The Novikov conjecture and extensions of coarsely embeddable groups

Let $1 \to N \to G \to G/N \to 1$ be a short exact sequence of countable discrete groups and let $B$ be any $G$-$C^*$-algebra. In this paper, we show that the strong Novikov conjecture with coefficients in $B$ holds for such a group $G$ when the normal subgroup $N$ and the quotient group $G/N$ are coarsely embeddable into Hilbert spaces. As a result, the group $G$ satisfies the Novikov conjecture under the same hypothesis on $N$ and $G/N$.

The Novikov conjecture is a consequence of the strong Novikov conjecture in the computation of the K-theory of group C * -algebras. Given a countable discrete group G, there is a universal proper G-space EG which is unique up to equivariant homotopy equivalence (see [2]). Let B be any C * -algebra equipped with a G-action by *automorphisms. The Baum-Connes assembly map for a countable discrete group G and a G-C * -algebra B is a group homomorphism is the equivariant K-homology with G-compact supports with coefficients in B of the universal space EG for proper G-actions, and K * (B ⋊ r G) is the K-theory of the reduced crossed product B ⋊ r G (see [15]). In the special case when B is the complex numbers C with trivial G-action, the Baum-Connes assembly map µ is a group homomorphism mapping each Dirac type operator to its higher index in K * (C * r (G)), where C * r (G) is the reduced group C * -algebra. The Baum-Connes conjecture with coefficients in B claims that µ is an isomorphism, while the strong Novikov conjecture with coefficients in B claims that µ is injective. When B is the complex numbers C, this reduces to the usual Baum-Connes conjecture and strong Novikov conjecture, respectively.
In [19], Oyono-Oyono established a group extension result for the Baum-Connes conjecture. Let N and G be countable discrete groups, and N a normal subgroup of G. Oyono-Oyono showed that if the quotient group G N and all subgroups of G containing N with finite index satisfy the Baum-Connes conjecture, then G satisfies the Baum-Connes conjecture. With this extension result, one can show that the Baum-Connes conjecture holds for a large class of groups. For instance, based on Higson and Kasparov's result on the Baum-Connes conjecture for a-T-menable groups ( [12]), the result of Oyono-Oyono implies that the Baum-Connes conjecture holds for all extensions of a-T-menable groups.
To obtain an extension result for the Novikov conjecture, one might attempt to show that coarse embeddability into Hilbert space is closed under taking group extensions, and apply Yu's result ( [28]). However, in [1], Arzhantseva and Tessera constructed a finitely generated group G which is not coarsely embeddable into Hilbert space, but has a normal subgroup N such that N and G N are coarsely embeddable into Hilbert spaces. Note that every subgroup of G containing N with finite index is also coarsely embeddable into Hilbert space. To obtain an analogue of the extension result of the Baum-Connes conjecture ( [19]), other techniques are needed to show that the group obtained from extension of coarsely embeddable groups satisfies the Novikov conjecture.
Our main results are the following.
Theorem 1.1. Let 1 → N → G → G N → 1 be a short exact sequence of countable discrete groups and B a G-C * -algebra. If N and G N are coarsely embeddable into Hilbert spaces, then the strong Novikov conjecture holds for G with coefficients in the G-C * -algebra B, that is, the Baum-Connes assembly map µ ∶ KK G * (EG, B) → K * (B ⋊ r G) is injective, where EG is the universal space for proper G-action, and B ⋊ r G is the reduced crossed product C * -algebra.
Let G be a countable discrete group with a coarsely embeddable normal subgroup N ≤ G. Assume that the quotient group G N is also coarsely embeddable into Hilbert space. It follows from Theorem 1.1 that the rational strong Novikov conjecture holds for G, that is, the Baum-Connes assembly map µ ∶ KK G * (EG) ⊗ Q → K * (C * r G) ⊗ Q is injective. We remark that the rational strong Novikov conjecture implies the Novikov conjecture on the homotopy invariance of higher signatures and the Gromov-Lawson-Rosenberg conjecture regarding the existence of positive scalar curvature on closed aspherical manifolds.
This paper is organized as follows. In Section 2, we introduce Roe algebras, the local index map and its relation to the Baum-Connes map. In Section 3, we recall the C *algebra associated with an infinite dimensional Euclidean space and the generalization to the field case. In Section 4, we define an a-T-menable groupoid associated to a coarsely embeddable group, and then define certain twisted Roe algebras and twisted localization algebras. In Section 5, we prove that the evaluation-at-zero map induces an isomorphism from the K-theory of twisted localization algebras to the K-theory of twisted Roe algebras for groups which are extensions of coarsely embeddable groups. In Section 6, we construct a geometric analogue of Higson-Kasparov-Trout's Bott map from the K-theory of localization algebras to the K-theory of twisted localization algebras. This geometric analogue of the Bott map is then used to reduce the Novikov conjecture to the twisted Baum-Connes conjecture for groups which are extensions of coarsely embeddable groups.

The Baum-Connes map and localization
In this section, we will first recall the definition of Roe algebras, and the Baum-Connes assembly map. We then move on to define the local index map and show the connection between the local index map and the Baum-Connes assembly map.
2.1. Roe algebras. Let G be a countable discrete group, and ∆ a locally compact metric space with a proper cocompact G-action. The action is proper if the map ∆ × G → ∆ × ∆, via (x, g) ↦ (x, gx), is a proper map. A G-action is said to be cocompact if there exists a compact subset ∆ 0 ⊂ ∆ such that G⋅∆ 0 = ∆. Let C 0 (∆) be the C * -algebra of all continuous functions on ∆, which vanish at infinity. Let B be any G-C * -algebra. (1) The support of T , denoted by supp(T ), is defined to be the complement (in ∆ × ∆) of all pairs (x, y) ∈ ∆ × ∆ for which there exist f, g ∈ C 0 (∆) with f (x) ≠ 0 and g(y) ≠ 0 such that π(f )T π(g) = 0. (2) The propagation of T is defined to be propagation(T ) = sup{d(x, y) ∶ (x, y) ∈ supp(T )}.
If propagation(T ) < ∞, the operator T is said to have finite propagation.
(3) The operator T is said to be locally compact if π(f )T and T π(f ) are in K(H) for all f ∈ C 0 (∆), where K(H) is defined to be the operator norm closure of all finite-rank operators on the Hilbert module H.
Let H be a countably generated Hilbert module over B, and U ∶ G → U(H) a unitary representation of G. A * -homomorphism π ∶ C 0 (∆) → B(H) is said to be covariant if π(γf ) = U γ π(f )U γ −1 , for all γ ∈ G, f ∈ C 0 (∆). The triple (C 0 (∆), G, π) is called a covariant system. An operator T ∈ B(H) is said to be G-invariant if U γ T U γ −1 = T , for all γ ∈ G. Let us also recall the definition of an admissible covariant system, more details can be found in [25].
where H ∆ and E are Hilbert spaces, E is isomorphic to 2 (F ) ⊗ H U as F -Hilbert spaces for some Hilbert space H U with a trivial F -action.
Let G be a countable discrete group, and ∆ a locally compact metric space with a proper cocompact G-action and let B be a G-C * -algebra. There is always an admissible covariant system. Chose an infinite-dimensional separable Hilbert space H 0 and a countable dense G-invariant subset X ⊂ ∆, then define The tensor product H is a Hilbert B-module with the B-valued inner product In addition, the Hilbert B-module is endowed with the diagonal action of G by where g, h ∈ G, a ∈ B, z ∈ X. Define an action of C 0 (∆) by pointwise multiplication Following Lemma 4.5.5 in [25], it is obvious that (C 0 (∆), G) is an admissible system. Now we are ready to define the Roe algebra, following Roe [21]. Definition 2.3. Let C 0 (∆), G, π be an admissible covariant system. The algebraic Roe algebra with coefficients in B, denoted by C * alg (∆, G, B), is defined to the algebra of all the G-invariant, locally compact operators in B(H) with finite propagation. The Roe algebra with coefficients in B, denoted by C * (∆, G, B), is the norm closure of C * alg (∆, G, B) under the operator norm on H.
Next, we will recall some basic properties of Roe algebras. Let ∆ 1 , ∆ 2 be two locally compact metric spaces with proper and isometric G-actions. A Borel map Given an equivariant coarse embedding, we will define an isometry between admissible Hilbert modules covering the map. Let (C 0 (∆ 1 ), π 1 , G) and (C 0 (∆ 2 ), π 2 , G) be admissible systems on A space ∆ 1 is said to be G-equivariantly coarsely equivalent to ∆ 2 , if there exist Gequivariant coarse embedding f ∶ ∆ 1 → ∆ 2 and g ∶ ∆ 2 → ∆ 1 , such that d(f g(y), y) < c for all y ∈ ∆ 2 and d(gf (x), x) < c for all x ∈ ∆ 1 , where c is a positive constant. Let us recall the result that the K-theory of Roe algebras with coefficients in any G-C *algebra B is invariant under equivariant coarse equivalence. For completeness, we also present the proof.
Proof. Since the G-action on ∆ 2 is proper and isometric, by Lemma A.2.8 in [24], one can find a Borel cover {U i } with mutually disjoint elements, such that K i has non-empty interior for all i, (3) the diameter of K i is uniformly bounded for all i. One obtains a cover {f −1 (U i )} of ∆ 1 . The representation of C 0 (∆ 1 ) on H 1 extends to a representation of the algebra of all bounded Borel functions on ∆ 1 . Thus, for each i, in return, gives us an isometry V = ⊕ i V i ∶ H 1 → H 2 . It follows from condition (3) above that the operator V T V * has finite propagation when T has finite propagation. Therefore, the map given by Ad(V )(T ) = V T V * is well-defined and induces a homomorphism on Ktheory . Similarly, the equivariant coarse map from ∆ 2 to ∆ 1 gives rise to an inverse map.
Remark 2.5. It is easy to check that when f is the identity map, the isometry above is a unitary, thus, there is an isomorphism between Roe algebras defined on different admissible covariant systems. As a result, the definition of Roe algebras is independent of the choice of the admissible covariant system.
The following result is essentially due to John Roe.
Proposition 2.6 ( [17]). Let G be a countable discrete group, and ∆ a locally compact metric space with a proper cocompact G-action. If C 0 (∆), G, π is an admissible covariant system, then the Roe algebra where K is the algebra of all compact operators on some infinite-dimensional separable Hilbert space.
2.2. The Baum-Connes assembly map. Let H be a G-Hilbert module over B. Let F be an operator in B(H), and let π ∶ C 0 (∆) → B(H) be a * -representation of The group KK G 0 (∆, B) is an abelian group consisting of the homotopy equivalence classes of KK-cycles. By Proposition 5.5 in [16], any class in KK G 0 (∆, B) can be represented by a KK-cycle (H, π, F ) such that the covariant system For any fixed > 0, let {U i } i∈I be a locally finite and G-equivariant open cover of ∆ such that diameter(U i ) < for all i ∈ I. An open cover is said to be G-equivariant Define an operator on H by where the sum converges in the strong operator topology.
Note that the propagation of F is smaller than , and (H, π, F ) is equivalent to (H, π, F ) in KK G 0 (∆, B), for any > 0. By the definition of F , F is a multiplier of C * (∆, G, B), and it is invertible modulo C * (∆, G, B). Let M (C * (∆, G, B)) be the multiplier algebra of C * (∆, G, B). Then we have the boundary map in K-theory We define the Baum-Connes assembly map for G . It is not difficult to check that the map µ is well-defined.
Similarly, we can define the Baum-Connes assembly map

This induces the Baum-Connes assembly map
is defined to be the inductive limit of KK G * (∆, B) over all Ginvariant and cocompact subspaces ∆ of EG. Later, we will show that these invariant and cocompact subspaces can be chosen to be finite-dimensional simplicial complexes, since there exist simplicial models for the universal space EG.
By Proposition 1.8 in [2], one can choose a model for the universal space EG for proper G-action as follows.
Definition 2.7. For each d > 0, we define the Rips complex, denoted by P d (G), to be the simplicial complex with vertex set G and such that a finite subset {γ i } n i=1 ⊂ G spans a simplex if and only if d(γ i , γ j ) ≤ d.
The simplex spanned by {γ i } n i=1 ⊂ G admits a metric induced by pullback of the standard Riemannian metric on the sphere. The spherical metric on the Rips complex P d (G) is the maximal metric such that it restricts to the above spherical metric on each simplex.
We can choose the union ⋃ d>0 P d (G) as a model of EG, where the union is equipped with the weak topology under which a subset C ⊂ ⋃ d>0 P d (G) is closed if and only if C ∩ P d (G) is closed for each d > 0.

Localization algebras and the local index map.
Let us now recall localization algebras and the local index map, and introduce some basic properties of the K-theory of localization algebras.
Let ∆ be the topological realization of a locally compact and finite-dimensional simplicial complex endowed with the simplicial metric. Let (C 0 (∆), G, π) be an admissible covariant system, where π ∶ C 0 (∆) → B(H) is a * -homomorphism for some Hilbert module H over B.
(1) The algebraic localization algebra C * L,alg (∆, G, B) is defined to be the algebra of all the bounded and uniformly continuous maps The localization algebra is an equivariant analogue of the algebra introduced by Yu in [26]. Note that, up to * -isomorphism, the localization algebra C * L (∆, G, B) is independent of the choice of the admissible covariant system by Remark 2.5. A Let { n } n∈N be a sequence of positive numbers with lim n→∞ n = 0. By the same argument as Proposition 2.4, for each k, there exists a G-equivariant isometry V k ∶ H 1 → H 2 between the Hilbert B-module, such that cos(πt 2) sin(πt 2) − sin(πt 2) cos(πt 2) .
Then V f (t) induces a homomorphism on unitization The family of maps Ad(V f (t)) induces a homomorphism, Ad(V f (t)) * ∶ K * (C * L (∆ 1 , G, B)) → K * (C * L (∆ 2 , G, B)) on the K-theory. Note that Ad(V f (t))(u(t) + cI) is uniformly continuous on t, even though V f (t) is not continuous. It is also easy to check the propagation condition of the path Ad(V f (t))(u(t) + cI). By Lemma 3.4 of [26], the homomorphism between the K-groups is independent of the choice of the family of isometries {V k } k . Definition 2.9. Let ∆ 1 and ∆ 2 be two proper metric spaces and f , g two Lipschitz maps from ∆ 1 to ∆ 2 . The map f is said to be strongly Lipschitz homotopy equivalent to g if there exists a continuous homotopy Definition 2.10. The metric space ∆ 1 is said to be strongly Lipschitz homotopy equivalent to ∆ 2 if there exist two Lipschitz coarse maps f ∶ ∆ 1 → ∆ 2 and g ∶ ∆ 2 → ∆ 1 such that f ○ g and g ○ f are respectively strongly Lipschitz homotopy equivalent to id ∆ 2 and id ∆ 1 .
The K-theory of localization algebras is invariant under strong Lipschitz homotopy equivalence. Following the proof of Proposition 3.7 in [26], it is not difficult to prove the following equivariant analogue. For ease of reference later, we include the proof.
Proposition 2.11 ([26]). Let f ∶ ∆ 1 → ∆ 2 be a strongly Lipschitz homotopy equivalence, then the map Remark 2.12. Let (C 0 (∆ 1 ), π, G) be an admissible covariant system on a Hilbert module It is easy to show that η induces an isomorphism on K-theory of localization algebras defined on different admissible covariant systems.
Proof. It suffices to show that the homomorphism Ad(V gf (t)) * is the identity map on K-theory. Let F (x, t) ∶ ∆ 1 ×[0, 1] → ∆ 2 be the strong Lipschitz homotopy equivalence with F (x, 0) = (gf )(x), F (x, 1) = x, for all x ∈ X. Fix a sequence of positive numbers { n } n∈N with lim n→∞ n = 0 and a sequence of non-negative numbers {t i,j } ∞ i,j=0 , satisfying • t 0,j = 0, t i+1,j ≥ t i,j , for all i, j ≥ 0, • for each j, ∃ N j , such that t i,j = 1 for all i ≥ N j , • d(F (x, t i,j ), F (x, t i+1,j )) ≤ j , and d(F (x, t i,j ), F (x, t i,j+1 )) ≤ j , for all x ∈ X. We shall prove that Ad * (V F (⋅,t 0,j ) (t)) = Id for K 1 -case. The K 0 -case can be dealt with in a similar way by a suspension argument. Notice that C L (∆ 1 , G, B) ⊗ M n (C) ≅ C * L (∆ 1 , G, B) for all n. Thus, every element in K * (C * L (∆ 1 , G, B)) can be represented by an invertible element u in C * L (∆ 1 , G, B) By the definitions of operators Ad F (⋅,t i,j ) and the sequence The following Mayer-Vietoris sequence is an equivariant analogue of the Mayer-Vietoris sequence introduced by Yu, and more details can be found in [14].
Proposition 2.13 ([14]). Let ∆ be a simplicial complex endowed with the spherical metric, and let G be a countable discrete group. Assume G acts on ∆ properly by isometries. Let X 1 , X 2 ⊂ ∆ be G-invariant simplicial subcomplexes endowed with subspace metric. Then we have the following six-term exact sequence: for brevity. Remark 2.14. It is easy to verify that the above exact sequence is natural with regard to the coefficient algebra B in the following sense.
Obviously, this map induces a homomorphism on the localization algebras ϕ L ∶ The exact sequence in Proposition 2.13 is natural with respect to coefficient algebras in the sense that the diagram Let us now define the local index map. For every positive integer n, let {U n,i } i∈I be a locally finite and G-equivariant open cover for ∆ with diameter(U n,i ) ≤ 1 n for all i. Let {φ n,i } i∈I be the partition of unity subordinate to the open cover for all t ∈ [n, n + 1), where the sum converges in the strong operator topology. Note that propagation(F (t)) → 0 as t → ∞. We obtain a multiplier ( ). The following result 1 established the relation between the K-homology and the K-theory of localization algebras.

Proposition 2.15 ([17]
). Let B be any G-C * -algebra, and ∆ a finite-dimensional simplicial complex endowed with a G-invariant metric. Then the local index map is an isomorphism.
Proof. This result is a consequence of the Mayer-Vietoris sequence and the Five Lemma (see [26]).
By choosing the model of EG as the union The above map induces a local index map It is not difficult to show that the map ind L is an isomorphism by Proposition 2.15. We will conclude this section by discussing the relation between the Baum-Connes assembly map and the local index map.
For each d > 0, it is natural to define an evaluation-at-zero map given by The evaluation-at-zero map induces a homomorphism on K-theory Following the argument in [26], it is easy to check that µ = ev * ○ ind L holds. In Section 6, we will show that the map ev * is an isomorphism under some assumptions. Combining the above relation and Proposition 2.15, it follows that the Baum-Connes assembly map is also an isomorphism under the same assumption.

C * -algebras associated to infinite dimensional Hilbert spaces
In this section, we will recall the C * -algebra associated with an infinite-dimensional Hilbert space (defined in [13]) and introduce a generalization of this C * -algebra associated with a continuous field of Hilbert spaces due to Tu (see [23]).

3.1.
The C * -algebra associated with an infinite-dimensional Euclidean space. 2 Let E be a separable, infinite-dimensional Euclidean space. Let E a , E b be any finitedimensional, affine subspaces of E. Let E 0 a be the finite-dimensional linear subspace of E consisting of differences of elements in E a . Let C(E a ) be the Z 2 -graded C * -algebra of continuous functions from E a to the complexified Clifford algebra of E 0 a which vanish at infinity. Let S be the Z 2 -graded C * -algebra of all continuous functions on R vanishing at infinity, where S is graded according to odd and even functions. Let Denote by X ∶ S → S the function of multiplication by x on R, viewed as a degree one, essentially selfadjoint, unbounded multiplier of S with domain the compactly supported functions in S.
(1) Let E a ⊂ E b be a pair of finite-dimensional affine subspaces of E. One can define a homomorphism where the direct limit is over all finite-dimensional affine subspaces.
Given any discrete group Γ, if Γ acts on the Euclidean space E by linear isometries, then the Γ-action on E induces a Γ-action on the C * -algebra A(E). Note that We define the Bott map to be the homomorphism induced by the asymptotic morphism The following result is due to Higson, Kasparov and Trout [8].
Theorem 3.3 (Infinite-dimensional Bott Periodicity [13]). Let Γ be a countable discrete group, E an infinite-dimensional Euclidean space with a Γ-action by linear isometries. Then the Bott map is an isomorphism.

3.2.
Generalization to the field cases. In the rest of this section, we will generalize the construction of Higson-Kasparov-Trout to the case of the continuous field. The following construction is essentially due to Tu [23]. The usage of the probability space is essentially due to Higson [10]. Let Γ be a countable, discrete group with identity element e ∈ Γ , and let X be a compact Hausdorff space admitting a Γ-action by homeomorphisms. Let us recall the definition of the transformation groupoid, denoted by X ⋊ Γ, associated with a group action Γ ↷ X. Definition 3.4. As a topological space, X ⋊ Γ = {(x, g) ∶ x ∈ X, g ∈ Γ} is equipped with the product topology. In addition, the topological space is endowed with the following structure.
(1) The product is given by ( Let us recall the definition of a continuous field of Hilbert spaces over a compact space. Let X be a compact topological space, and let H x x∈X be a family of Banach Definition 3.5. Let X be a compact space. A continuous field of Banach spaces over X is a family of Banach spaces H x x∈X , with a set of sections Θ(X, H), such that and every > 0, there exists a section s ′ ∈ Θ(X, H) such that s(y) − s ′ (y) < for all y in some neighborhood of x, then s ∈ Θ(X, H).
If every fiber H x is a Hilbert space, we will say H x x∈X is a continuous field of Hilbert spaces. If every fiber is a C * -algebra and the collection of sections is closed under the * -operation and the multiplication, the continuous field is called a continuous field of C * -algebras.
Let X ⋊ Γ be a transformation groupoid associated with the right group action Γ ↷ X, and let H x x∈X be a continuous field of Hilbert spaces over X. Let us recall the concept of the affine isometric action of X ⋊ Γ on the continuous field of Hilbert spaces H x x∈X . Definition 3.6. Let H x x∈X be a continuous field of Hilbert spaces over X. We say that the transformation groupoid X ⋊ Γ acts on H x x∈X by affine isometries if Since each fiber of the continuous field is a Hilbert space, we can define a C *algebra A(H x ) associated with each fiber H x . Then we obtain a bundle of C * -algebras A(H x ) x∈X . Next, we will introduce a structure of a continuous field of C * -algebras for the bundle A(H x ) x∈X , some more details can also be found in [11].
for each x ∈ X, > 0, there exists a neighborhood x ∈ U ⊂ X, such that for each y ∈ U , there is a linearly isometric embedding φ y ∶ R n → H y satisfying that The above definition gives rise to a continuous field structure on A(H x ) x∈X . For any x ∈ X, note that there exists a collection of sections e n ∈ Θ(X, H), such that {e n (x)} n∈N is an orthonormal basis of the fiber bundle H x . Indeed, let {a n ∈ Θ(X, H)} n∈N be the collection of continuous sections such that {a n (x)} is a basis for H x . For each positive integer n, there exists a neighborhood x ∈ U ′ n , such that the collection {a i (y) ∶ 1 ≤ i ≤ n} is linearly independent for each y ∈ U ′ n . By the Gram-Schmidt process, one can find a family of continuous local sections {e ′ i ∶ 1 ≤ i ≤ n} over U ′ n , such that the collection of vectors {e ′ i (y) ∶ 1 ≤ i ≤ n} are mutually orthogonal with norm one for each y ∈ U ′ n . By Urysohn Lemma, we can find a open set U n ⊂ U ′ n and a function f ∶ X → [0, 1], such that supp(f ) ⊂ U ′ n and f (y) = 1 for each y ∈ U n . For each i, we obtain a global section e i by setting e i (y) = f (y)e ′ n (y) for all y ∈ U ′ n and extending by zero outside U ′ n . By induction on n, we obtain such a collection of continuous global sections {e n } n∈N .
The C * -algebra of all continuous sections of the continuous field of C * -algebras is denoted by A(X, H).
Define a Γ-action on A(X, H) as follows. For each γ ∈ Γ, we have an isometry The associated linear subspace consisting of differences of elements in E a (y) is denoted by E 0 a (y) = span{v 1 (y), v 2 (y), ⋯, v k (y)}. We have a continuous local distribution of C * -algebras Cliff(E 0 a (y) y∈U . Because (V (xγ −1 ,γ) x∈X is a collection of continuous isometries for each γ ∈ Γ , we obtain another continuous local affine distribution for all y ∈ U . Thus, for every γ ∈ Γ, we get a homomorphism Lemma 3.7. The following diagram is commutative, where the maps β U,ba and β γ γU,ba are defined fiber-wise.
Proof. Since S is generated by g 0 (e) = e −x 2 and g 1 (x) = xe −x 2 by the Stone-Weierstrass theorem, it suffices to show that γ(β U,ba (g⊗h)) = β γ γU,ba (γ(g⊗h)), for g equal to g 0 or g 1 and h ∶ U → Cliff(E a ) a continuous local section over U . We will prove the lemma for g = g 0 ; the case for g = g 1 can be proved similarly. Let C ba and C γ ba be the Clifford multiplications on C(E a ) and C(γE a ) under the transformation ϕ a,y ∶ E a (y) → γE a (y) for each y ∈ U . Since we have Because of the definition of A(X, H), and the above commutative diagram, we have a Γ-action on A(X, H).
Define a fiber-wise Bott map β t ∶ C(X)⊗S → A(X, H) as follows. Viewing an element in C(X)⊗S as a continuous function f ∶ X → S, we obtain an element β t (f x ) ∈ A(H x ) for each x ∈ X. By the definition of the continuous field structure, it is easy to check that β t (f x ) x∈X ∈ A(X, H). Thus, we have an asymptotic morphism We have an affine isometric action of Γ on the continuous field of Hilbert spaces H x x∈X , and it is easy to check that (β t ) t∈[1,∞) is an asymptotic Γ-equivariant morphism. This asymptotic morphism induces a map on K-theory of the reduced crossed products for every finite subgroup Γ 0 ≤ Γ. Following the argument in [13], we also have a fiber-wise defined Dirac map. Let us briefly recall the definition of the Dirac map on is the linear space of differences between pairs of vectors in E a (x), and the norm on Cliff(E 0 a (x)) is obtained by fixing an orthonormal basis on E 0 Using the similar method of construction of the continuous field structure on A(H x ) x∈X , we obtain a continuous field structure on K(V (x)) x∈X , where K(V (x)) is the algebra of all compact operator on V (x) for each x ∈ X. Let V = ⊔ x∈X V (x). Define K(X, V ) to be the C * -algebra of all continuous sections of the continuous field K(V (x)) x∈X . By the structure of the continuous field of K(V (x)) x∈X , we have that the K-theory of the C * -algebra C(X) is the same as the K-theory of K(X, V ).
Denote by s(x) = lim → s a (x) the direct limit of the Schwartz subspaces s a (x) ⊂ V a (x). If E a (x) is a finite-dimensional affine subspace, then the Dirac operator D a (x) is defined by for every homogeneous element ξ ∈ s(E a (x)), where {v 1 , v 2 , . . . , v n } is an orthonormal basis for E 0 a (x), and {x 1 , x 2 , . . . , x n } are the dual coordinates to {v 1 , v 2 , . . . , v n }. The Clifford operator on E a (a) is given by ). This infinite sum is well-defined since any vector in the Schwartz space s(x) can be approximated by the one which has only finitely many nonzero terms in its infinite series. It is well-known that the operators B n,t (x) are essentially selfadjoint. Following the argument in [13], we obtain an asymptotic morphism α n from A(E 0 (x) ⊕ E 1 (x) ⊕ ⋯ ⊕ E n (x)) to S⊗K(V (x)) by and M ht is the operator of left multiplication by the function h t . Moreover, the diagram is asymptotically commutative. As a result, we get an asymptotic morphism α(x) ∶ A(H x ) → S⊗K(V (x)). Moreover, we obtain an asymptotic morphism α ∶ A(X, H) → S⊗K(X, V ).
Following the argument in [13], the map induced by α on K-theory is the inverse map of the fiber-wise defined Bott map. Consequently, we have the following result.
Theorem 3.8. Let Γ be a countable discrete group and X a compact Hausdorff space with Γ-action. Assume the associated transformation groupoid X ⋊ Γ acts on a continuous field of Hilbert spaces H x x∈X by affine isometries. Then for each finite subgroup Γ 0 ≤ Γ, the Bott map induces an isomorphism on K-theory.
An affine isometric action of a transformation groupoid X ⋊ Γ on a continuous field of Hilbert spaces H x x∈X is said to be proper, if for any R > 0, the set {g ∈ Γ ∶ ∃x ∈ X such that V (x,g) (B(xg, R)) ∩ B(x, R) ≠ ∅} is finite, where B(x, R) is the set of all elements in H x with norm less than R. Due to Tu (see [23]), the a-T-menability of the transformation groupoid guarantees the existence of a proper affine isometric action on a continuous field of Hilbert spaces H x x∈X .
, and x ∈ X. A conditionally negative definite function ϕ ∶ X ⋊ Γ → R is said to be proper if for any R > 0, the number of elements in the set {g ∈ Γ ∶ ∃x ∈ X, such that ϕ(x, g) ≤ R} is finite.
The concept of a-T-menability for groupoids was introduced by Tu in [23]. Now, let us recall the construction of the transformation groupoid from a coarsely embeddable group, by Skandalis, Tu, and Yu in [22].
the groupoid X ⋊ Γ has a proper continuous conditionally negative definite function.
Let us describe the construction of the topological space X. For any fixed element γ ∈ Γ, we define a bounded function f γ ∶ Γ → R by for any y ∈ Γ.
Let c 0 (Γ) be the C * -subalgebra of ∞ (Γ) consisting all functions vanishing at infinity. We define a Γ-action on ∞ (Γ) by (γ ⋅ f )(x) = f (xγ), for each f ∈ ∞ (Γ), x, γ ∈ Γ. Let X ′ be the spectrum of the unital commutative Γ-invariant C * -subalgebra of ∞ (Γ) generated by all constant functions, c 0 (Γ) functions, and all functions of the form f γ together with their translations by group elements in G. It is obvious that every function f γ extends continuously to X ′ . Indeed, the space X ′ is a compactification of Γ, and it admits a right action of Γ induced by the Γ-action on C(X ′ ) where C(X ′ ) is viewed as a C * -subalgebra of ∞ (Γ).
Let X be the probability space of X ′ . It is a second countable, compact space equipped with the weak- * topology (A reference of weak- * topology is Chapter 1 in [5]) and it admits a Γ-action induces by the action of Γ on X ′ . We define a conditionally negative definite function on X ⋊ Γ by for any m ∈ X.
Proposition 3.12. Let Γ → H 0 be the coarse embedding as above. The continuous map ϕ ∶ X ⋊ G → R defined above is a proper conditionally negative definite function.
Proof. It is obvious that condition (1) in the Definition 3.9 is satisfied. Let us verify condition (2). For each (x, g) ∈ X ⋊ Γ, we have The third equality follows from ϕ ′ (yg, g −1 ) = ϕ(y, g) for all (y, g) ∈ X ⋊ Γ. Condition (3) follows from the fact that ∑ The properness of ϕ follows from the definition of ϕ ′ and the fact that the map h ∶ Γ → H 0 is a coarse embedding.
The space X × Γ is equipped with the product topology. Let C c (X × Γ) be the C * -algebra of all complex valued functions on X × Γ with compact support. Define Let ϕ ∶ X ⋊ Γ → R be a continuous, proper conditional negative definite function. Then we can define a continuous field of Hilbert spaces as follows.
For each x ∈ X, consider a linear space C 0 c (Γ) ∶= f ∈ C c (Γ) ∶ ∑ g∈Γ f (g) = 0 , and define a sesquilinear form for all ξ, η ∈ C 0 c (Γ). Since ϕ is conditionally negative definite type, the form above turns out to be positive semidefinite and one can quotient out by the zero subspace, denoted by E x . Then complete E x to a Hilbert space, denoted by H x . For any function f ∈ C 0 c (X × Γ), we can view it as a continuous map (1) ξ(x) ∈ H x , for every x ∈ X, (2) ∀ x ∈ X, ∀ > 0, there exists an element ξ ′ ∈ C 0 c (X × Γ), such that ξ(y) − ξ ′ (y) Hy < for all y in some neighborhood of x.
The affine isometric action of X ⋊ Γ is defined as follows. For every γ ∈ Γ, and every x ∈ X, the unitary U (x,γ) ∶ H xγ → H x is defined by U (x,γ) (f )(g) = f (γ −1 g) for all f ∈ E x , and for all g ∈ Γ, γ ∈ Γ, then extends to a unitary U (x,γ) ∶ H xγ → H x . The cocycle b(x, g) is defined to be the element in E x represented by the function δ g − δ e . Let V (x,γ) (v) = U (x,γ) (v)+b(x, γ) for all v ∈ H xγ , (x, γ) ∈ X ⋊Γ. It is easy to check that the collection of affine isometries V (x,γ) (x,γ)∈X⋊Γ consists of a proper affine isometric action of X ⋊ Γ on the continuous field of Hilbert spaces H x x∈X .

Twisted Roe algebras and twisted localization algebras
Let 1 → N → G → G N → 1 be a short exact sequence of countable discrete groups. In this section we will construct twisted Roe algebras and twisted localization algebras with coefficients in some G-C * -algebra, and prove that the twisted Baum-Connes conjecture with coefficients holds for the group G, under the assumption that both G and G N are coarsely embeddable into Hilbert spaces.
Fix a left invariant proper metric on G. This metric restricts to every subgroup of G and the quotient group G N is endowed with the quotient metric.

Some Geometric Constructions.
In this subsection, we will construct a compact topological G-space Y , such that (1) for every subgroup N ′ ≤ G containing N with finite index, i.e., N ′ N < ∞, the transformation groupoid X ⋊ N ′ is a-T-menable, (2) for every finite subgroup G 0 ≤ G, the space Y is G 0 -contractible.
Let N ′ ≤ G be a subgroup containing N with finite index. The fact that N ′ N < ∞ implies that N ′ is coarsely equivalent to N . Since N is coarsely embeddable into a Hilbert space H 0 , we have that N ′ is aslo coarsely embeddable into the Hilbert space Let S ⊂ G be a set of the representatives of the left cosets G N ′ . Then we have a decomposition G = ⊔ g∈S gN ′ , and the coarse embedding where g ∈ S, n ∈ N ′ . Since every element g ′ ∈ G can be uniquely written as g ′ = gn for some g ∈ S, n ∈ N ′ , the extension is well-defined, and it is not a coarse embedding in general.
For any fixed element n ∈ N ′ , define a function f n ∶ G → R by By the coarse embeddability of N ′ , f n is a bounded function on G for each n ∈ N ′ . Unfortunately, f n could be an unbounded function on G if the element n ∈ G is not in N ′ . Let Y ′ N ′ be the spectrum of the unital commutative G-invariant C * -subalgebra of ∞ (G) generated by all c 0 (G) functions, all constant functions and all functions of the form f n together with their right translations. The right action of G on ∞ (G) is defined by (γf )(g) = f (gγ), for all γ, g ∈ G, all f ∈ ∞ (G). Accordingly, the compact space Y ′ N ′ admits a right action induced by the restriction of the right G-action on ∞ (G) to the C * -subalgebra which is * 2 , for all g ∈ G, n ∈ N ′ . For each n ∈ N ′ , the bounded function φ ′ N ′ (⋅, n) ∶ G → R extends to a continuous function on Y N ′ by the definition of Y N ′ . As a result, we obtain a continuous conditionally negative definite function The space Y N ′ is a second countable compact space equipped with the weak- * topology (c.f. [5]). The G-action on Y ′ N ′ induces a right action on Y N ′ . We define a continuous function Proposition 4.1. The countinuous function φ N ′ ∶ Y N ′ ⋊ N ′ → R is a proper conditionally negative definite function.
Proof. By Proposition 3.12, we have that φ N ′ is a continuous conditionally negative definite functionon the transformation groupoid Y N ′ ⋊ N ′ .
By the definition of the map Remark 4.2. For each subgroup N ′ ≤ G, we can find a compact space Y N ′ with a right G-action, such that the groupoid Y N ′ ⋊ N ′ is a-T-menable in the sense that Y N ′ ⋊ N ′ admits a proper, continuous conditionally negative definite function.
Let F be the set of all subgroups of G containing N with finite index. For each N ′ ∈ F, we can find a compact space Y N ′ . We then define a compact topological space The topology on Y is the product topology, and the G-action is then defined by Proof. Define a continuous conditionally negative definite function on the transfor- It is easy to check that this map is a proper conditionally negative definite function.

Remark 4.5. The domain of the conditionally negative definite function is
Assume the quotient group G N coarsely embeds into a Hilbert space. In Section 3, we obtained a compact metrizable space X such that X ⋊ G N is a-T-menable, and a proper G N -C * -algebra, denoted by A(X, H). In the rest of this section, we will formulate the twisted Baum-Connes conjecture for G with coefficients in C(Y )⊗A(X, H)⊗B.
For each d > 0, let P d (G) be the Rips complex endowed with the spherical metric as defined in Section 2. Take a countable dense subset Let H x x∈X be the continuous field of Hilbert spaces such that the transformation groupoid X ⋊ G N acts properly on H x x∈X by affine isometries. For every (x, g) ∈ X ⋊ G N , there is an affine isometry V (x,g) ∶ H xg → H x and a continuous section b ∶ X × G N → H with b(x, g) ∈ H x , such that for every v ∈ H (x,g) , H) is called the cocycle associated with the groupoid action of X ⋊ G N on the continuous field of Hilbert spaces H x x∈X . By the construction of the continuous field of Hilbert spaces, b(x, e) = 0 ∈ H x , for all x ∈ X. By coarse embeddability and the definition of b, we have that inf x∈X b(x, g) Hx → ∞ as g → ∞.
Let Θ(X, H) be the space of all continuous sections associated with the continuous field of Hilbert spaces H x x∈X . We will define a second countable, locally compact topological space W , and a proper G N -action on W . As a set, denote W = R + × ⊔ x∈X H x , where R + is the set of all non-negative numbers. A topology can be defined as follows. Let The topology on W can also be characterized in terms of its base consisting of the following open sets. For each point The topology on W is generated by sets of the above forms. The space W is a second countable, locally compact and Hausdorff space. By the construction of the space X, it is obvious that X is second countable and separable. We obtain a countable basis for the topology on W by taking and t 0 in the rational numbers Q, x 0 in a countable dense subset of X, and v 0 in countable dense subset in each fiber H x in the definition of the above open subsets. As a consequence, the space W is second countable. For local compactness, for each R > 0, the subset Hx ≤ R 2 is compact. To see this, we first choose a net Hx ≤ R 2 . Since {t i } i is bounded, a convergent subnet exists. Without loss of generality, we assume {t i } i converges to t 0 . We can also assume {x i } i converges to x 0 due to the compactness of the space X. Fix an orthonormal basis {e n } n≥0 for the fiber H x 0 , one can find a sequence of continuous sections {e n ∈ Θ(X, H)} such that e n (x 0 ) = e n , for all n ≥ 0. By the diagonal argument, we can find a subnet For every g ∈ G N , every continuous section s ∈ Θ(X, H), it follows that g ⋅ s is also a continuous section. In addition, every function on W of the form (t, As a result, the action of G N on W is well-defined.
According to [2], the properness of the G N -action on W is equivalent to the fact that the set {g ∈ G N ∶ g ⋅ K ∩ K ≠ ∅} is finite for each compact subset K ⊂ W . Proof. For each positive integer n > 0, let Hx ≤ n . Since K n is compact for each n > 0, and W = ⋃ n>0 K n , it suffices to show that the set {g ∈ G N ∶ g ⋅ K n ∩ K n ≠ ∅} is finite for each n > 0. Let (t, x, v) ∈ K n , g ⋅ (t, x, v) = (t, xg −1 , V (xg −1 ,g) (v)) for all g ∈ G N . Since V (xg −1 ,g) (v) = U (xg −1 ,g) (v) + b(xg −1 , g) and inf x∈X b(x, g) Hx → ∞ as g → ∞, there exists some R > 0, such that Hx for all g > R. Since U (xg −1 ,g) is an isometry for each (xg −1 , g) ∈ X ⋊G N , so g⋅(t, x, v) ∉ K n . By the properness of the metric on G N , the set {g ∈ G N ∶ g ⋅ K n ∩ K n ≠ ∅} is finite for each n > 0. Thus, the G N -action on W is proper.
Note that the C * -algebra C 0 (W ) of all continuous functions on W vanishing at infinity is contained in the center of the C * -algebra A(X, H), see [23] for more details. For each open subset U ⊂ X and a continuous affine distribution {E a (y)} y∈U over U , we have that where E a = ⊔ y∈U E a (y) and W (U,Ea) ∶= {(t, y, v) ∶ t ∈ R + , y ∈ U, v ∈ E 0 a (y)} ⊂ W . If U ⊂ V are open subset of X, and {E a (y)} y∈U and {E b (y)} y∈V are continuous affine distributions over U and V respectively, with E a (y) ⊂ E b (y) for each y ∈ U , then the fiber-wise defined Bott map β U,ba takes C(W (U,Ea) ) into C 0 (W (V,E b ) ). Accordingly, the C * -algebra C 0 (W ) can be viewed as a direct limit lim → C 0 (W U,Ea ). As a result, C 0 (W ) ⋅ A(X, H) is dense in A(X, H). We have an action of G N on the C * -algebra A(X, H) as defined in Section 3. The C * -algebra C 0 (W ) is contained in the center of the C * -algebra, and the properness of the G N -action on W implies that the action of G N on A(X, H) is proper. In the rest of this section, we lift G N -actions on W and A(X, H) to G-actions via the quotient map G → G N , and we use the same notations for G and G N actions.

Twisted Roe algebras and twisted localization algebras.
In the rest of this section, we will define the twisted version of Roe algebras and localization algebras. Let d > 0, and let P d (G) be the simplical complex at scale d, endowed with the simplicial metric. Let Z d be the countable dense G-invariant subset of P d (G) consisting of all linear combinations ∑ g∈G c g g with c g ∈ Q and c g ≠ c ′ g for any pair g ≠ g ′ . Note that for each subcomplex C ⊂ P d (G), the intersection C ∩ Z d is non-empty and C = Ğ C ∩ Z d . Since the left translation action G ↷ P d (G) is proper and cocompact, one can define a coarse G-equivariant map J ∶ P d (G) → G. This map is defined as follows.
for every z ∈ Z d , there exist unique x ∈ ∆ d and unique g ∈ G, such that z = g ⋅x, By condition (2), for every element z ∈ Z d , there exists a unique element g ∈ G, such that g −1 z ∈ ∆ d . The map J ∶ Z d (G) → G can be defined as J(z) = g, where z ∈ Z d , and z = gx, for some g ∈ G and x ∈ ∆ d . Note that the map J is G-equivariant.
Define an open set We need the following lemma to define twisted Roe algebras.
Proof. For any (t, x, v) ∈ O R (z), the action is given by Let H 0 be any fixed separable complex Hilbert space with infinite dimension. Let K G be the algebra of all compact operators on H 0⊗ 2 (G), and B a G-C * -algebra. Define a G-action on H 0⊗ 2 (G) by first defining γ(v⊗δ g ) = v⊗δ γg , for all γ, g ∈ G and v ∈ H 0 , and then extending linearly to H 0⊗ 2 (G). The algebra B⊗K G is equipped with a unitary G-action by γ ⋅ (b⊗T ) = γ ⋅ b⊗γT γ * , for all b ∈ B, T ∈ K G , γ ∈ G.
Definition 4.8. For an element S ∈ C(Y )⊗A(X, H)⊗B⊗K G , we can define the support of S, denoted by supp(S), to be the complement of the set of (t, x, v) ∈ R + × H such that there exists f ∈ C 0 (R + × H) with f (t, x, v) ≠ 0, (1 Y⊗ f⊗k) ⋅ S = 0, for all k ∈ B⊗K G . Definition 4.9. Define the algebraic twisted Roe algebra, denoted by there exists L > 0, such that #{y ∶ T (y, z) ≠ 0} < L and #{z ∶ T (y, z) ≠ 0} < L, (4) there exists r 1 ≥ 0, such that T (y, z) = 0, for any y, z ∈ Z d with d(y, z) > r 1 , (5) there exists r 2 > 0, such that supp(T y,z ) = O r 2 (J(y)), (6) the operator T is G-invariant, i.e., γ(T γ −1 y,γ −1 z ) = T y,z , for all γ ∈ G, y, z ∈ Z d . The algebraic twisted Roe algebra is equipped with a * -algebra structure by the matrix operations. The action of G is given by According to the definition of the algebraic twisted Roe algebra, the * -representation on E is well-defined. The twisted Roe algebra, denoted by is defined to be the completion of the algebraic twisted Roe algebra under the operator norm in B(E), where B(E) is the C * -algebra of all adjointable module homomorphisms.
Let C * L,alg (P d (G), C(Y )⊗A(X, H)⊗B) G be the set of all bounded, uniformly normcontinuous functions Taking the completion with respect to the norm we have the twisted localization algebra, denoted by C * L (P d (G), C(Y )⊗A(X, H)⊗B) G . Remark 4.11. By Proposition 2.4 and Remark 2.5, we have that the twisted Roe algebras and the twisted localization algebras are independent of the choice of the countable dense subset Z d for each d > 0. Indeed, if we have two countable dense There is a natural evaluation-at-zero map Obviously, it is a * -homomorphism. The evaluation-at-zero map induces a homomorphism on K-theory

The K-theory of twisted Roe algebras and twisted localization algebras
In this section, we will prove the following twisted Baum-Connes conjecture for groups which are extensions of coarsely embeddable groups.
Theorem 5.1. Let 1 → N → G → G N → 1 be a short exact sequence of countable discrete groups. Assume N and G N can be coarsely embedded into Hilbert spaces. The map induced by the evaluation-at-zero map is an isomorphism.
To prove this theorem, the main idea is to decompose the twisted Roe algebra into ideals whose K-theory can be easily computed, and then use the Mayer-Vietoris sequence and the Five Lemma to piece them all together. The decomposition of the twisted Roe algebra relies on the structure of the G-action on the space W (Proposition 4.6). Let us recall an equivalent definition of proper action of a discrete group on a topological space (Definition 1.3 in [2]).
Definition 5.2. Let Γ be a countable discrete group, X a topological space equipped with a continuous action of Γ. The Γ-action is called proper if (1) X is second countable, for every x ∈ X there is a Γ-invariant neighborhood U of x and a finite subgroup Γ 0 of Γ, such that there exists a continuous Γ-map U → Γ Γ 0 .
In Section 4, we obtained a proper G N -action on a topological space W (Proposition 4.6) and lifted the G N -action to a G-action on W via the quotient map Hx < R 2 } and V i is N i -invariant, for some N i a subgroup of G containing N with finite index. We will consider the restriction of the G-action on W to an N i -action on V i . Let G ⋅ V i denote the union ⊔ g∈G N i g ⋅ V i .

Definition 5.3.
(1) For any element T ∈ C * (P d (G), C(Y )⊗A(X, H)⊗B) G , the support of T is defined to be the set (2) For any element a ∈ C * L (P d (G), C(Y )⊗A(X, H)⊗B) G , the support of a is defined to be supp(a) = ⋃ For each open subset We can also define the localization algebra, denoted by C * L (P d (G), C(Y )⊗A(X, H⊗B) G U , to be the C * -subalgebra of C * L (P d (G), C(Y )⊗A(X, H)⊗B) G generated by all paths a(t) with support contained in Z d × Z d × U . We have an evaluation-at-zero map , induced by the evaluation-at-zero map on K-theory is an isomorphism.
In order to prove this theorem, we need some lemmas. For each N i and d > 0, we define the Roe algebra with coefficients in C(Y )⊗A(X, H) V i⊗ B as follows. Since N i ≤ G, we can assume P d (N i ) ⊂ P d (G) has the restricted spherical metric of P d (G).
By the definition of Z d , the set Z 1 d is non-empty for all i. The algebraic Roe algebra, denoted by is defined to be the set of all matrices T y,z y,z∈Z i d which represent bounded, finite propagation, locally compact, and N i -invariant operators on the Hilbert module where K G is the algebra of all the compact operators on 2 (G)⊗H endowed with the tensor product unitary representation of N i . Define the Roe algebra to be the norm closure of the algebraic Roe algebra C * For each i, we can define an inclusion homomorphism between the C * -algebras by Similarly, we can define the inclusion map on the localized version, The following result is crucial to reducing twisted Roe algebras and twisted localization algebras of G to twisted Roe algebras and twisted localization algebras associated with its subgroups N i .
where T g is a Z d ×Z d -matrix with all entries supported in g⋅V i and the map J ∶ Z d → G is the map defined in Section 4. Since every element For each g ∈ G, define A g, * alg to be the algebra of all bounded, locally compact operators T = T y,z y,z∈Z d on the Hilbert module 2 (Z d )⊗C(Y )⊗A(X, H) g⋅V i⊗ B⊗H G , satisfying: (1) supp(T y,z ) ⊂ g ⋅ V i for all y, z ∈ P d (G); (2) there exists some C > 0, such that T y,z ≠ 0 implies that d(y, Taking closure under the operator norm over the Hilbert module gives rise to a C * -algebra, denoted by A g, * .
Similarly, we can define a localized version of the above algebra, denoted by A g, * L . Let us define It is not difficult to check that According to the definition of twisted Roe algebras and coarse embeddability of the group G N i , for any T ∈ A e, * , there is a constant M such that T x,y ≠ 0 implies that For the Roe algebra case, there is a * -isomorphism For the localization algebra case, it suffices to show that for d > 0 large enough, the C * -algebra A e, * L,M has the same K-theory as C * L (P d (B G (N i , M )), C(Y )⊗A(X, H)⊗B) N i for any fixed M . By Proposition 2.11, it suffices to show that P d (B G (N i , M )) is strongly N i -homotopy equivalent to P d (N i ) when d is large enough.
When d is large enough, we can define a strong Lipschitz homotopy equivalence between the subcomplexes P d (B G (N i , M )) and P d (N i ) as follows. For any element g ∈ G with d(g, N i ) ≤ M , there exists an element s ∈ G with s ≤ M such that gs −1 ∈ N i , and the number of elements in the M -ball of the group G is finite. Let B G (N i , M ) = ⊔ n 0 k=1 N i s k where s k ≤ M , and {s 1 , s 2 , ⋯, s n 0 } is a subset of representatives for the right cosets G N i . We can define a map ρ ∶ B G (N i , M ′ ) → N i by ρ(g) = g ′ where g = g ′ s is the unique product with g ′ ∈ N i , and s ∈ {s 1 , s 2 , ⋯, s n 0 }. Uniqueness of the product is guaranteed by the fact that {s 1 , s 2 , ⋯, s n 0 } is a subset of the representatives for the right cosets of G N i . It is easy to check that the map ρ is well-defined and N i -invariant.
We can define a strong Lipschitz homotopy equivalence where t ∈ [0, 1], and ∑ i c i g i ∈ P d (B G (N i , M )). By Proposition 2.11, the localization algebra version is done.
Lemma 5.6. For all i, and all d > 0, we have a commutative diagram We need the following result due to Tu (see [23]).
Theorem 5.7. Let Y ⋊ Γ be an a-T-menable transformation groupoid. Then for any Γ-C * -algebra A, the Baum-Connes conjecture with coefficients in C(Y )⊗A holds for Γ, i.e., the map on K-theory induced by the evaluation-at-zero map is isomorphic.
In [23], Tu constructed a continuous field of C * -algebras which admits a proper Y ⋊ Γ-action by the a-T-menability of the groupoid X ⋊ Γ. The continuous field defined by Tu of C * -algebras is essentially the same as the one that we described in Section 3. Then the Baum-Connes conjecture for the groupoid X ⋊ Γ is reduced from the Baum-Connes conjecture for X ⋊ Γ with coefficients in the continuous field of C * -algebras by the Dirac-dual-Dirac method. In fact, the Baum-Connes conjecture with coefficients in C(X) is actually the Baum-Connes conjecture for the groupoid X ⋊ Γ.
Combining Lemma 5.5 with Theorem 5.7, we have the following result.
. We obtain the following decomposition for the twisted localization algebras and the twisted Roe algebras.
The proof of Lemma 5.11 is similar to that of Lemma 6.3 in [28], and is therefore omitted.
Note that, for any 1 is of the same form as V ′ × N ′ G for some open subset V ′ ⊂ W and some subgroup N ′ ≤ G containing N with finite index. So by the Mayer-Vietoris sequence and the Five Lemma, we have the following result.
Proposition 5.12. Let 1 → N → G → G N → 1 be an extension of countable discrete groups. Assume N and G N are coarsely embeddable into Hilbert spaces. The evaluation-at-zero map By taking the direct limit over R, we have the twisted Baum-Connes conjecture with coefficients in C(Y )⊗A(X, H)⊗B.
The proof of Theorem 5.1. Since we have it suffices to show that the map is an isomorphism, where we set for brevity. By Lemma 5.11 and O R = ⋃ i G ⋅ V i (R), the proof is completed using the Mayer-Vietoris sequence and the Five Lemma.

Proof of the main theorem
In this section, we will prove the main result of this paper. We are going to define a geometric analogue of the infinite-dimensional Bott map introduced by Higson, Kasparov and Trout in [13], and then prove the Novikov conjecture from the twisted Baum-Connes conjecture.
Let X be the compact space and H x x∈X the continuous field of Hilbert spaces defined in Section 4. Every element in C(X)⊗S can be viewed as a continuous Svalued map. The action of G on C(X)⊗S is given by In Section 3, we obtain the fiber-wise defined Bott map β t ∶ C(X)⊗S → A(X, H), by β t ((f x ) x∈X ) = (β t (f x )) x∈X where f x ∈ S, for any x ∈ X. It induces an isomorphism on the K-theory level.
Each element T ∈ C * (P d (G), G, C(Y )⊗C(X)⊗B)⊗S can be expressed as where f ∈ S, T y,z ∈ C(Y )⊗C(X)⊗B⊗K G , for all y, z ∈ Z d . We define the Bott map between the algebraic Roe algebras and algebraic twisted localization algebras, β t (T )(y, z) = T y,z⊗ s y,z , where s y,z ∈ A(X, H) is a section with s y,z (x) = β t (b(x, J(y))(f ), and is the Bott map induced by the inclusion {b(x, J(y))} → H x , for t ∈ [0, ∞).
We are going to show that the Bott map β t is well-defined. It suffices to show that, for each T⊗f = T y,z y, Let s g ∶ X → H = ⊔ x∈X H x be the continuous section defined by s g (x) = b(x, g), for all g ∈ G. By Lemma 3.7, we have that β(g ⋅ s) = g ⋅ β(s) for all s ∈ A(X, H). For each g ∈ G, y, z ∈ Z d , it follows from the definition of T⊗f that g(T g −1 y,g −1 z ) = T y,z .
It suffices to show that g ⋅ (β t (s J(g −1 y) )(f )) = β(s J(y) )(f ). For each x ∈ X, we have For the section s J(y) , we have for all x ∈ X. Therefore g ⋅ s J(g −1 y) = s J(y) , for all y ∈ Z d , and hence it follows that g(β(s J(g −1 y) )(f )) = β t s J(y) (f ).
Therefore, g(β t (T⊗f ) g −1 y,g −1 z ) = β t (T⊗f ) y,z . As a consequence, the map Proposition 6.1. The family of maps z a z converges in norm .
For every g ∈ S, define a bounded module homomorphism N g ∶ E → E given by for all ∑ z∈Z d a z [z]. It is easy to check that for all g ∈ S and T ∈ C * alg (P d (G), G, C(Y )⊗C(X)⊗B), where T⊗1 is a bounded module homomorphism from E to E by By the definition of Roe algebras, the map β t extends to a linear map satisfying β t (T⊗g) ≤ g T for all g ∈ S, and T ∈ C * alg (P d (G), G, C(Y )⊗C(X)⊗B). By the definition of C * (P d (G), G, C(Y )⊗C(X)⊗B) and the properness of the G N -action on the C * -algebra C(Y )⊗A(X, H)⊗B, one can verify that β t is an asymptotic morphism from C * (P d (G), G, C(Y )⊗C(X)⊗B)⊗ alg S to C * (P d (G), C(Y )⊗A(X, H)⊗B) G . Hence β t extends to a homomorphism As a consequence of the nuclearity of S, β t extends to a homomorphism We define the Bott map on K-theory, as that induced by the asymptotic morphism Similarly, we can define the localized version of the asymptotic morphism Since this is an asymptotic morphism, it induces the Bott map on K-theory . Note that, for every d > 0, the following diagram is commutative. Let C * L (P d (G), C(Y )⊗A(X, H)⊗B) G be the twisted localization algebra. Let C 1 ⊂ C 2 ⊂ P d (G) be G-invariant closed subsets such that C i = Ğ C i ∩ Z d for i = 1, 2. Assume that the inclusion map i ∶ C 1 → C 2 is a strong Lipschitz homotopy equivalence. Let C * L (C i , C(Y )⊗A(X, H)⊗B) G be the C * -subalgebra consisting of all the operators Then the map i ∶ C 1 ↪ C 2 induces a map on the twisted Roe algebras otherwise, for all T = (T y,z ) y,z∈C 1 ∈ C * (C 1 , C(Y )⊗A(X, H)⊗B) G . This map is well-defined, because the inclusion map i ∶ C 1 ↪ C 2 is isometric. Similarly, one can define a homomorphism between the localization algebras The following result is a twisted analogue of the result that the K-theory of twisted localization algebras is invariant under strong Lipschitz homotopy equivalence. The proof is similar to that of Proposition 2.11. Lemma 6.2. Assume the inclusion map i ∶ C 1 ↪ C 2 is a strong Lipschitz homotopy equivalence. The map (i L ) * ∶ K * (C * L (C 1 , C(Y )⊗A(X, H)⊗B) G ) → K * (C * L (C 2 , C(Y )⊗A(X, H)⊗B) G ) induced by i L on K-theory is an isomorphism.
Similarly, we have a twisted version of the Mayer-Vietoris sequence for twisted localization algebras, as in Proposition 2.13. The proof will be similar, so it is omitted. Lemma 6.3. Let ∆ be a simplicial complex endowed with the spherical metric, and let G be a countable discrete group. Assume G acts on ∆ properly by isometries. Let C 1 , C 2 ⊂ ∆ be G-invariant simplicial subcomplexes endowed with the subspace metric. Then we have the following six-term exact sequence where we set L C 1 = C * L (X 1 , C(Y )⊗A(X, H)⊗B) G , L C 2 = C * L (X 1 , C(Y )⊗A(X, H)⊗B) G , L C 1 ∩C 2 = C * L (X 1 ∩X 1 , C(Y )⊗A(X, H)⊗B) G and L C 1 ∪C 2 = C * L (X 1 ∪C 2 , C(Y )⊗A(X, H)⊗B) G for short. Proof. By induction on the dimension of the space P d (G), the theorem is a consequence of the fact that the K-theory of localization algebras is invariant under the strong Lipschitz homotopy equivalence (see Lemma 6.2), Theorem 3.8, the twisted Mayer-Vietoris sequence (see Lemma 6.3) and the Five Lemma.
Let us recall the commutative diagram which is obvious from the definition of twisted localization algebras and twisted Roe algebras. In the above diagram, the vertical map β L, * and the bottom horizontal map are isomorphisms. As a result, we have that the Novikov conjecture holds for G with coefficients in C(Y )⊗C(X)⊗B. induced by the evaluation-at-zero map on K-theory is injective.
In the rest of this section, we will reduce the Novikov conjecture to Lemma 6.5. By identifying the C * -algebra C(Y )⊗C(X) with C(Y × X), one can define a map c ∶ C → C(Y )⊗C(X) mapping each constant s ∈ C to the constant function with value s on Y × X. By tensoring with the identity map on compact operators, we can define a map c⊗1 ∶ K G → C(Y )⊗C(X)⊗K G .
Note that Y ×X admits a G-action and it is G 0 -contractible for any finite subgroup G 0 ≤ G. The map c ⊗ 1 induces a homomorphism between the Roe algebras c ∶ C * (P d (G), G, B) → C * (P d (G), G, C(Y )⊗C(X)⊗B) given byc (T )(x, y) = c⊗1(T x,y ), for x, y ∈ Z d , T = T x,y x,y∈Z d . Similarly, one can define a localized version of the homomorphism c L ∶ C * L (P d (G), G, B) → C * L (P d (G), G, C(Y )⊗C(X)⊗B), byc L (g)(t) =c(g(t)), where g ∈ C * L (P d (G), G, B). Lemma 6.6. Let G 0 ≤ G be a finite subgroup, and B any G-C * -algebra. If V ⊂ P d (G) is a G 0 invariant and G 0 -contractible subcomplex such that G ⋅ V is homeomorphic to the space V × G 0 G, then we have K * (C * L (V × G 0 G, G, B)) ≅ K * (C * L (V, G 0 , B)). Proof. Define a homomorphism ı ∶ C * L (V, G 0 , B) → C * L (V 1 × G 0 G, G, B) by ı(b(t)) x,y = g −1 (b(t)) gx,gy if ∃ g ∈ G such that gx, gy ∈ V , 0 otherwise, for all b(t) ∈ C * L (V, G 0 , B). Any element in K 1 (C * L (V × G 0 G, G, B)) can be represented by an invertible element a + I, for some a ∈ C * L (V × G 0 G, G, B). Since the propagation of a(t) approaches 0 as t → ∞, we can find a large constant T 0 such that supp(a(t)) ⊂ ⊔ g∈G G 0 gV × gV for all t ≥ T 0 . By uniform continuity of the path a(t), a s (t) = a(t + sT 0 ) (s ∈ [0, 1]) is a homotopy between a(t) and a(t + T 0 ). Thus, any element in K 1 (C * L (V × G 0 G, G, B)) can be represented by an invertible element b + I ∈ C * L (V × G 0 G, G, B) + with supp(b(t)) ⊂ ⊔ g∈G G 0 gV 1 × gV 1 . Since b(t) is G-invariant for all t ∈ [0, ∞), we can find an element b ′ ∈ C * L (V, G 0 , B) such that ı * ([b ′ + I]) = [b + I]. Consequently, the map ı is onto, and we can similarly prove that it is injective. Therefore we have G 0 , B)). The K 0 case can be dealt with by a suspension argument.
The following result was proved originally using E-theory ( [8], Lemma 12.11). We now give an alternative proof using localization algebras. Proof. When d is large enough, there exist finitely many precompact open subsets V i , and finite subgroups G i ≤ G, i = 1, . . . , k, such that P d (G) = ⋃ k i=1 V i × G i G and each V i is G i -contractible by a strong Lipschitz G i -homotopy equivalence. The existence of subsets V i and finite subgroups G i is guaranteed by properness of the G-action on P d (G). By the Mayer-Vietoris sequence, it suffices to show that the map (c L ) * ∶ K * (C * L (V i × G i G, G, B)) → K * (C * L (V i × G i G, G, C(Y × X)⊗B)) is an isomorphism for each i. Without loss of generality, it suffices to show this for i = 1.
According to Proposition 2.15, the following result implies Theorem 1.1. Since the map (c L ) * is an isomorphism, and the lower horizontal map is injective, the upper horizontal map is injective.