adjoint module homomorphism.

The above definition of the twisted Roe algebra is similar to that in [Y00].

LetC_L,alg^∗ (P_d(Γ),A) be the set of all bounded, uniformly norm-continuous functions

g: R+→C_alg^∗ (P_d(Γ),A)

such that

1. there exists a bounded function R(t) : R+→R+ with lim

t→∞R(t) = 0 such that (g(t))(x, y) = 0 whenever d(x, y)> R(t);

2. there existsL >0 such that for everyz∈P_d(Γ), the number of elements in the following set

{(x, y)∈Γd×Γd: d(x, z)≤3R, d(y, z)≤3R, g(t)(x, y)6= 0}

is less than Lfor every t∈R+.

3. there existsr >0 such that Supp((g(t))(x, y))⊂B(f(x), r) for allt∈R+,x, y∈Γd, wheref :P_d(Γ)→M is the extension of the coarse embedding f : Γ→M and B(f(x), r) ={p∈M :d(p, f(x))< r}.

Definition 50. The twisted localization algebra C_L^∗(Pd(Γ),A) is defined to be the norm completion of C_L,alg^∗ (Pd(Γ),A), where C_L,alg^∗ (Pd(Γ),A) is endowed with the norm

kgk_∞= sup

t∈R+

kg(t)k_C∗(P_d(Γ),A).

The above definition of the twisted localization Roe algebra is similar to that in [Y00].

The evaluation homomorphism efrom C_L^∗(P_d(Γ),A) to C^∗(P_d(Γ),A) defined by e(g) =g(0) induces a homomorphism at K-theory level:

e∗ : lim

d→∞K∗(C_L^∗(Pd(Γ),A))→ lim

d→∞K∗(C^∗(Pd(Γ),A)).

Theorem 28. Let Γ be a discrete metric space with bounded geometry which admits a coarse embedding f : Γ→M into a simply connected, complete Riemannian manifold M of non-positive sectional curvature. Then the homomorphism

e∗ : lim

d→∞K∗(C_L^∗(P_d(Γ),A))→ lim

d→∞K∗(C^∗(P_d(Γ),A)) is an isomorphism.

The proof of Theorem 28 is similar to the proof of Theorem 6.8 in [Y00]. To begin with, we need to discuss ideals of the twisted algebras associated to open subsets of the manifold M.

Definition 51.

1. The support of an element T in C_alg^∗ (Pd(Γ),A) is defined to be

Supp(T) = n

(x, y, p)∈Γ_d×Γ_d×M : p∈Supp(T(x, y))o

= n

(x, y, p)∈Γ_d×Γ_d×M : (T(x, y))(p)6= 0o

;

2. The support of an element g in C_L,alg^∗ (Pd(Γ),A) is defined to be

[

t∈R+

Supp(g(t)).

LetO ⊂M be an open subset of M. Define C_alg^∗ (Pd(Γ),A)O to be the subalgebra of C_alg^∗ (P_d(Γ),A) consisting of all elements whose supports are contained in Γ_d×Γ_d×O, i.e.,

C_alg^∗ (P_d(Γ),A)O = {T ∈C_alg^∗ (P_d(Γ),A) : Supp(T(x, y))⊂O, ∀x, y∈Γ_d}.

Define C^∗(P_d(Γ),A)_O to be the norm closure of C_alg^∗ (P_d(Γ),A)_O. Similarly, let

C_L,alg^∗ (P_d(Γ),A)O= n

g∈C_L,alg^∗ (P_d(Γ),A) : Supp(g)⊂Γ_d×Γ_d×O o

and defineC_L^∗(P_d(Γ),A)_O to be the norm closure ofC_L,alg^∗ (P_d(Γ),A)_O under the norm kgk_∞= sup_t∈R+kg(t)k_C∗(Pd(Γ),A).

Note thatC^∗(P_d(Γ),A)_O andC_L^∗(P_d(Γ),A)_O are closed two-sided ideals of C^∗(P_d(Γ),A) and C_L^∗(P_d(Γ),A), respectively. We also have an evaluation homomorphism

e:C_L^∗(P_d(Γ),A)O→C^∗(P_d(Γ),A)O given by e(g) =g(0).

Lemma 23. For any two open subsets O₁, O₂ of M, we have

C^∗(Pd(Γ),A)O1+C^∗(Pd(Γ),A)O2 =C^∗(Pd(Γ),A)O1∪O2,

C^∗(P_d(Γ),A)_O1∩C^∗(P_d(Γ),A)_O2 =C^∗(P_d(Γ),A)_O1∩O2, C_L^∗(Pd(Γ),A)O1 +C_L^∗(Pd(Γ),A)O2 =C_L^∗(Pd(Γ),A)O1∪O2,

C_L^∗(P_d(Γ),A)_O1 ∩C_L^∗(P_d(Γ),A)_O2 =C_L^∗(P_d(Γ),A)_O1∩O₂.

Consequently, we have the following commuting diagram connecting two Mayer-Vietoris sequences at K-Theory level:

AL0 //

BL0 //

CL0

e∗

{{xxxxxxxx

CL₁

;;x

xx xx xx x

e∗

BL₁

oo

AL₁

oo

A0 //B0 //C0

{{xxxxxxxxx

C₁

;;x

xx xx xx xx

B₁

oo A₁^oo

where, for∗= 0,1,

AL∗=K∗

C_L^∗(P_d(Γ),A)O1∩O2

, CL∗ =K∗

C_L^∗(P_d(Γ),A)O1∪O2

,

A∗=K∗

C^∗(P_d(Γ),A)_O1∩O₂

, C∗ =K∗

C^∗(P_d(Γ),A)_O1∪O₂

,

BL∗=K∗

C_L^∗(P_d(Γ),A)_O1 M K∗

C_L^∗(P_d(Γ),A)_O2 ,

B∗=K∗

C^∗(Pd(Γ),A)O1

MK∗

C^∗(Pd(Γ),A)O2

.

Proof. We shall prove the first equality. Other equalities can be proved similarly. Then the two Mayer-Vietoris exact sequences follow from Lemma 2.4 of [HRY].

To prove the first equality, it suffices to show that

C_alg^∗ (P_d(Γ),A)_O1∪O₂ ⊆C_alg^∗ (P_d(Γ),A)_O1+C_alg^∗ (P_d(Γ),A)_O2.

Now suppose T ∈C_alg^∗ (P_d(Γ),A)O1∪O2. Take a continuous partition of unity{ϕ₁, ϕ2} on O1∪O2 subordinate to the open over{O₁, O2} ofO1∪O2. Define two functions

T₁, T₂: Γ_d×Γ_d−→A⊗K

by

T₁(x, y)(p) =ϕ₁(p)

T(x, y)(p) ,

T2(x, y)(p) =ϕ2(p)

T(x, y)(p)

forx, y∈Γ_dand p∈M. Then T₁ ∈C_alg^∗ (P_d(Γ),A)_O1,T₂ ∈C_alg^∗ (P_d(Γ),A)_O2, and

T =T1+T2∈C_alg^∗ (P_d(Γ),A)_O1+C_alg^∗ (P_d(Γ),A)_O2

as desired.

It would be convenient to introduce the following notion associated with the coarse embedding f : Γ→M.

Definition 52. Let r >0. A family of open subsets {O_i}_i∈J of M is said to be (Γ, r)-separate if

1. O_i∩O_j =∅ ifi6=j;

2. there exists γi ∈Γ such that Oi ⊆B(f(γi), r)⊂M for each i∈J. Lemma 24. If {O_i}_i∈J is a family of (Γ, r)-separate open subsets of M, then

e∗: lim

d→∞K∗(C_L^∗(P_d(Γ),A)t_i∈JOi)→ lim

d→∞K∗(C^∗(P_d(Γ),A)t_i∈JOi) is an isomorphism, where t_i∈JOi is the (disjoint) union of{O_i}_i∈J.

We will prove Lemma 24 in the next section. Granting Lemma 24 for the moment, we are able to prove Theorem 28.

Proof of Theorem 28. [Y00]. For any r >0, we defineO_r⊂M by

Or= [

γ∈Γ

B(f(γ), r),

wheref : Γ→M is the coarse embedding andB(f(γ), r) ={p∈M :d(p, f(γ))< r}.

For anyd >0, ifr < r⁰ thenC^∗(P_d(Γ),A)_Or ⊆C^∗(P_d(Γ),A)_Or0 and C_L^∗(P_d(Γ),A)_Or ⊆C_L^∗(P_d(Γ),A)_Or0. By definition, we have

C^∗(Pd(Γ),A) = lim

r→∞C^∗(Pd(Γ),A)Or, C_L^∗(P_d(Γ),A) = lim

r→∞C_L^∗(P_d(Γ),A)_Or.

On the other hand, for any r >0, ifd < d⁰ then Γ_d⊆Γ_d⁰ inP_d(Γ)⊆P_d⁰(Γ) so that we have natural inclusions C^∗(Pd(Γ),A)Or ⊆C^∗(P_d⁰(Γ),A)Or and

C_L^∗(Pd(Γ),A)Or ⊆C_L^∗(Pd⁰(Γ),A)Or. These inclusions induce the following commuting diagram

K∗(C_L^∗(P_d0(Γ),A)_Or) ^e∗ //

K∗(C^∗(Pd⁰(Γ),A)Or)

K∗(C_L^∗(P_d(Γ),A)O_r)

55j

jj jj jj jj jj jj

jj _e∗

//

K∗(C^∗(Pd(Γ),A)Or)

55j

jj jj jj jj jj jj jj

K∗(C_L^∗(P_d0(Γ),A)O_r0) e∗ //K∗(C^∗(P_d0(Γ),A)O_r0)

K∗(C_L^∗(Pd(Γ),A)O_r0) ^e∗ //

55j

jj jj jj jj jj jj jj

K∗(C^∗(Pd(Γ),A)O_r0)

55j

jj jj jj jj jj jj jj

which allows us to change the order of limits from lim

d→∞ lim

r→∞ to lim

r→∞ lim

d→∞ in the second piece of the following commuting diagram

d→∞lim K∗(C_L^∗(P_d(Γ),A))

∼=

e∗ // lim

d→∞K∗(C^∗(P_d(Γ),A))

∼=

d→∞lim lim

r→∞K∗(C_L^∗(P_d(Γ),A)_Or)

∼=

e∗ // lim

d→∞ lim

r→∞K∗(C^∗(P_d(Γ),A)_Or)

∼=

r→∞lim lim

d→∞K∗(C_L^∗(P_d(Γ),A)_Or) ^e∗ // lim

r→∞ lim

d→∞K∗(C^∗(P_d(Γ),A)_Or) So, to prove Theorem 28, it suffices to show that, for anyr >0,

e∗: lim

d→∞K∗(C_L^∗(P_d(Γ),A)_Or)→ lim

d→∞K∗(C^∗(P_d(Γ),A)_Or)

is an isomorphism.

Letr >0. Since Γ has bounded geometry and f : Γ→M is a coarse embedding, there exist finitely many mutually disjoint subsets of Γ, say Γ_k:={γ_i :i∈J_k} with some index set J_k fork= 1,2,· · ·, k₀, such that Γ =F_k0

k=1Γ_k and, for eachk,d(f(γ_i), f(γ_j))>2r for distinct elementsγi, γj in Γ_k.

For eachk= 1,2,· · ·, k0, let

Or,k= [

i∈J_k

B(f(γi), r).

Then O_r =Sk0

k=1O_r,k and eachO_r,k, or an intersection of severalO_r,k, is the union of a family of (Γ, r)-separate (Definition 52) open subsets ofM. Now Theorem 28 follows from Lemma 24 together with a Mayer-Vietoris sequence argument by using Lemma 23.