A Lichnerowicz Vanishing Theorem for the Maximal Roe Algebra
aa r X i v : . [ m a t h . K T ] N ov A LICHNEROWICZ VANISHING THEOREM FOR THE MAXIMALROE ALGEBRA
HAO GUO, ZHIZHANG XIE, AND GUOLIANG YU
Abstract.
We show that if a countable discrete group acts properly and isomet-rically on a spin manifold of bounded Riemannian geometry and uniformly positivescalar curvature, then, under a suitable condition on the group action, the maximalhigher index of the Dirac operator vanishes in K -theory of the maximal equivariantRoe algebra. The group action is not assumed to be cocompact. A key step in theproof is to establish a functional calculus for the Dirac operator in the maximalequivariant uniform Roe algebra. This allows us to prove vanishing of the index ofthe Dirac operator in K -theory of this algebra, which in turn yields the result forthe maximal higher index. Contents
1. Introduction 22. Maximal equivariant Roe algebras 32.1. Geometric conditions 32.1.1. Bounded geometry 32.1.2. A condition on the Γ-action 42.2. Operator algebras 52.3. Estimating the maximal norm 72.3.1. Discretizing H M Mathematics Subject Classification. .2. Vanishing of the maximal uniform index 204.3. Vanishing of the maximal index 21References 231. Introduction
The connection between index theory and the existence question for metrics ofpositive scalar curvature on spin manifolds goes back to the classical version of theLichnerowicz vanishing theorem [6], which states that if a closed spin manifold admitsa metric of positive scalar curvature, then the Fredholm index of the Dirac operatorvanishes. By taking into account the fundamental group, one can define a morerefined invariant called the higher index . For cocompact group actions, this indextakes values in the K -theory of the (reduced or maximal) group C ∗ -algebra. By apioneering result of Rosenberg [11], if a closed spin manifold admits a positive scalarcurvature metric, then the higher index of the Dirac operator lifted to the universalcover vanishes.For group actions that are not necessarily cocompact, the higher index of the Diracoperator takes values in the K -theory of the equivariant Roe algebra [10], which forcocompact manifolds is isomorphic to the K -theory of the reduced group C ∗ -algebra.In [3], Gong, Wang, and Yu introduced a version of this index that takes values inthe K -theory of the maximal equivariant Roe algebra, in analogy to the situation forgroup C ∗ -algebras. The maximal equivariant Roe algebra is well-defined when themanifold and group action satisfy certain geometric conditions that we shall makeprecise. One of the advantages of working with the maximal version of the higherindex is that it enjoys better functoriality properties than its reduced counterpart,making it in some cases more computable [7, 13, 14, 2]. The maximal higher indexis used in an essential way in the recent work [1] of Chang, Weinberger, and Yu,who establish a new index theory for non-compact manifolds and use it to provideexamples of manifolds with exotic scalar curvature behavior.A basic notion on which the results of [1] rest is that the maximal higher index ofthe spin-Dirac operator vanishes in the presence of uniformly positive scalar curva-ture. However, it was pointed out in [12] that, compared to the reduced setting, aLichnerowicz-type argument for such a vanishing result needs to be carried out withmore care, due to analytical difficulties that arise in connection with the maximalequivariant Roe algebra.Our intention in this paper is to prove that, under a natural geometric assumptionon the group action, which in particular in satisfied in the setting considered in [3], his index does indeed vanish in the presence of uniformly positive scalar curvature.More precisely, we prove: Theorem 1.1.
Let ( M n , g ) be a Riemannian manifold with bounded Riemanniangeometry and Γ a countable discrete group acting properly and isometrically on M ,satisfying Assumption 2.6. Suppose that M has a Γ -equivariant spin structure, with S and D the spinor bundle and Dirac operator respectively. If M has uniformly positivescalar curvature, then index max ( D ) = 0 ∈ K n ( C ∗ max ( M ) Γ ) , where C ∗ max ( M ) Γ is the maximal equivariant Roe algebra of M . We remark that this result is new even when Γ is the trivial group.Let us give a brief overview of the paper. In section 2, we recall some standardnotions from higher index theory, as well as describe the two geometric conditionsalluded to above. We show that, under these geometric conditions, it is possibleto define various related versions of the maximal equivariant Roe algebra of M , inparticular the maximal equivariant uniform Roe algebra . In section 3, we show thatthis algebra can be viewed as a Hilbert module on which the operator D acts. A keypoint is that this operator is regular and essentially self-adjoint and hence admits afunctional calculus. This allows us to define, in section 4, the maximal equivariant uniform index of D , which we show vanishes under the positive scalar curvatureassumption. This, together with the fact that the equivariant uniform Roe algebra isa subalgebra of the maximal equivariant Roe algebra, implies Theorem 1.1.2. Maximal equivariant Roe algebras
Geometric conditions.
We now make precise the two geometric conditions alluded to above. The first isthat the manifold M has bounded Riemannian geometry. Under this assumption,it was established in [3] that, when the acting group Γ is trivial, the maximal Roealgebra (see Definition 2.13) is well-defined. However, to prove the analogous resultfor general Γ, we will need a second hypothesis (even if the action is free). This isgiven by Assumption 2.6.2.1.1. Bounded geometry.
Definition 2.1.
A Riemanniann manifold M is said to have bounded Riemanniangeometry if it has positive injectivity radius, and the curvature tensor and each of itscovariant derivatives is uniformly bounded across M .There is also a notion of bounded geometry for discrete metric spaces: efinition 2.2. A discrete metric space (
X, d ) is said to have bounded geometry iffor any r > N r > x ∈ X , B r ( x ) = { x ′ ∈ X | d ( x, x ′ ) ≤ r } ≤ N r . Suppose M has bounded Riemannian geometry, and let d be the Riemannian dis-tance. Then the metric space ( M, d ) contains a countable discrete subspace X M withbounded geometry such that, for some constants c > r >
0, we have: • B c ( X M ) = M , or that X M is c -dense ; • for all x, y ∈ X M , B r ( x ) ∩ B r ( y ) = ∅ = ⇒ x = y. (1)For any r ≥
0, let us define the following two quantities: U r := inf x ∈ M { vol B r ( x ) } , V r := sup x ∈ M { vol B r ( x ) } . Note that bounded Riemannian geometry implies that for each r > < U r ≤ V r < ∞ . The fact that the Ricci curvature of M is bounded from below means that it satisfiesthe following volume estimate [8, Lemma 7.1.3]: Lemma 2.3.
There exist constants C , C > such that for any r ≥ , V r ≤ C e C r . It follows from this, and the definition of X M , that: Lemma 2.4.
There exists a constant C such that for each x ∈ M and R > , B R ( x ) ∩ X M ) ≤ C V R /U r . A condition on the Γ -action. To state the second geometric condition, we use the notion of a basic domain . Ob-serve that, since Γ acts properly on M , each x ∈ M is contained in a Γ-invariantneighborhood of the form Γ V ∼ = Γ × Γ x V . Here V is an open Γ x -invariant neighbor-hood of x with compact closure, Γ x is the stabilizer of x , and Γ × Γ x V denotes thequotient of Γ × V by the relation ( g, y ) ∼ ( gh − , hy ), for h ∈ Γ x , g ∈ Γ, and y ∈ V ;the isomorphism is given by sending gy to the class [ g, y ]. From this it follows that M can be written as a disjoint union M ∼ = ⊔ i Γ N i , where each N i is a Borel subset of M that is preserved by the action of an isotropy subgroup F i , and for each i we havea Γ-equivariant homeomorphism Γ N i ∼ = Γ × F i N i . Definition 2.5.
We call N := ∪ i N i ⊆ M the basic domain for the decomposition M ∼ = ⊔ i Γ N i given above. or the rest of this paper, we will make the following standing assumption: Assumption 2.6.
There exists a basic domain N for the Γ -action on M such that l ( g ) → ∞ = ⇒ d ( N, gN ) → ∞ , where l : Γ → N is a fixed length function and d is the Riemannian distance. We remark that Assumption 2.6 is satisfied when the Γ-action on M is cocompact,or when M is a cocompact manifold to which an infinite cylinder is attached. Inparticular, it is satisfied in the situation studied in [1]. It is also satisfied by anyaction of a finite group on a manifold of bounded Riemannian geometry.Suppose the Γ-action on M satisfies Assumption 2.6. Then we have the followingtwo easy consequences. Corollary 2.7. M decomposes into finitely many orbit types. Corollary 2.8.
For each x ∈ M and R > , there exists C R > such that B R ( x ) ∩ G · x ) ≤ C R . Proof.
Observe that Assumption 2.6 implies that this relation holds for x ∈ N . Thegeneral statement follows by observing that for any g ∈ Γ, B R ( x ) ∩ G · x ) = B R ( g · x ) ∩ G · x ) . (cid:3) Operator algebras.
We now recall the definitions of geometric modules and Roe algebras, with the goalof proving that, under the previously stated geometric assumptions, the maximalequivariant Roe algebra is well-defined. We also provide an estimate of the maximalnorm that will be important in section 3 (see Lemma 2.19).
Notation.
Let M , S , and Γ be as in the introduction. We will use B ( M ), C b ( M ),and C ( M ) to denote the C ∗ -algebras of complex-valued functions on M that are,respectively: bounded Borel, bounded continuous, and continuous and vanishing atinfinity. A superscript ‘ ∞ ’ may be added to the latter two algebras to indicate theadditional requirement that its elements be smooth.We will use d to denote the Riemannian distance on M , and B r ( S ) to denote theopen ball in M of radius r around a subset S ⊆ M . For any two sets A and B , wewill write pr and pr for the projections of the cartesian product A × B , or subsetsthereof, onto A and B respectively.The Γ-action on M naturally determines a Γ-action on spaces of functions over M :for a function f and g ∈ Γ, let g · f be the function given by g · f ( x ) = f ( g − x ). Moregenerally, for a section s of a Γ-vector bundle over M , define g · s ( x ) = g ( s ( g − x )).The action of Γ on l (Γ) will always be given by the left-regular representation. efinition 2.9. An M -Γ-module is a separable Hilbert space H equipped with anon-degenerate ∗ -representation ρ : C ( M ) → B ( H ) and a unitary representation U : Γ → U ( H ) such that for all f ∈ C ( M ) and g ∈ Γ, U g ρ ( f ) U ∗ g = ρ ( g · f ).For brevity, we will omit ρ from the notation when it is clear from context. Definition 2.10.
Let H be an M -Γ-module and T ∈ B ( H ). • The support of T , denoted supp( T ), is the complement in M × M of the setof ( x, y ) for which there exist f , f ∈ C ( M ) with f ( x ) = 0, f ( y ) = 0, and f T f = 0; • The propagation of T is the extended real numberprop( T ) = sup { d ( x, y ) | ( x, y ) ∈ supp( T ) } ; • T is locally compact if f T and T f ∈ K ( H ) for all f ∈ C ( M ); • T is Γ -invariant if U g T U ∗ g = T for all g ∈ Γ.Let C [ M ; H ] Γ ⊆ B ( H ) be the ∗ -subalgebra of Γ-invariant, locally compact operatorswith finite propagation.We will work with certain maximal completions of C [ M ; H ] Γ . In order show thatsuch completions are well-defined, we restrict ourselves to those modules H thatsatisfy an additional admissibility condition. To state this, we need the following fact:if H is a Hilbert space and ρ : C ( M ) → B ( H ) a non-degenerate ∗ -representation,then ρ extends uniquely to a ∗ -representation ˜ ρ : B ( M ) → B ( H ) subject to theproperty that, for a uniformly bounded sequence in B ( M ) converging pointwise, thecorresponding sequence in B ( H ) converges in the strong topology. Definition 2.11 ([16]) . An M -Γ-module H is admissible if:(i) For any non-zero f ∈ C ( M ), π ( f ) / ∈ K ( H );(ii) For any finite subgroup F of Γ and any F -invariant Borel subset E of M , thereis a Hilbert space H ′ equipped with the trivial F -representation such that˜ π ( E ) H ′ ∼ = l ( F ) ⊗ H ′ as F -representations, where ˜ π is defined by extending π as above.If an M -Γ-module H is admissible, we will write C [ M ] Γ := C [ M ; H ] Γ , noting that C [ M ] Γ is independent of the choice of admissible module.In this paper we will use two M -Γ-modules. The first is L ( S ), equipped with thenatural Γ-action and C ( M )-representation. In general, L ( S ) is not admissible (seealso Remark 2.12). The second is the space H M := L ( S ) ⊗ l (Γ) , quipped with the multiplicative action of C ( M ) on the first factor and the diagonalΓ-action. One verifies that L ( S ) ⊗ l (Γ) is an admissible M -Γ-module.We can view L ( S ) as a submodule of H M in the following way. Let χ ∈ C ∞ ( M ) bea cut-off function, meaning that supp( χ ) has compact intersection with every Γ-orbitand that for all x ∈ M , we have P g ∈ Γ χ ( gx ) = 1 . Note that this sum is finite byproperness of the action. Then the map j : L ( S ) → H M , j ( s )( x, g ) = χ ( g − x ) s ( x )is a Γ-equivariant isometric embedding. Let p : H M → j ( L ( S )) be the orthogonalprojection associated to j . One calculates that j ∗ p ( t )( x ) := X g ∈ Γ χ ( g − x ) t ( x, g ) , for t ∈ L ( S ) ⊗ l (Γ). On operators, j induces a map taking T jT j − p , and wewill denote this by ⊕ B ( L ( S )) → B ( H M ). It is an injective ∗ -homomorphism thatpreserves Γ-equivariance, local compactness, as well as finiteness of propagation, andhence restricts to an injective ∗ -homomorphism C [ M, L ( S )] Γ ֒ → C [ M ] Γ . Remark 2.12.
When the Γ-action M is both proper and free, L ( S ) is itself anadmissible M -Γ-module, and the discussion above simplifies. Definition 2.13.
For an operator T ∈ C [ M ] Γ , its maximal norm is || T || max := sup φ,H ′ n k φ ( T ) k B ( H ′ ) | φ : C [ M ] Γ → B ( H ′ ) is a ∗ -representation o . The maximal equivariant Roe algebra of M , denoted C ∗ max ( M ) Γ , is the completion of C [ M ] Γ with respect to || · || max .2.3. Estimating the maximal norm.
To make sense of Definition 2.13, one needs to show that for any T ∈ C [ M ] Γ , thereexists a constant C bounding the norm of T in any ∗ -representation. We now showthat this is the case under the geometric conditions in subsection 2.1, namely: Proposition 2.14.
Suppose that M has bounded geometry and that Assumption 2.6holds. Then for any T ∈ C [ M ] Γ and any ∗ -homomorphism φ : C [ M ] Γ → B ( H ′ ) , for H ′ a Hilbert space, we have k φ ( T ) k B ( H ′ ) < ∞ . In order to perform the estimates required to prove this, we will work with themodule H M in a discretized form that we now describe. .3.1. Discretizing H M . Let us consider the case when S is the trivial line bundle over M , with the generalcase being analogous. Let X M , c , and r be as in 2.1.1. The fact that X M is c -dense,together with (1), implies that there exists a Borel cover U of M such that, for each U ∈ U , there exists x ∈ X M with B r ( x ) ⊆ U ⊆ B c ( x ).Let π : M → M/ Γ denote the projection onto orbits. Using the cover U , we canconstruct a subset X of X M with the following properties: • B c (Γ · X ) = M ; • There exists a constant
C > x ∈ X and R > π ( B R (Γ · x ) ∩ (Γ · X )) ≤ CV R /U r . First, let us use the Γ-invariant set Z := Γ · X to rewrite the module H M ∼ = L ( M × Γ) in the following way. Since the (diagonal)action of Γ on M × Γ is free, there exists a fundamental domain D ⊆ M × Γ forthis action. We may choose D so that pr ( D ) ⊆ N , where N is the basic domain inAssumption 2.6. Then the set D := D ∩ ( Z × Γ) is a fundamental domain for theΓ-subspace Z × Γ ⊆ M × Γ. We may choose D in such a way that it contains D asa c -dense subset. This defines for us a a unitary isomorphism L ( D ) ∼ = l ( D ) ⊗ H for some separable Hilbert space H , and in turn a Γ-equivariant unitary isomorphism H M ∼ = L ( M × Γ) ∼ = L ( D ) ⊗ l (Γ) ∼ = l ( D ) ⊗ l (Γ) ⊗ H ∼ = l ( Z × Γ) ⊗ H, (2)where H is equipped with the trivial Γ-representation.In order to state the final form of H M that will be useful to us, we need some morepreparation. Given a point y ∈ Z , let O y ⊆ Z and F y < Γ denote its orbit andstabilizer respectively. Set-theoretically, we may identify O y with Γ /F y and thus abijective map ϕ y : O y × F y ∼ = −→ Γ /F y × F y , defined in the obvious way. Choose a section φ y : Γ /F y → Γ. We have a bijection˜ φ y : Γ → Γ /F y × F y ,g ( gF y , g − φ y ( gF y )) . Let Γ act on Γ /F y × F y simply by the pushforward of the Γ-action on itself along ˜ φ y .Now consider the collection of orbits W := π ( Z ) ⊆ M/ Γ. For each O ∈ W , choose arepresentative in N , and let Y be the collection of representatives so obtained as O anges over W . Define the sets˜ Z := G y ∈ Y ( O y × F y ) , E := G y ∈ Y (Γ /F y × F y ) , and equip them with piecewise Γ-actions. Upon taking a disjoint union of the maps ϕ y , we obtain a Γ-equivariant bijection ϕ : ˜ Z ∼ = −→ E. This in turn gives equivalent Γ-representations on the Hilbert spaces M y ∈ Y l ( O y × F y ) ∼ = M y ∈ Y l (Γ /F y × F y ) . We can now state the following:
Proposition 2.15.
There is a Γ -equivariant unitary isomorphism H M ∼ = l ( ˜ Z ) ⊗ H, where H is a Hilbert space equipped with the trivial Γ -representation.Proof. By (2), we have H M ∼ = M y ∈ Y l ( O y × Γ) ⊗ H ∼ = M y ∈ Y l (Γ /F y × Γ) ⊗ H. (3)For each y ∈ Y , let ν y : F y \ Γ → Γ be a section. Let φ y and ˜ φ y be given as above.One verifies that the following map is a Γ-equivariant bijection: ϑ y : Γ /F y × Γ → Γ /F y × F y × F y \ Γ , ( g F y , g ) ( g F y , ν y ( F y g − g ) g − φ y ( g ν y ( F y g − g ) − F y ) , F y g − g ) , where Γ acts on the left diagonally, while on the right it acts on Γ /F y × F y by pushingforward the left-action of Γ on itself along ˜ φ y , and trivially on F y \ Γ. Indeed, ϑ y canbe written as a composition of maps as follows. Define the bijection˜ ν y : Γ → F y × F y \ Γ , g ( gν y ( F y g ) − , F y g ) . Then ϑ y is the following composition of bijections:Γ /F y × Γ → Γ × F y Γ → (Γ × F y F y ) × F y \ Γ → Γ × F y \ Γ → Γ /F y × F y × F y \ Γ , ( g F y , g ) [( g , g − g )] ([ g , pr (˜ ν y ( g − g ))] , pr ( ˜ φ y ( g − g ))) ( g ν y ( F y g − g ) − , F y g − g ) ( ˜ φ y ( g ν y ( F y g − g ) − ) , F y g − g ) , here Γ × F y Γ and Γ × F y F y are, respectively, the quotients of Γ × Γ and Γ × F y bythe equivalence relation ( g , g ) ∼ ( g k − , kg ) , for g ∈ Γ, k ∈ F y , and g in either Γor F y . For each y ∈ Y , ϑ y induces a Γ-equivariant isomorphism l (Γ /F y × Γ) ∼ = l (Γ /F y × F y ) ⊗ l ( F y \ Γ) , where Γ acts trivially on l ( F y \ Γ). Combining this with (3) gives H M ∼ = M y ∈ Y l (Γ /F y × F y ) ⊗ l ( F y \ Γ) ⊗ H ∼ = M y ∈ Y l ( O y × F y ) ⊗ l ( F y \ Γ) ⊗ H. Now, for each y , pick an identification of l ( F y \ Γ) ⊗ H with H to give H M ∼ = M y ∈ Y l ( O y × F y ) ⊗ H ∼ = l ( ˜ Z ) ⊗ H. (cid:3) The isomorphism constructed in the above proof gives a Γ-equivariant identificationbetween operators in B ( H M ) and ˜ Z × ˜ Z -matrices with entries in B ( H ). Moreover,this construction strongly relates the propagation of an operator to the off-diagonalsupport of the corresponding matrix. The following notion makes this more precise. Definition 2.16.
The matricial support of T ∈ B ( H M ) is the setmatsupp( T ) = { ( w, z ) ∈ ˜ Z × ˜ Z | T wz = 0 } . Using the map pr : ˜ Z pr −−→ G y ∈ Y O y ֒ → M, we can relate the matsupp( T ) to its M × M -support (Definition 2.10). In particular,it follows from the fact that the set Z is c -dense in M thatmatsupp( T ) ≤ prop( T ) + c. Before we proceed to estimates of the maximal norm, let us make one more obser-vation, namely that the subset F := n ( x, e ) ∈ ˜ Z | x ∈ Y o . is a fundamental domain for the Γ-action on ˜ Z . Thus if T ∈ B ( H M ) is Γ-invariant,it is determined by its entries in ˜ Z × F , while if T also has finite propagation, thenone only needs to know the entries in B ˜ Z prop( T )+ c ( F ) × F , where B ˜ ZR ( S ) := { z ∈ ˜ Z | d ˜ Z ( z, S ) < R } , or a subset S ⊆ ˜ Z and R >
0. Finally, let us introduce one more convenient piece ofnotation. If
S, S ′ are subsets of ˜ Z and R >
0, then we will write d ˜ Z ( S, S ′ ) := d (pr( S ) , pr( S ′ )) . (If S = { z } , we will write z instead.)2.3.2. Norm estimation.
We now proceed with the proof of Proposition 2.14. The first observation is:
Lemma 2.17.
There exists a constant C such that for any z ∈ F and R > , B ˜ ZR ( z ) ≤ CV R . Proof.
By Corollary 2.7, the size of the stabilizer F x of any point x ∈ M is uniformlybounded. Thus it suffices to show that there exists C such that B ˜ ZR ( z )) ≤ CV R for any z ∈ F and R >
0. In other words, it suffices to show that for any x ∈ N , B R ( x ) ∩ Z ) ≤ CV R , where Z is as in 2.3.1 To this end, let U be the Borel coverfrom 2.3.1. Lemma 2.4 implies that there exists C such that { U ∈ U : U ∩ B R ( x ) = ∅} ≤ C V R /U r . Meanwhile, Corollary 2.8 implies that each element of U contains at most C c pointsfrom any single Γ-orbit O ⊆ M (here C c is the constant C R from Corollary 2.8 with R = 2 c ). Thus B R ( x ) ∩ O ) ≤ V R + c C c /U r . By construction, the number of orbits in the set Z that intersect B R ( x ) is boundedabove by C V R /U r for some constant C . It follows from Lemma 2.3 that B R ( x ) ∩ Z ) ≤ CV R , where C is independent of R . (cid:3) Lemma 2.18.
There exists a constant C such that for any z ∈ F and R > , B ˜ ZR ( z ) ∩ F ) ≤ CV R . Proof.
This follows from the proof of Lemma 2.17, but without the need to considerthe orbit direction. (cid:3)
We are led to the following lemma, which in particular implies Proposition 2.14: emma 2.19. For any T ∈ C [ M ] Γ and any ∗ -homomorphism φ : C [ M ] Γ → B ( H ′ ) ,where H ′ is a Hilbert space, k φ ( T ) k B ( H ′ ) ≤ CC T V T ) , where C T := sup w,z ∈ ˜ Z k T wz k and C is a constant independent of T .Proof. Let T denote the operator whose ˜ Z × ˜ Z -matrix entries are equal to those of T on B ˜ Z prop( T )+ c ( F ) × F , with all others being zero. It follows from the proof of [3,Lemma 3.4] that we can write F as a disjoint union of subsets F , F , . . . , F L +1 withthe property that if w, z ∈ F i for some i , then d ˜ Z ( w, z ) > T ) + 3 c . Let Q = { ( z ′ , z ) ∈ ˜ Z × F | d ˜ Z ( z ′ , z ) ≤ prop( T ) + c } . Write Q i = Q ∩ ( ˜ Z × F i ). By Lemmas 2.17, 2.18, and 2.3, there exist C , C > z ∈ F , B ˜ Z T )+3 c ( z ) ∩ F ) ≤ C V T ) , B ˜ Z prop( T )+ c ( z ) ≤ C V T ) . Note that C and C are independent of T . Setting L := (cid:4) C V T ) (cid:5) , L := (cid:4) C V T ) (cid:5) , one sees that for any z ∈ F i , there are at most L elements z ′ ∈ ˜ Z such that( z ′ , z ) ∈ Q i . Thus there exists a disjoint union decomposition Q = ( L +1) L G i =1 P i , where the sets P i have the property that, for any two distinct elements ( w, z ) and( w ′ , z ′ ) ∈ P i , we have d ˜ Z ( z, z ′ ) > T ) + 3 c , and hence w = w ′ . This gives adecomposition T = ( L +1) L X i =1 T i , matsupp( T i ) ⊆ P i . Observe that, for each i , the operator T ∗ i T i is Γ-invariant and has matricial supportconfined to the diagonal of F × F and, as such, has norm at most C T . Let l ∞ ( ˜ Z, K ( H )) Γ ⊆ C [ M ] Γ denote the ∗ -subalgebra of Γ-invariant operators whosematrix entries belong to the diagonal of ˜ Z × ˜ Z . Since it is a C ∗ -algebra, the norm f any operator T ′ ∈ l ∞ ( ˜ Z, K ( H )) Γ contracts under any ∗ -representation of C [ M ] Γ .Applying this to T ′ = T ∗ i T i , and using Lemma 2.3, we get: k φ ( T ) k B ( H ′ ) ≤ ( L +1) L X i =1 k φ ( T i ) k B ( H ′ ) ≤ ( L + 1) L C T ≤ CC T V T ) , for some C independent of T . (cid:3) Maximal Roe algebras.
The results of the previous section allow us to define several versions of the maximalequivariant Roe algebra.The first is the algebra C ∗ max ( M ) Γ defined on the admissible module H M as thecompletion of C [ M ] Γ in the norm k·k max (see Definition 2.13).We also have: Definition 2.20.
The maximal equivariant Roe algebra on the M -Γ-module L ( S ),denoted by C ∗ max ( M ; L ( S )) Γ , is the completion of C [ M ; L ( S )] Γ under the normpulled back under the injective ∗ -homomorphism given by the composition C [ M ; L ( S )] Γ ⊕ −−→ C [ M ; L ( S )] ⊕ ֒ → C ∗ max ( M ) Γ . Finally, we define a uniform version of the maximal Roe algebra on the module L ( S ) as the completion in C ∗ max ( M ; L ( S )) Γ of a certain space Schwartz kernels. Thisalgebra will play a key role in the results of the next section. Recall: Definition 2.21.
A section k of End( S ) ∼ = S ⊠ S ∗ has finite propagation if thereexists an R > x, y ∈ M , d ( x, y ) > R = ⇒ k ( x, y ) = 0 . The infimum of such R is called the propagation of k , denoted by prop( k ). Definition 2.22.
Let S Γ u denote the ∗ -subalgebra of B ( L ( S )) whose elements aregiven by Schwartz kernels k ∈ C ∞ b ( S ⊠ S ∗ ) that satisfy:(i) k has finite propagation;(ii) k ( x, y ) = k ( gx, gy ) for all g ∈ Γ;(iii) Each covariant derivative of k ( x, y ) is uniformly bounded over M .Note that properties (i) and (ii) imply that S Γ u is a ∗ -subalgebra of C [ M ; L ( S )] Γ . Definition 2.23.
The maximal equivariant uniform Roe algebra of M on L ( S ),denoted by C ∗ max ,u ( M ; L ( S )) Γ , is the completion of S Γ u in C ∗ max ( M ; L ( S )) Γ . emark 2.24. Elements of S Γ u are approximable on each local piece of the manifoldby finite-rank operators in a way that is uniform across the manifold. The completionof S Γ u in the operator norm on B ( L ( S )) is referred to as the reduced equivariantuniform Roe algebra on the module L ( S ).3. Functional calculus
We now use the estimates established in the previous section to complete a keystep in the proof of Theorem 1.1, namely to establish a functional calculus for theunbounded operator D on the maximal equivariant uniform Roe algebra. A basicreference for the material used in this section is [5].3.1. A Hilbert module operator.
We view the C ∗ -algebra C ∗ max ,u ( M ; L ( S )) Γ as a right Hilbert module over itself. Theinner product and right action on C ∗ max ,u ( M ; L ( S )) Γ are defined naturally throughmultiplication: for a, b ∈ C ∗ max ,u ( M ; L ( S )) Γ , h a, b i = a ∗ b, a · b = ab, where the adjoint is defined on the kernel algebra S Γ u in the usual way. The algebra ofcompact operators on this Hilbert module can be identified with C ∗ max ,u ( M ; L ( S )) Γ via left multiplication. Similarly, the algebra of bounded adjointable operators canbe identified with the multiplier algebra M of C ∗ max ,u ( M ; L ( S )) Γ .We first show that D can be viewed as an unbounded operator on this Hilbertmodule. The Dirac operator D acts naturally on smooth sections of M × M asfollows: for each s ∈ S Γ u , define Ds to be the section( x, y ) D x s ( x, y ) , where D x means the operator D acting on the x -coordinate. One verifies easily that D is symmetric with respect to the inner product structure defined above. Furthermore, D preserves the space S Γ u , hence it makes sense to consider the operator D l : S Γ u → S Γ u for each positive integer l . In keeping with the usual notion of an unbounded op-erator on a Hilbert module, we need to ensure that the domain of D l is a right C ∗ max ,u ( M ; L ( S )) Γ -module. To do this, let ( C ∗ max ,u ( M ; L ( S )) Γ ) + be the unitizationof C ∗ max ,u ( M ; L ( S )) Γ . Then the right ideal S Γ u · ( C ∗ max ,u ( M ; L ( S )) Γ ) + contains S Γ u and admits a right action by C ∗ max ,u ( M ; L ( S )) Γ . We can extend the action of D l in a natural way to S Γ u · ( C ∗ max ,u ( M ; L ( S )) Γ ) + by setting, for each a ∈ S Γ u and ∈ C ∗ max ,u ( M ; L ( S )) Γ , D l ( a · b ) := lim n →∞ D l a · b n = ( D l a ) · b, where b n is a sequence in S Γ u converging to b . Thus we obtain a family of denselydefined unbounded C ∗ max ,u ( M ; L ( S )) Γ -linear operators D l : C ∗ max ,u ( M ; L ( S )) Γ → C ∗ max ,u ( M ; L ( S )) Γ . (4)Since the action of D l on S Γ u · ( C ∗ max ,u ( M ; L ( S )) Γ ) + is determined by its action on S Γ u , in practice we may just work with the latter.3.2. Regularity and essential self-adjointness.
Let us state main result of this section:
Theorem 3.1.
There exists a real number µ = 0 such that the operators D ± µi : C ∗ max ,u ( M ; L ( S )) Γ → C ∗ max ,u ( M ; L ( S )) Γ have dense range. We will proceed to the proof of this theorem shortly. First observe the followingconsequence ([5, Lemmas 9.7 and 9.8]):
Corollary 3.2.
The operator D on the Hilbert module C ∗ max ,u ( M ; L ( S )) Γ is regularand essentially self-adjoint. Regular and essentially self-adjoint operators on a Hilbert C ∗ -module admit a func-tional calculus that satisfy the following set of properties [5, Theorem 10.9], [4, Propo-sition 16]: Theorem 3.3.
Let B be a C ∗ -algebra and N a Hilbert B -module. Let C ( R ) be the ∗ -algebra of complex-valued continuous functions on R . For any regular, essentiallyself-adjoint operator T on N , there is a ∗ -preserving linear map π T : C ( R ) → R B ( N ) ,f f ( T ) := π T ( f ) , where R B ( N ) denotes the regular operators on N , such that:(i) π T restricts to a ∗ -homomorphism C b ( R ) → L B ( N ) ;(ii) If | f ( t ) | ≤ | g ( t ) | for all t ∈ R , then dom( g ( T )) ⊆ dom( f ( T )) ;(iii) If ( f n ) n ∈ N is a sequence in C ( R ) for which there exists F ∈ C ( R ) such that | f n ( t ) | ≤ F ( t ) | for all t ∈ R , and if f n converge to a limit function f ∈ C ( R ) uniformly on compact subsets of R , then f n ( T ) x f ( T ) x for each x ∈ dom( f ( T )) ;(iv) T ) = T . n the rest of this section, we will finish the proof of Theorem 3.1.Let f µ : R → C be the function x ( x + µi ) − . Let K f µ denote the Schwartzkernel of the bounded operator f µ ( D ). Since K f µ is pseudodifferential, it is smoothon the complement of the diagonal. Furthermore, it satisfies the following estimate: Lemma 3.4.
There exists C µ > such that for all x, y ∈ M with d ( x, y ) ≥ , (cid:13)(cid:13) K f µ ( x, y ) (cid:13)(cid:13) ≤ C µ e − µ d ( x,y ) , where k·k denotes the fiberwise norm on S ⊠ S ∗ .Proof. Let x, y ∈ M with λ := dist( x, y ) ≥
1. Choose a smooth function φ : R → R such that φ ( ξ ) = 1 for | ξ | ≥ φ ( ξ ) = 0 if | ξ | ≤ . Let φ λ ( ξ ) := φ (cid:0) ξλ (cid:1) . Let g µ bethe function on R with Fourier transform b g µ ( ξ ) = φ λ ( ξ ) b f µ ( ξ ) . Let K g µ denote the Schwartz kernel of g µ ( D ). By [15, Lemma 3.5], for all x, y ∈ M ,there exists C > (cid:13)(cid:13) K g µ ( x, y ) (cid:13)(cid:13) ≤ C · sup l ≤ dim M +3 (cid:13)(cid:13) D l g µ ( D ) (cid:13)(cid:13) B ( L ( S )) . For a given l ≤ dim M + 3, we can estimate the right-hand side as follows. Let ψ l be the function given by ψ l ( s ) = s l g µ ( s ). We haveˆ ψ l ( ξ ) = (cid:18) i ddξ (cid:19) n ( φ λ ˆ f )( ξ ) . By the Fourier inversion formula g µ ( D ) = 12 π Z ∞−∞ b g µ ( ξ ) e iξD dξ, and the fact that φ λ is supported on | ξ | ≥ λ , we have: (cid:13)(cid:13) D l g µ ( D ) (cid:13)(cid:13) B ( L ( S )) = k ψ l ( D ) k B ( L ( S )) ≤ π Z | ξ |≥ λ (cid:12)(cid:12)(cid:12) b ψ l ( ξ ) (cid:12)(cid:12)(cid:12) dξ ≤ C l X j =0 Z | ξ |≥ λ (cid:12)(cid:12)(cid:12) b f ( l − j ) µ ( ξ ) (cid:12)(cid:12)(cid:12) dξ ≤ C Z | ξ |≥ λ e − µξ (0 , ∞ ) dξ ≤ C e − µλ , or some C , C , C >
0, and where we have used that b f µ ( ξ ) = πµi e − µξ (0 , ∞ ) , with being the indicator function. It follows that for all x, y with d ( x, y ) ≥
1, thereexists C µ > (cid:13)(cid:13) K g µ ( x, y ) (cid:13)(cid:13) ≤ C µ e − µ d ( x,y ) for a constant C µ . Now a standardfinite-propagation argument for the wave operator shows that K f µ ( x, y ) = K g µ ( x, y )whenever d ( x, y ) ≥
1, and we conclude. (cid:3)
Corollary 3.5.
For any k ∈ S Γ u , there exists a constant C k such that (cid:13)(cid:13) ( D + µi ) − k ( x, y ) (cid:13)(cid:13) ≤ C k e − µ d ( x,y ) , where k·k denotes the fiberwise norm on S ⊠ S ∗ Proof.
For any y ∈ M , set L y = { z ∈ M | k ( z, y ) = 0 } . Observe that sup y ∈ M { diam( L y ) } ≤ prop( k ). By bounded Riemannian geometry, wehave sup y ∈ M { vol( L y ) } ≤ C ′ k for some constant C ′ k . Let B be the set of ( x, y ) ∈ M × M with d ( x, y ) <
1, and let B c denote its complement. Let K < := K f µ B and K ≥ := K f µ B c . Thus K f µ = K < + K ≥ . We have, for all x, y ∈ M , (cid:13)(cid:13)(cid:13)(cid:13)Z M K ≥ ( x, z ) k ( z, y ) dz (cid:13)(cid:13)(cid:13)(cid:13) ≤ Z L y k K ≥ ( x, z ) k ( z, y ) k dz ≤ C µ Z L y e − µ ( d ( x,y ) − d ( z,y )) k k ( z, y ) k dz ≤ C µ e − µ d ( x,y ) Z L y e µ d ( z,y ) k k ( z, y ) k dz ≤ C µ e − µ d ( x,y ) · C ′ k e µ prop( k ) k k k ∞ = C ′′ k e − µ d ( x,y ) , for a new constant C ′′ k . Now, we have (cid:13)(cid:13)(cid:13)(cid:13)Z M K < ( x, z ) k ( z, y ) dz (cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13) ( D + µi ) − k ( x, y ) (cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)Z M K ≥ ( x, z ) k ( z, y ) dz (cid:13)(cid:13)(cid:13)(cid:13) ≤ C + C ′′ k , for a constant C > M . Also note that the function( x, y ) Z M K < ( x, z ) k ( z, y ) dz s supported on some ball B of finite radius around the diagonal, so there exists C k > (cid:13)(cid:13) ( D + µi ) − k ( x, y ) (cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)Z M K ≥ ( x, z ) k ( z, y ) dz (cid:13)(cid:13)(cid:13)(cid:13) + ( C + C ′′ k ) B ( x, y ) ≤ C k e − µ d ( x,y ) . (cid:3) Proof of Theorem 3.1.
We will argue in the case of the operator D + µi , with thecase of D − µi being entirely analogous. Let k ∈ S Γ u . Fix a countable open cover M whose elements have uniformly bounded diameter. Let { ρ j } j ∈ N be a smooth partitionof unity subordinate to this cover. We can write k ( x, y ) = X j ρ j ( x ) k ( x, y ) . Note that the Schwartz kernel k j := ( D + µi ) − ( ρ j k ) is smooth but may not becompactly supported in M × M . However, since it is compactly supported in the y -direction, the sum ˜ k ( x, y ) := X j k j ( x, y ) = ( D + µi ) − k ( x, y )still makes sense at each point ( x, y ) ∈ M × M and moreover is Γ-invariant. Let∆ M × M denote the diagonal in M × M . Consider the open cover V := { V l } l ∈ Z ≥ of M × M consisting of the sets V l := ( B (∆ M × M ) if l = 0, B l + (∆ M × M ) \ B l − (∆ M × M ) otherwise,where the balls are taken on M × M . Under the diagonal action of Γ on M × M ,each V l is Γ-stable and has cocompact closure, hence there exists a partition of unity { ψ l } l ∈ Z ≥ consisting of smooth Γ-invariant functions subordinate to V . For each l ,the kernel ˜ k l ( x, y ) := ψ l ˜ k ( x, y )is an element of S Γ u with propagation at most 2 l + 1. Let T ˜ k l denote the operatoron L ( S ) given by the kernel ˜ k l . It remains to show that, for any ∗ -representation φ : C [ M ] Γ → B ( H ′ ), we have convergence of the sum φ ( T ˜ k ⊕
0) = X l φ ( T ˜ k l ⊕ . By Lemmas 2.19 and 2.3, there exist constants C , C , C , and C l such that (cid:13)(cid:13) φ (cid:0) T ˜ k l ⊕ (cid:1)(cid:13)(cid:13) B ( H ′ ) ≤ C l C V l ≤ C l C e C l , here C , C , and C are C l is bounded above by a multiple of (cid:13)(cid:13) ˜ k l (cid:13)(cid:13) ∞ . Since ˜ k l issupported away from B l − (∆ M × M ), Lemma 3.4 implies that there exists a constant C such that (cid:13)(cid:13) ˜ k l (cid:13)(cid:13) ∞ ≤ C e − µl , whence, by choosing µ sufficiently large, we have absolute convergence. (cid:3) Proof of the main theorem
In this section, we first define the maximal uniform index of D and show that underthe positive scalar curvature assumption, this index vanishes. We then use this toprove our main result, Theorem 1.1.4.1. The maximal uniform index.
Let M := M ( C ∗ max ,u ( M ; L ( S )) Γ ) denote the multiplier algebra of C ∗ max ,u ( M ; L ( S )) Γ .The short-exact sequence of C ∗ -algebras0 → C ∗ max ,u ( M ; L ( S )) Γ → M → M /C ∗ max ,u ( M ; L ( S )) Γ → K -theory: K ( C ∗ max ,u ( M ; L ( S )) Γ ) / / K ( M ) / / K ( M /C ∗ max ,u ( M ; L ( S )) Γ ) ∂ (cid:15) (cid:15) K ( M /C ∗ max ,u ( M ; L ( S )) Γ ) ∂ O O K ( M ) o o K ( C ∗ max ,u ( M ; L ( S )) Γ ) . o o Let χ : R → R be a continuous odd function with limit 1 at ∞ (called a nor-malizing function ). We now apply the functional calculus of Theorem 3.3 with N = C ∗ max ,u ( M ; L ( S )) Γ and T = D to form the bounded adjointable operator χ ( D ) : C ∗ max ,u ( M ; L ( S )) Γ → C ∗ max ,u ( M ; L ( S )) Γ . In the remainder of this section we prove:
Proposition 4.1.
The operator χ ( D ) is invertible modulo C ∗ max ,u ( M ; L ( S )) Γ and sodefines a class [ χ ( D )] ∈ K n +1 ( M /C ∗ max ,u ( M ; L ( S )) Γ ) that is independent of the choice of normalizing function χ . Definition 4.2.
The maximal uniform index of D , denoted index max ,u ( D ), is theimage of [ χ ( D )] under the boundary map ∂ : K n +1 ( M /C ∗ max ,u ( M ; L ( S )) Γ ) → K n ( C ∗ max ,u ( M ; L ( S )) Γ ) . roof of Proposition 4.1. Without loss of generality, let us work in the case when n is even. By Theorem 3.1, we have χ ( D ) ∈ M ( C ∗ max ,u ( M ; L ( S )) Γ ) . To see that χ ( D ) defines a class in K ( M /C ∗ max ,u ( M ; L ( S )) Γ ) , it suffices to show that for any f ∈ C ( R ), we have f ( D ) ∈ C ∗ max ,u ( M ; L ( S )) Γ , since χ − ∈ C ( R ). Since M has bounded Riemannian geometry, [9, Proposition 2.10] implies that for any f ∈S ( R ) with compactly supported Fourier transform, the operator f ( D ) is given by asmooth Schwarz kernel that is uniformly bounded along with all of its derivatives.The fact that S ( R ) is dense in C ( R ), together continuity of the functional calculushomomorphism (part (i) of Theorem 3.3) shows this is true for general f ∈ C ( R ).Finally, since the difference of any two normalizing functions lies in C ( R ), the class[ χ ( D )] is independent of the choice of χ . (cid:3) Vanishing of the maximal uniform index.
Suppose that the scalar curvature function κ of ( M n , g ) is uniformly positive; that is,there exists c > κ ≥ c . Recall that the Lichnerowicz formula states that D = ∇ ∗ ∇ + κ , where ∇ is the lift of the Levi-Civita connection to S . Proposition 4.3.
There exists ǫ > such that spec( D ) ⊆ R \ ( − ǫ, ǫ ) .Proof. It suffices to show that the operator D on C ∗ max ,u ( M ; L ( S )) Γ , defined in (4),is strictly positive with respect to the C ∗ max ,u ( M ) Γ -valued inner product. Since κ ≥ c ,we only need to show that ∇ ∗ ∇ is a non-negative operator with respect to this innerproduct - that is, for any Schwartz kernel k ∈ S Γ u , we have( ∇ ∗ ∇ k ) ∗ k ≥ ∈ C ∗ max ,u ( M ; L ( S )) Γ . This follows directly from the relation( ∇ ∗ ∇ k ) ∗ k = ( ∇ k ) ∗ ( ∇ k ) . (cid:3) To see that this implies vanishing of index max,u ( D ), let us suppose that the dimen-sion n is odd; a similar argument applies in the even case. Consider the function χ on R \{ } given by χ ( x ) = x/ | x | . Since D has a spectral gap at 0, we may use the functional calculus for regularoperators on Hilbert modules to form the operator χ ( D ) ∈ M . Define f ( D ) := χ ( D ) + 12 . hen one can verify directly that f ( D ) is in fact a projection in M . Thus by definitionof the map index max ,u ,index max ,u ( D ) = (cid:2) e πif ( D ) (cid:3) = 0 ∈ K ( C ∗ max ,u ( M ; L ( S )) Γ ) . This yields a version of Theorem 1.1 for the uniform maximal Roe algebra:
Theorem 4.4.
Let ( M n , g ) be a complete Γ -spin Riemannian manifold with boundedRiemannian geometry. Let Γ be a countable discrete group acting and properly andisometrically on M , satisfying Assumption 2.6. If M has uniformly positive scalarcurvature, then index max ,u ( D ) = 0 ∈ K n ( C ∗ max ,u ( M ; L ( S )) Γ ) . Vanishing of the maximal index.
We can now complete the proof of our main result, Theorem 1.1. Let us first recallthe definition of the maximal higher index of D [3, 4.14]. We will work in the casewhen the dimension n is even, with the odd case being analogous.Given a normalizing function χ , we can form the operator χ ( D ) using the functionalcalculus for self-adjoint operators on L ( S ). Pick a locally finite Γ-invariant open cover { U i } i ∈ N of M with the property thatsup i ∈ N { diam( U i / Γ) } < C for some C >
0. Let { φ i } i ∈ N be a continuous partition of unity subordinate to { U i } i ∈ N .Then the sum F D := X i φ i χ ( D ) φ i defines a bounded, Γ-invariant, locally compact operator on L ( S ) with finite propa-gation. Consider the matrix of bounded operators W D = (cid:18) F D (cid:19) (cid:18) − F ∗ D (cid:19) (cid:18) F D (cid:19) (cid:18) −
11 0 (cid:19) . Each entry of W D has finite propagation, and one verifies that P D := W D (cid:18) (cid:19) W − D − (cid:18) (cid:19) is a projection in M ( C [ M ; L ( S )] Γ ). We define the maximal higher index of D on L ( S ) to be the K -theoretic class of P D :index L ( S )max ( D ) := [ P D ] ∈ K ( C ∗ max ( M ; L ( S )) Γ ) . ow the map ⊕ C [ M ; L ( S )] Γ → C [ M ] Γ extends to an injective ∗ -homomorphismbetween the maximal completions of both sides that we will still denote by ⊕ C ∗ max ( M ; L ( S )) Γ → C ∗ max ( M ) Γ . This induces a homomorphism on K -theory: ⊕ ∗ : K ( C ∗ max ( M ; L ( S )) Γ ) → K ( C ∗ max ( M ) Γ ) . The maximal higher index of D is the image of index L ( S )max ( D ) under this map:index max ( D ) := ⊕ ∗ (index L ( S )max ( D )) ∈ K ( C ∗ max ( M ) Γ ) . Equivalently, the elements index L ( S )max ( D ) and index max ( D ) can be obtained in thefollowing way. Let χ ′ be a normalizing function with compactly supported Fouriertransform. Then χ ′ ( D ) has finite propagation, and the matrix W ′ D = (cid:18) χ ′ ( D )0 1 (cid:19) (cid:18) − χ ′ ( D ) ∗ (cid:19) (cid:18) χ ′ ( D )0 1 (cid:19) (cid:18) −
11 0 (cid:19) defines a projection P ′ D = W ′ D (cid:18) (cid:19) ( W ′ D ) − − (cid:18) (cid:19) in M ( C [ M ; L ( S )]) Γ . One then verifies thatindex L ( S )max ( D ) = [ P D ] = [ P ′ D ] , whence they give rise to the same element index max ( D ) ∈ K ( C ∗ max ( M ) Γ ). Proof of Theorem 1.1.
Assume, without loss of generality, that n is even. By defini-tion of the maximal equivariant uniform Roe algebra on L ( S ), we have an inclusionmap ψ : C ∗ max ,u ( M ; L ( S )) Γ → C ∗ max ( M ; L ( S )) Γ . The composition C ∗ max ,u ( M ; L ( S )) Γ ψ −→ C ∗ max ( M ; L ( S )) Γ ⊕ −−→ C ∗ max ( M ) Γ induces a composition of group homomorphisms K ( C ∗ max ,u ( M ; L ( S )) Γ ) ψ ∗ −→ K ( C ∗ max ( M ; L ( S )) Γ ) ⊕ ∗ −−→ K ( C ∗ max ( M ) Γ ) . Choose a normalizing function χ ′ with compactly supported Fourier transform. Then χ ′ ( D ) has finite propagation, and its associated projection P ′ D represents index max,u ( D ).By Theorem 4.4, uniform positive scalar curvature implies thatindex max,u ( D ) = 0 . Consequently, we haveindex max ( D ) = ⊕ ∗ ◦ ψ ∗ (index max,u ( D )) = 0 . (cid:3) eferences [1] S. Chang, S. Weinberger, and G. Yu. Positive scalar curvature and a new index theory fornoncompact manifolds. arXiv:1506.03859.[2] X. Chen, Q. Wang, and G. Yu. The maximal coarse Baum-Connes conjecture for spaces whichadmit a fibred coarse embedding into Hilbert space. Adv. Math. , 249:88–130, 2013.[3] G. Gong, Q. Wang, and G. Yu. Geometrization of the strong Novikov conjecture for residuallyfinite groups.
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Clay Math. Proc. , pages 649–657. Amer. Math. Soc., Providence, RI, 2010.(Hao Guo)
Department of Mathematics, Texas A&M University
E-mail address : [email protected] (Zhizhang Xie) Department of Mathematics, Texas A&M University
E-mail address : [email protected] (Guoliang Yu) Department of Mathematics, Texas A&M University
E-mail address : [email protected]@math.tamu.edu