Functoriality for higher rho invariants of elliptic operators
aa r X i v : . [ m a t h . K T ] M a y FUNCTORIALITY FOR HIGHER RHO INVARIANTS OFELLIPTIC OPERATORS
HAO GUO, ZHIZHANG XIE, AND GUOLIANG YU
Abstract.
Let N be a closed spin manifold with positive scalar curvature and D N the Dirac operator on N . Let M and M be two Galois covers of N suchthat M is a quotient of M . Then the quotient map from of M to M naturallyinduces maps between the geometric C ∗ -algebras associated to the two mani-folds. We prove, by a finite-propagation argument, that the maximal higher rhoinvariants of the lifts of D N to M and M behave functorially with respect tothe above quotient map. This can be applied to the computation of higher rhoinvariants, along with other related invariants. Contents
1. Introduction and motivation 2Acknowledgements 52. Preliminaries 52.1. Notation 52.2. Geometric C ∗ -algebras 52.3. Geometric setup 73. Folding maps 83.1. Definition of the folding map 93.2. Description of the folding map on invariant sections 163.3. Induced maps on geometric C ∗ -algebras and K -theory 174. Functional calculus and the wave operator 184.1. Functional calculus on the maximal Roe algebra 194.2. The wave operator 205. Functoriality for the higher index 235.1. Higher index 235.2. Functoriality 246. Functoriality for the higher rho invariant 27 Mathematics Subject Classification.
Introduction and motivation
An elliptic differential operator on a closed manifold has a Fredholm index. Whensuch an operator is lifted to a covering space, one can, by taking into account thegroup of symmetries, define a far-reaching generalization of the Fredholm index,called the higher index [2, 3, 6, 16, 27]. Often referred to as a primary invariant dueto its invariance under homotopy, the higher index plays a fundamental role in thestudy of geometry and topology through the Novikov conjecture [7, 14, 16, 31, 32]on homotopy invariance of higher signatures and the Gromov-Lawson conjecture[10, 11] on the existence of Riemannian metrics with positive scalar curvature. TheBaum-Connes Conjecture [2, 3] proposes an algorithm for computing the higherindex.The higher index serves as an obstruction to the existence of invariant metrics ofpositive scalar curvature. In the case that such a metric exists, so that the higherindex of the lifted operator vanishes, a secondary invariant called the higher rhoinvariant [21, 15] can be defined. The higher rho invariant is an obstruction to theinverse of the operator being local [4]. For some recent applications of the higherindex and higher rho invariant to problems in geometry and topology, we refer thereader to [5, 23, 24, 26, 28, 29, 30].The main purpose of this paper is to prove that the higher rho invariant behavesfunctorially under maps between covering spaces. This result can be applied to theproblem of computing the higher rho invariant, along with other related invariants.For instance it is a key tool in the recent work of Wang, Xie, and Yu [25], whocomputed the delocalized eta invariant on the universal cover of a closed manifold byshowing that, under suitable conditions, it can be expressed as a limit of delocalizedeta invariants associated to finite-sheeted covering spaces.To motivate our main result, recall that by a well-known theorem of Atiyah [1],the Fredholm index of an elliptic differential operator D N on a closed manifold N is equal to the L -index of its lift e D N to the universal cover e N . One can interpretthis result in the framework of higher index theory as follows. Let C π N denote thegroup algebra of the fundamental group π N . The higher index of e D N takes valuesin K -theory of the associated group C ∗ -algebra. The L -index is then the image ofthe higher index under a tracial map on K -theory. UNCTORIALITY FOR HIGHER RHO INVARIANTS OF ELLIPTIC OPERATORS 3
More precisely, let C ∗ r ( π N ) denote the reduced group C ∗ -algebra of π N . Thevon Neumann trace τ : C π N → C , k X i =1 c γ i γ i c e induces a map τ ∗ : K • ( C ∗ r ( π N )) → C . We then haveInd D N = τ ∗ (cid:0) Ind π N e D N (cid:1) , where Ind π N e D N is the higher index of e D N in K • ( C ∗ r ( π N )), and the right-handside is the L -index of e D N .There is an analogue of this result for the maximal version of the higher index of e D N , which takes values in K -theory of the maximal group C ∗ -algebra C ∗ max ( π N ).To state it, consider the ∗ -homomorphism a : C π N → C , k X i =1 c γ i γ i k X i =1 c γ i . This induces a K -theoretic map a ∗ : K • ( C ∗ max ( π N )) → K • ( C ) ∼ = Z . The maximal analogue of Atiyah’s L -index theorem then statesInd D N = a ∗ (cid:0) Ind π N, max e D N (cid:1) , where Ind π N, max e D N ∈ K • ( C ∗ max ( π M )) is the maximal version of the higher indexof e D N . This can for instance be established using KK -theory (see also Theorem 1.1below). We thus have the following commutative diagram:Ind π M, max e D N Ind π M e D N Ind D N Ind D Na ∗ τ ∗ where the top arrow is given by the K -theoretic map induced by the quotient map C ∗ max ( π M ) → C ∗ r ( π M ).More generally, let M and M be two Galois covers of N with deck transformationgroups Γ and Γ ∼ = Γ /H , where H is a normal subgroup of Γ . Then the quotienthomomorphism Γ → Γ induces a natural surjective ∗ -homomorphism betweengroup algebras, which we denote by α : C Γ → C Γ , k X i =1 c γ i γ i k X i =1 c γ i [ γ i ] , where [ γ ] is the class of an element γ ∈ Γ in Γ /H .The following functoriality result for the higher index is a consequence of a moregeneral result of Valette [19, Theorem 1.1]: HAO GUO, ZHIZHANG XIE, AND GUOLIANG YU
Theorem 1.1.
Let N be a closed Riemannian manifold. Let D N be a first-order,self-adjoint elliptic differential operator acting on a bundle E N → N . Let M and M be Galois covers of N with deck transformation groups Γ and Γ ∼ = Γ /H respectively, for a normal subgroup H of Γ . Let D and D be the lifts of D N to M and M . Then the map on K -theory induced by α relates the maximal higherindices of D and D : α ∗ (Ind Γ , max D ) = Ind Γ , max D ∈ K • (cid:0) C ∗ max (Γ ) (cid:1) . Theorem 1.1 serves as our motivation to prove a corresponding functoriality resultfor the higher rho invariant. To achieve this, we work with the quotient map atthe manifold level, namely π : M → M , along with various geometric group C ∗ -algebras , which we will review in section 2.The map π induces a family of ∗ -homomorphisms between geometric group C ∗ -algebras associated to M and M , called folding maps , which we will examine indetail in section 3. In particular, let us denote the folding map between maximalequivariant Roe algebras on M and M byΨ : C ∗ max ( M ) Γ → C ∗ max ( M ) Γ . A important geometric property of Ψ is that it preserves small propagation of opera-tors , and as such gives rise to a folding map Ψ L, at the level of obstruction algebras C ∗ L, , max ( M ) Γ → C ∗ L, , max ( M ) Γ . This is crucial to the proof of our main result: Theorem 1.2.
Let N be a closed, spin Riemannian manifold with positive scalarcurvature. Let D N be the Dirac operator on N . Let M and M be Galois coversof M with deck transformation groups Γ and Γ ∼ = Γ /H respectively, for a normalsubgroup H of Γ . Let D and D be the lifts of the Dirac operator on N . Then themap on K -theory of the obstruction algebra induced by the folding map Ψ L, relatesthe maximal higher rho invariants of D and D : (Ψ L, ) ∗ ( ρ max ( D )) = ρ max ( D ) ∈ K • ( C ∗ L, , max ( M ) Γ ) . In order to prove Theorem 1.2, we will first give a new proof of Theorem 1.1.This proof makes essential use of the properties of the folding map Ψ, together withfinite propagation of the wave operator.One feature of our approach is that it emphasizes the geometric content of theabove results while keeping the exposition reasonably self-contained. As such, thesame method can be adapted to more general geometric settings. As an example,we give a generalization of Theorem 1.1 to the non-cocompact setting for operatorsthat are invertible at infinity (see Theorem 7.2).
Overview.
The paper is organized as follows. We begin in section 2 by recalling the geometricand operator-algebraic setup we work with. In section 3 we define the folding maps
UNCTORIALITY FOR HIGHER RHO INVARIANTS OF ELLIPTIC OPERATORS 5 and establish some of their basic properties. In section 4 we develop the analyticalproperties of the wave operator in the maximal setting. These tools are then put touse in section 5, where we give a new proof of Theorem 1.1 using the folding map.This serves as an intermediate step toward our main result, Theorem 1.2, which weprove in section 6. We conclude in section 7 by giving a generalization of Theorem1.1 to the non-cocompact setting for operators that are invertible at infinity.
Acknowledgements.
The authors are grateful to Peter Hochs for pointing us to the reference [19, Theorem1.1] of Alain Valette mentioned above.2.
Preliminaries
In this section, we fix some notation before introducing the necessary operator-algebraic background and geometric setup for our results.2.1.
Notation.
For X a Riemannian manifold, we will write B ( X ), C b ( X ), C ( X ), and C c ( X ) todenote the C ∗ -algebras of complex-valued functions on X that are, respectively:bounded Borel, bounded continuous, continuous and vanishing at infinity, and con-tinuous with compact support. A superscript ‘ ∞ ’ may be added where appropriateto indicate the additional requirement of smoothness.We will write d X for the Riemannian distance function on X and S for thecharacteristic function of a subset S .For any C ∗ -algebra A , we will denote its unitization by A + and its multiplieralgebra by M ( A ). We will view A as an ideal of M ( A ).If a group Γ acts on X , there is a naturally induced Γ-action on spaces of functionson X : given g ∈ Γ and a function f on X , the function g · f is given by g · f ( x ) = f ( g − x ). More generally, for a section s of a Γ-vector bundle over X , the section g · s is defined by g · s ( x ) = g ( s ( g − x )). We will say that an operator on sections ofa bundle is Γ-equivariant if it commutes with the Γ-action.2.2. Geometric C ∗ -algebras. We now recall the notions of geometric modules and their associated C ∗ -algebras.Throughout this subsection, X is a Riemannian manifold equipped with a properisometric action by a discrete group G . Definition 2.1. An X - G -module is a separable Hilbert space H equipped with anon-degenerate ∗ -representation ρ : C ( X ) → B ( H ) and a unitary representation U : G → U ( H ) such that for all f ∈ C ( X ) and g ∈ G , we have U g ρ ( f ) U ∗ g = ρ ( g · f ).For brevity, we will omit ρ from the notation when it is clear from context. HAO GUO, ZHIZHANG XIE, AND GUOLIANG YU
Definition 2.2.
Let H be an X - G -module and T ∈ B ( H ). • The support of T , denoted supp( T ), is the complement of all ( x, y ) ∈ X × X for which there exist f , f ∈ C ( X ) such that f ( x ) = 0, f ( y ) = 0, and f T f = 0; • The propagation of T is the extended real numberprop( T ) = sup { d X ( x, y ) | ( x, y ) ∈ supp( T ) } ; • T is locally compact if f T and T f ∈ K ( H ) for all f ∈ C ( X ); • T is G -equivariant if U g T U ∗ g = T for all g ∈ G ;The equivariant algebraic Roe algebra for H , denoted C [ X ; H ] G , is the ∗ -subalgebraof B ( H ) consisting of G -equivariant, locally compact operators with finite propaga-tion.We will work with the maximal completion of the equivariant algebraic Roe al-gebra. To ensure that this completion is well-defined, we require that the module H satisfy an additional admissibility condition. To define what this means, we needthe following fact: if H is a Hilbert space and ρ : C ( X ) → B ( H ) is a non-degenerate ∗ -representation, then ρ extends uniquely to a ∗ -representation e ρ : B ( M ) → B ( H )subject to the property that, for a uniformly bounded sequence in B ( X ) convergingpointwise, the corresponding sequence in B ( H ) converges in the strong topology. Definition 2.3 ([33]) . An X - G -module H is admissible if:(i) For any non-zero f ∈ C ( X ) we have π ( f ) / ∈ K ( H );(ii) For any finite subgroup F of G and any F -invariant Borel subset E ⊆ X ,there is a Hilbert space H ′ equipped with the trivial F -representation suchthat e π ( E ) H ′ ∼ = l ( F ) ⊗ H ′ as F -representations, where e π is defined byextending π as above.If an X - G -module H is admissible, we will write C [ X ] G in place of C [ X ; H ] G ,observing that C [ X ; H ] G is independent of the choice of admissible module – see[27, Chapter 5]. Remark 2.4.
When G acts freely and properly on X , the Hilbert space L ( X ) isan admissible X - G -module. For non-free actions, L ( X ) can always be embeddedinto a larger admissible module. Definition 2.5.
The maximal norm of an operator T ∈ C [ X ] G is || T || max := sup φ,H ′ (cid:8) k φ ( T ) k B ( H ′ ) | φ : C [ X ] G → B ( H ′ ) is a ∗ -representation (cid:9) . The maximal equivariant Roe algebra of M j , denoted C ∗ max ( X ) G , is the completionof C [ X ] G in the norm || · || max . UNCTORIALITY FOR HIGHER RHO INVARIANTS OF ELLIPTIC OPERATORS 7
Remark 2.6.
To make sense of Definition 2.5 for general X and G , one first needsto establish finiteness of the quantity k · k max . It was shown in [9] that if G actson X freely and properly with compact quotient, the norm k · k max is finite. Thiswas generalized in [13] to the case when X has bounded geometry and the G -actionsatisfies a suitable geometric assumption. Remark 2.7.
Equivalently, one can obtain C ∗ max ( X ) G by taking the analogousmaximal completion of the subalgebra S G of C [ X ] G consisting of those operatorsgiven by smooth Schwartz kernels. Definition 2.8.
Consider the ∗ -algebra L of functions f : [0 , ∞ ) → C ∗ max ( X ) G thatare uniformly bounded, uniformly continuous, and such thatprop( f ( t )) → t → ∞ . (i) The maximal equivariant localization algebra , denoted by C ∗ L, max ( X ) G , is the C ∗ -algebra obtained by completing L with respect to the norm k f k := sup t k f ( t ) k max ;(ii) The map L → C [ X ] G given by f f (0) extends to the evaluation map ev : C ∗ L, max ( X ) G → C ∗ max ( X ) G ;(iii) The maximal equivariant obstruction algebra is C ∗ L, , max ( X ) G := ker(ev).2.3. Geometric setup.
We will work with the following geometric setup.Let (
N, g N ) be a closed Riemannian manifold. Let D N be a first-order essentiallyself-adjoint elliptic differential operator on a bundle E N → N . We will assumethroughout that if N is odd-dimensional then D N is an ungraded operator, while if N is even-dimensional then D N is odd-graded with respect to a Z -grading on E N .Let p : M → N and p : M → N be two Galois covers of N with deck trans-formation groups Γ and Γ respectively. We will assume throughout this paperthat Γ ∼ = Γ /H for some normal subgroup H of Γ , so that M ∼ = M /H . Let π : M → M be the projection map. Note that for j = 1 ,
2, the group Γ j acts freelyand properly on M j .For j = 1 or 2, let g j be the lift of the Riemannian metric g N to M j . Let E j be the pullback of E along the covering map M j → M , equipped with the naturalΓ j -action. Since D acts locally, it lifts to a Γ j -equivariant operator D j on C ∞ ( E j ).We will apply the notions in subsection 2.2 with X = M j and G = Γ j . The Hilbertspace L ( E j ), equipped with the natural Γ j -action on sections and the C ( M j )-representation defined by pointwise multiplication, is a M j -Γ j -module in the senseof Definition 2.2. HAO GUO, ZHIZHANG XIE, AND GUOLIANG YU
Moreover, the fact that the Γ j -action on M j is free and proper implies that L ( E j )is an admissible module in the sense of Definition 2.3. This can be seen as follows.Choosing a compact, Borel fundamental domain D j for the Γ j -action on M j suchthat for each γ ∈ Γ j , the restriction p j | γ ·D j : γ · D j → N is a Borel isomorphism, where the projection maps p j are as in subsection 2.3. Foreach j = 1 , j : L ( E j ) → l (Γ j ) ⊗ L ( E j | D j ) ,s X γ ∈ Γ j γ ⊗ γ − χ γ D j s. (2.1)This is a Γ j -equivariant unitary isomorphism with respect to the tensor product ofthe left-regular representation on l (Γ j ) and the trivial representation on L ( E j | D j ).Conjugation by Φ j induces a ∗ -isomorphism C [ M j ] Γ j ∼ = C Γ j ⊗ K ( L ( E j | D j )) . (2.2) Remark 2.9.
It follows from (2.2) that for any r >
0, there exists a constant C r such that for all T ∈ C [ M j ] Γ j with prop( T ) ≤ r , we have k T k max ≤ C r k T k B ( L ( E j )) . Thus the maximal equivariant Roe algebra C ∗ max ( M j ) Γ j from Definition 2.5 is well-defined in our setting. Moreover, letting C ∗ max (Γ j ) denote the maximal group C ∗ -algebra of Γ j , we have C ∗ max ( M j ) Γ j ∼ = C ∗ max (Γ j ) ⊗ K , so that the K -theories of both sides are isomorphic.3. Folding maps
In this section, we define certain natural ∗ -homomorphisms, called folding maps ,between geometric C ∗ -algebras of covering spaces. These maps play a central rolein the formulation and proof of our main result, Theorem 1.2.To begin, let us provide some motivation at the level of groups. As mentioned insection 1, the quotient homomorphism Γ → Γ ∼ = Γ /H induces a natural surjective ∗ -homomorphism between group algebras, α : C Γ → C Γ , k X i =1 c γ i γ i k X i =1 c γ i [ γ i ] , UNCTORIALITY FOR HIGHER RHO INVARIANTS OF ELLIPTIC OPERATORS 9 where [ γ ] is the class of an element γ ∈ Γ in Γ /H . If one views elements of C Γ j as kernel operators on the Hilbert space l (Γ j ), then the map α takes a kernel k : Γ × Γ → C to the kernel α ( k ) : Γ × Γ → C , (cid:0) [ γ ] , [ γ ′ ] (cid:1) X h ∈ H k ( hγ, γ ′ ) . Remark 3.1.
When Γ = π N and Γ is the trivial group, α reduces to the map a : C π N → C considered at the start of section 1.There is an analogous map between kernels at the level of the Galois covers M and M ∼ = M /H . Given a smooth, Γ -equivariant Schwartz kernel k ( x, y ) withfinite propagation on M , one can define a smooth, Γ -equivariant Schwartz kernel ψ ( k ) with finite propagation on M by the formula ψ ( k )([ x ] , [ y ]) = X h ∈ H k ( hx, y ) . (3.1)Note that this sum is finite by properness of the H -action on M . The formula (3.1)defines a ∗ -homomorphism ψ : S Γ → S Γ , where the kernel algebras S Γ and S Γ are as in Remark 2.7.3.1. Definition of the folding map.
We now define a more general version of the map (3.1), called the folding map
Ψ,at the level of finite-propagation operators.Let B fp ( L ( E j )) Γ j denote the ∗ -algebra of bounded, Γ j -equivariant operatorson L ( E j ) with finite propagation. The folding map Ψ is a ∗ -homomorphism B fp ( L ( E )) Γ → B fp ( L ( E )) Γ with the following properties: • For any T ∈ B fp ( L ( E )) Γ , we have prop (cid:0) Ψ( T ) (cid:1) ≤ prop( T ); • Ψ is surjective B fp ( L ( E )) Γ → B fp ( L ( E )) Γ ; • Ψ restricts to a surjective ∗ -homomorphism C [ M ] Γ → C [ M ] Γ .These properties are proved in Propositions 3.4, 3.5, and 3.6 below.To define Ψ, it will be convenient to use partitions of unity on M and M thatare compatible with the Γ and Γ -actions, as follows. Since N is compact, thereexists ǫ > x ∈ N , the ball B ǫ ( x ) is evenly covered by both M and M . Let U N := { U i } i ∈ I (3.2)be a finite open cover of N such that each U i has diameter at most ǫ . Let { φ i } i ∈ I be a partition of unity subordinate to U N .For each i ∈ I , let U ei be a lift of U i to M via the covering map p : M → N .Similarly, let φ ei be the lift of φ i to U ei . For each g ∈ Γ , let U gi and φ gi be the g -translates of U ei and φ ei in M respectively. Then U M := (cid:8) U gi (cid:9) i ∈ I, g ∈ Γ , { φ gi } i ∈ I, g ∈ Γ (3.3)define a locally finite, Γ -invariant open cover of M and a subordinate partitionof unity. This partition of unity is Γ -equivariant in the sense that for each i ∈ I , g ∈ Γ, and x ∈ supp( φ ei ), we have φ gi ( gx ) = φ ei ( x ).After taking a quotient by the H -action, we obtain a Γ -invariant open cover of M , together with a subordinate Γ -equivariant partition of unity: U M := (cid:8) U [ g ] i (cid:9) i ∈ I, [ g ] ∈ Γ , (cid:8) φ [ g ] i (cid:9) i ∈ I, [ g ] ∈ Γ . (3.4)For each i and g , we have the following commutative diagram of isomorphisms oflocal structures involving the covering map π : E | U gi U gi E | U [ g ] i U [ g ] i . π We will write π ∗ and π | ∗ U gi for the maps relating a local section of E to the corre-sponding local section of E . That is, if u is a section of E over U gi , and v is thecorresponding section of E over U [ g ] i , then we have u v π ∗ π | ∗ Ugi If w is any vector in the bundle E , we will also write π ∗ ( w ) for its image under theprojection E → E .We are now ready for the definition of the folding map Ψ. Choose a set S ⊆ Γ of coset representatives for Γ ∼ = H \ Γ . Given an operator T ∈ B fp ( L ( E )) Γ , wedefine Ψ( T ) to be the operator on L ( E ) that takes a section u ∈ L ( E ) to thesection Ψ( T )( u ) := X g ∈ Γ ,s ∈ S X i,j ∈ I π ∗ (cid:0) φ gi · T ◦ π | ∗ U sj ◦ φ sj ( u ) (cid:1) ∈ L ( E ) . (3.5)To clarify the presentation of this equation and others like it, we adopt the followingnotational convention for moving between equivalent local sections of E and E : Convention 3.2.
For v ∈ L ( E ) and u ∈ L ( E ), we will use the short-hand • φ [ g ] i v to denote π ∗ ( φ gi v ); • φ gi u to denote π | ∗ U gi ( φ [ g ] i u ).(Note that the meaning of φ [ g ] i v depends on the representative g .) UNCTORIALITY FOR HIGHER RHO INVARIANTS OF ELLIPTIC OPERATORS 11
Using this convention, the formula (3.5) readsΨ( T )( u ) := X g ∈ Γ ,s ∈ S X i,j ∈ I φ [ g ] i T ( φ sj u ) ∈ L ( E ) . (3.6)We call Ψ the folding map . The fact that the above definition is independent of thechoices of the set S and compatible partitions of unity is proved in Proposition 3.4.In order to establish the basic properties of Ψ, let us record a few straightfowardidentities that will help us navigate through notational clutter:(i) If v is a local section of E supported in some neighborhood U gj , then forany g ′ ∈ Γ we have g ′ (cid:0) φ gi v (cid:1) = φ g ′ gi gv, (3.7) π ∗ ( g ′ v ) = [ g ′ ] · π ∗ ( v ) . (3.8)(ii) If u is a local section of E supported in some neighborhood U [ g ] j , then forany g, g ′ ∈ Γ and j ∈ I we have g ′ (cid:0) π | ∗ U gj u (cid:1) = π | ∗ U g ′ gj ([ g ′ ] · u ) , (3.9) u = X i ∈ I,γ ∈ Γ φ [ γ ] i ( π | ∗ U gj u ) . (3.10)We will also make use of the following lemma, which provides a convenient point-wise formula for Ψ( T ) u : Lemma 3.3.
For any u ∈ L ( E ) , x ∈ M , and y ∈ π − ( x ) , we have (cid:0) Ψ( T ) u )( x ) = π ∗ (cid:16) X j ∈ I,g ∈ Γ T ( φ gj u )( y ) (cid:17) , (3.11) where the sum on the right-hand side is finite.Proof. Observe that for any j ∈ I and g ∈ Γ , the summand T ( φ gj u )( y ) may benon-zero only if the set U gj ∩ B prop T ( y ) = ∅ . Since T has finite propagation andthe Γ -action is proper, the set { ( g, j ) | U gj ∩ B prop T ( y ) = ∅} is finite. Hence the sum in question is finite for every x ∈ M and y ∈ π − ( x ).To prove (3.11), let us first rewrite the right-hand side of (3.12) as π ∗ (cid:16) X j ∈ I X h ∈ H,s ∈ S T ( φ hsj u )( y ) (cid:17) . (3.12) Next by (3.9), (3.10), and Γ -equivariance of T , each summand in (3.12) equals T ( φ hsj u )( y ) = T (cid:0) h ( φ sj u ) (cid:1) ( y )= h (cid:0) T ( φ sj u ) (cid:1) ( y )= h · (cid:0) T (cid:0) φ sj u (cid:1) ( h − y ) (cid:1) = (cid:16) h · X i ∈ I,g ∈ Γ φ gi T φ sj u (cid:17) ( y ) . (3.13)Now observe that if v is a section of E supported in some open set U gi , then forany x ∈ M we have ( π ∗ v )( x ) = π ∗ (cid:0) X h ∈ H ( hv )( y ) (cid:1) , (3.14)where y is an arbitrary point in the inverse image π − ( x ). Keeping this in mindand taking a sum of (3.13) over j ∈ I , s ∈ S , and h ∈ H , we see that the right-handside of (3.12) equals π ∗ (cid:16) X h,g,s,i,j (cid:0) h ( φ gi T φ sj u ) (cid:1) ( y ) (cid:17) = (cid:16) π ∗ X g,s,i,j ∈ I φ gi · T ( φ sj u ) (cid:17) ( x ) , which is equal to (Ψ( T ) u )( x ) by (3.6). (cid:3) Proposition 3.4.
The folding map Ψ defined in (3.6) is a ∗ -homomorphism Ψ : B fp ( L ( E )) Γ → B fp ( L ( E )) Γ and is independent of the choices of the set S of coset representatives and compatiblepartitions of unity used to define it.Proof. Boundedness of Ψ( T ) follows from boundedness and finite propagation of T ,via the formula (3.11). To see that Ψ( T ) is Γ -equivariant, note that by (3.11), wehave for all [ γ ] ∈ Γ and u ∈ L ( E ) that([ γ ]Ψ( T )[ γ − ] u )( x ) = [ γ ] · π ∗ X j ∈ I,g ∈ Γ T ( φ gj [ γ − ] u )( γ − y )= [ γ ] · π ∗ X j,g T γ − ( φ γgj u )( γ − y ) . Using (3.8) and Γ -equivariance of T , this is equal to π ∗ X j,g γT γ − ( φ γgj u )( y ) = π ∗ X j,g T ( φ γgj u )( y ) = (Ψ( T ) u )( x ) , where we used a change-of-variable for the last equality. UNCTORIALITY FOR HIGHER RHO INVARIANTS OF ELLIPTIC OPERATORS 13
To see that Ψ is a ∗ -homomorphism, let S be another operator in B fp ( L ( E )) Γ .Let x ∈ M and y ∈ π − ( x ). By (3.11), and the fact that both S and T have finitepropagation, one sees that there exists a finite subset F ⊆ Γ such that(Ψ( ST ) u )( x ) = π ∗ X j ∈ I,g ∈ F ST ( φ gj u )( y )= π ∗ (cid:16) S X j,g T ( φ gj u ) (cid:17) ( y )= π ∗ (cid:16) S X i ∈ I,γ ∈ F φ γi X j,g T ( φ gj u ) (cid:17) ( y )= π ∗ (cid:16) X i,γ S (cid:0) φ γi X j,g T ( φ gj u ) (cid:1)(cid:17) ( y )= (Ψ( S ) ◦ Ψ( T ) u )( x ) . One checks directly that Ψ respects ∗ -operations.That (3.6) is independent of the choice of coset representatives S ⊆ Γ is impliedby (3.11). To see that (3.6) is independent of partitions of unity, let { ϕ i } , { ϕ [ γ ] i } ,and { ϕ γi } be another set of compatible partitions of unity for N , M , and M withthe same properties as { φ i } , { φ [ γ ] i } , and { φ γi } . Let us writeΨ { φ } ( T ) and Ψ { ϕ } ( T )to distinguish the operators defined by (3.6) with respect to these two sets of par-titions of unity. Since T has finite propagation, there exists a finite subset F ′ ⊆ Γ such that if g / ∈ F , then supp( φ gj ) ∩ B prop T ( y ) = ∅ and supp( ϕ gj ) ∩ B prop T ( y ) = ∅ for any j ∈ I . By (3.11), for each x ∈ M we have(Ψ { φ } ( T ) u )( x ) = π ∗ X j ∈ I, g ∈ Γ T ( φ gj u )( y )= π ∗ X j ∈ I, g ∈ F T ( B prop T ( y ) φ gj u )( y ) (3.15)= π ∗ (cid:16)(cid:0) T X j,g B prop T ( y ) φ gj u (cid:1) ( y ) (cid:17) , (3.16)where B prop T ( y ) is the characteristic function of the set B prop T ( y ), and we haveused that T commutes with finite sums. Now observe that X j,g B prop T ( y ) φ gj u and X j,g B prop T ( y ) ϕ gj u are both equal to the lift of the section u to M restricted to the compact subset B prop T ( y ). Thus (3.15) is equal to π ∗ (cid:16)(cid:0) T X j,g B prop T ( y ) ϕ gj u (cid:1) ( y ) (cid:17) = (Ψ { ϕ } ( T )( u ))( x ) . (cid:3) The next proposition shows that the folding map Ψ preserves locality of oper-ators. In particular, this means that Ψ induces a map at the level of localizationalgebras (see Definition 3.9), which will be crucial when we prove functoriality forthe maximal higher rho invariant in section 6.
Proposition 3.5.
For any operator T ∈ B fp ( L ( E )) Γ , we have prop (cid:0) Ψ( T ) (cid:1) ≤ prop( T ) . Proof.
Take a section u ∈ C c ( E ), and write it as a finite sum P j, [ g ] φ [ g ] j u . By (3.11), (cid:0) Ψ( T ) u )( x ) = π ∗ X j ∈ I,g ∈ Γ T ( φ gj u )( y ) , where y is a lift of x to M . Now if d M ( x, supp u ) > prop( T ), then d M (cid:0) y , supp( φ gj u ) (cid:1) > prop( T )for any j ∈ I and g ∈ Γ , since distances between points do not increase under theprojection π . Thus if d M ( x, supp u ) > prop( T ), then Ψ( T ) u ( x ) = 0. Since thisholds for any section u ∈ C c ( E ), we have prop(Ψ( T )) ≤ prop( T ). (cid:3) Proposition 3.6.
The map Ψ is surjective and preserves local compactness of op-erators. Moreover, it restricts to a surjective ∗ -homomorphism Ψ : C [ M ] Γ → C [ M ] Γ . Proof.
Let S ⊆ Γ be the set of coset representatives used in the definition of Ψ.For any T ∈ B fp ( L ( E )) Γ , define an operator T on L ( E ) by the formula T ( v ) := X g ∈ Γ i ∈ I X s ∈ Sj ∈ I φ gsj T (cid:0) φ [ g ] i v (cid:1) , (3.17)for v ∈ L ( E ). We claim that T ∈ B fp ( L ( E )) Γ and that Ψ( T ) = T . Indeed,for each fixed i and j , the sum of operators X g ∈ Γ X s ∈ S φ gsj T φ [ g ] i converges strongly in B ( L ( E )). Since I is a finite indexing set, it follows that T is bounded. To see that T is Γ -equivariant, note that for any γ ∈ Γ , we have, by UNCTORIALITY FOR HIGHER RHO INVARIANTS OF ELLIPTIC OPERATORS 15 the identities (3.7) and (3.8), that γ − T γ ( v ) = X g,i X s,j γ − (cid:0) φ gsj T ( φ [ g ] i γv ) (cid:1) = X g,i X s,j γ − (cid:0) φ gsj T ( π ∗ ( γ · ( φ γ − gi v ))) (cid:1) = X g,i X s,j γ − (cid:0) φ gsj T ([ γ ] · ( φ [ γ − g ] i v )) (cid:1) , where we observe Convention 3.2 as usual. Using Γ -equivariance of T and (3.9),one finds this to be equal to X g,i X s,j γ − (cid:0) φ [ gs ] j ([ γ ] T ( φ [ γ − g ] i v )) (cid:1) = X g,i X s,j φ γ − gsj T ( φ γ − gi v )= X γ − g,i X s,j φ γ − gsj T ( φ [ γ − g ] i v )= T ( v ) , using the definition of T and a change of variable for the last equality.Next, finite propagation of T follows from finite propagation of T . Indeed, forany v ∈ C c ( E ), g ∈ Γ , and i ∈ I , the set S T := n s ∈ S (cid:12)(cid:12) φ [ gs ] j T (cid:0) φ [ g ] i v (cid:1) = 0 for some j ∈ I o is finite, thus the sum over S in equation (3.17) reduces to a finite sum over S T .By the same equation, and the fact that the Γ -action is isometric, we haveprop( T ) ≤ sup s ∈ S T i,j ∈ I { d M (cid:0) U gi , U gsj (cid:1) } = sup s ∈ S T i,j ∈ I (cid:8) d M (cid:0) U ei , U sj (cid:1)(cid:9) < ∞ . We now show that Ψ( T ) = T . By (3.17) and (3.5) we have, for any u ∈ C c ( E ),Ψ( T ) u = X γ ∈ Γ ,t ∈ S X i,j ∈ I φ [ γ ] i T φ tj ( u )= X γ ∈ Γ ,t ∈ S X i,j ∈ I φ [ γ ] i (cid:16) X g ∈ Γ k ∈ I X s ∈ Sl ∈ I φ gsl T (cid:0) φ [ g ] k ( φ tj u ) (cid:1)(cid:17) . Finite propagation of T and compact support of u imply that all of these sums arefinite, so by (3.10) this is equal to X g,s,k,l X γ,t,i,j φ [ γ ] i (cid:0) φ gsl T (cid:0) φ [ g ] k ( φ tj u ) (cid:1)(cid:1) = X g,s,k,l X t,j φ [ gs ] l T (cid:0) φ [ g ] k ( φ tj u ) (cid:1) . Since { gs | s ∈ S } is a set of coset representatives for H \ Γ for any g ∈ Γ , theidentity (3.10) implies that the above is equal to X t,j T (cid:16) X g,k φ [ g ] k ( φ tj u ) (cid:17) = X t,j T ( φ [ t ] j u ) = T u. Thus Ψ( T ) = T as bounded operators on L ( E ).Finally, that Ψ preserves local compactness of operators follows directly from(3.11), so that Ψ restricts to a map C [ M ] Γ → C [ M ] Γ . To see that this map issurjective, suppose T is locally compact. Then for any f ∈ C c ( M ), we have f T = f X g ∈ Γ s ∈ S X i,j ∈ I φ gsj T φ [ g ] i . The sum over Γ reduces to a sum over the finite set F f := n g ∈ Γ (cid:12)(cid:12) U gsj ∩ supp( f ) = ∅ for some j ∈ I o , hence f T is equal to X g ∈ F f s ∈ S X i,j ∈ I f φ gsj π ∗ (supp f ) T φ [ g ] i . Since the subset π ∗ (supp f ) ⊆ M is compact, local compactness of T means thatthis is a finite sum of compact operators and hence compact. A similar argumentshows that the operator T f is compact. Thus the operator T is locally compact,so Ψ restricts to a surjective ∗ -homomorphism C [ M ] Γ → C [ M ] Γ . (cid:3) Description of the folding map on invariant sections.
The folding map Ψ admits an equivalent description using N -invariant sections of E . Such sections, while not in general square-integrable over M , form a space thatis naturally isomorphic to L ( E ) Γ , as we now describe. This allows computationsinvolving Ψ to be carried out entirely on M . Thus it may be a useful perspectivefor some applications. The discussion below slightly generalizes the averaging mapfrom [12, subsection 5.2] applied in the context of discrete groups.Let c : M → [0 ,
1] be a function whose support has compact intersections withevery H -orbit, and such that for all x ∈ M , X h ∈ H c ( hx ) = 1 . Note that this sum is finite by properness of the Γ -action.Let C tc ( E ) H denote the space of H -transversally compactly supported sectionsof E , defined as the space of continuous, H -invariant sections of E whose supportshave compact images in M ∼ = M /H under the quotient map. Let L T ( E ) H denote UNCTORIALITY FOR HIGHER RHO INVARIANTS OF ELLIPTIC OPERATORS 17 the Hilbert space of H -invariant, H -transversally L -sections of E , defined as thecompletion of C tc ( E ) H with respect to the inner product( s , s ) L T ( E ) N := ( cs , cs ) L ( E ) . Then one checks that:
Lemma 3.7.
The space L T ( E ) H is naturally unitarily isomorphic to L ( E ) andis independent of the choice of c . For any T ∈ B fp ( L ( E )) Γ , the operator Ψ( T ) on L ( E ) from (3.6) can bedescribed equivalently by its action on the isomorphic space L T ( E ) H as follows.Take a partition of unity { φ gi } of M as in (3.3). Let s ∈ L T ( E ) H be an H -transversally L -section of E . For any y ∈ M , set(Ψ( T ) s )( y ) := X i ∈ I,g ∈ Γ ( T ( φ gi s ) ( y )) . (3.18)The fact that T has finite propagation means that this pointwise sum is finite, andthat c (Ψ( T ) s ) ∈ L ( E ). Further, Ψ( T ) s is an element of L T ( E ) H , since for any h ∈ H and y ∈ M we have( h (Ψ( T ) s ))( y ) = X i ∈ I,g ∈ Γ ( T ◦ h ( φ gi s ) ( y )) = X i ∈ I,g ∈ Γ T ( φ hgi s )( y ) = (Ψ( T ) s )( y ) , where we have used the equivariance properties of T and s . Finally, a direct com-parison shows that this definition is equivalent with that given by equation (3.11).3.3. Induced maps on geometric C ∗ -algebras and K -theory. By Proposition 3.6, the folding map Ψ restricts to a surjective ∗ -homomorphismΨ : C [ M ] Γ → C [ M ] Γ . By the defining property of maximal completions of ∗ -algebras, Ψ extends to asurjective ∗ -homomorphism between maximal equivariant Roe algebras,Ψ : C ∗ max ( M ) Γ → C ∗ max ( M ) Γ . In this paper we will make use of a number natural extensions of this map toother geometric C ∗ -algebras, as well as the induced maps on K -theory. We refer tothese maps collectively as folding maps .To begin, we have the following elementary lemma: Lemma 3.8.
Let A and B be C ∗ -algebras and let φ : A → B be a surjective ∗ -homomorphism. Then φ extends to a ∗ -homomorphism e φ : M ( A ) → M ( B ) .Proof. Since φ is surjective, we may define e φ by requiring e φ ( m ) φ ( a ) = φ ( ma ) , φ ( a ) e φ ( m ) = φ ( am ) , for m ∈ A . Concretely, by way of a faithful non-degenerate representation σ : B →B ( H ) on some Hilbert space H , we may identify B with a subalgebra of B ( H ). Since φ is surjective, the composition σ ◦ φ : A → B ( H ) is a non-degenerate representationof A . Thus the above formulas define a representation of M ( A ) with values in theidealizer of σ ( B ), which one identifies with M ( B ). (See also [8] Lemma I.9.14.) (cid:3) By Lemma 3.8 and Proposition 3.6, the map Ψ extends to a ∗ -homomorphism M (cid:0) C ∗ max ( M ) Γ (cid:1) → M (cid:0) C ∗ max ( M ) Γ (cid:1) . Viewing C ∗ max ( M j ) Γ j as an ideal in M , we will still denote this extended map by Ψ.This map will be essential when we apply the functional calculus for the maximalRoe algebra in sections 5 and 6.Next, suppose we have a path r : [0 , ∞ ) → C ∗ max ( M ) Γ that satisfiesprop( r ( t )) → t → ∞ . Then by Proposition 3.5, the same is true of the path Ψ ◦ r : [0 , ∞ ) → C ∗ max ( M ) Γ .This allows the folding map to pass to the level of localization algebras: Definition 3.9.
Define the mapΨ L : C ∗ L, max ( M ) Γ → C ∗ L, max ( M ) Γ r Ψ ◦ r. Then Ψ L restricts to a ∗ -homomorphism between obstruction algebras:Ψ L, : C ∗ L, , max ( M ) Γ → C ∗ L, , max ( M ) Γ . Each of the above maps Ψ, Ψ L , and Ψ L, induces a map at the level of K -theory.In this paper, we will make use of two of them:Ψ ∗ : K • (cid:0) C ∗ max ( M ) Γ (cid:1) → K • (cid:0) C ∗ max ( M ) Γ (cid:1) , (Ψ L, ) ∗ : K • (cid:0) C ∗ L, , max ( M ) Γ (cid:1) → K • (cid:0) C ∗ L, , max ( M ) Γ (cid:1) . Functional calculus and the wave operator
We now develop the analytical properties of the wave operator on the maximalRoe algebra that will form the basis of our work in subsequent sections.Let us begin by recalling the functional calculus for the maximal Roe algebra,which was developed in a more general setting in [13]. The discussion here is spe-cialized to the cocompact setting.Throughout this section we will work in the geometric situation in subsection 2.3.To simplify notation, in this subsection and the next we will write M, Γ for either M , Γ or M , Γ , and D, E for either D , E or D , E . In other words, Γ acts freelyand properly on M with compact quotient, and D is a Γ-equivariant operator onthe bundle E → M . UNCTORIALITY FOR HIGHER RHO INVARIANTS OF ELLIPTIC OPERATORS 19
Functional calculus on the maximal Roe algebra.
We shall view the C ∗ -algebra C ∗ max ( M ) Γ as a right Hilbert module over itself. Theinner product and right action on C ∗ max ( M ) Γ are defined naturally through multi-plication: for a, b ∈ C ∗ max ( M ) Γ , h a, b i = a ∗ b, a · b = ab. (4.1)The algebra of compact operators on this Hilbert module can be identified with C ∗ max ( M ) Γ via left multiplication. Similarly, the algebra of bounded adjointableoperators can be identified with the multiplier algebra M of C ∗ max ( M ) Γ .The operator D defines an unbounded operator on this Hilbert module in thefollowing way. First note that D acts on smooth sections of the external tensorproduct E ⊠ E ∗ → M × M by taking a section s to the section Ds defined by( Ds )( x, y ) = D x s ( x, y ) , (4.2)where D x denotes D acting on the x -variable.Let S Γ be the ∗ -subalgebra of C [ M ] Γ defined in Remark 2.7. That is, an elementof S Γ is an operator T κ given by a smooth kernel κ ∈ C ∞ b ( E ⊠ E ∗ ) that:(i) is Γ-equivariant with respect to the diagonal Γ-action;(ii) has finite propagation, meaning that there exists a constant c κ ≥ d ( x, y ) > c κ then κ ( x, y ) = 0.The operator D acts on elements of S Γ by acting on the corresponding smoothkernels as in (4.2). In this way, D becomes a densely defined operator on the Hilbertmodule C ∗ max ( M ) Γ that one verifies is symmetric with respect to the inner productin (4.1). Theorem 4.1 ([13] Theorem 3.1) . There exists a real number µ = 0 such that theoperators D ± µi : C ∗ max ( M ) Γ → C ∗ max ( M ) Γ have dense range. Consequently, the operator D on the Hilbert module C ∗ max ( M ) Γ is regular andessentially self-adjoint, and so admits a continuous functional calculus (see [18, The-orem 10.9] and [17, Proposition 16]): Theorem 4.2.
For j = 1 or , there is a ∗ -preserving linear map π : C ( R ) → R (cid:0) C ∗ max ( M ) Γ (cid:1) ,f f ( D ) := π ( f ) , where C ( R ) denotes the continuous functions R → C , and R (cid:0) C ∗ max ( M ) Γ (cid:1) denotesthe regular operators on C ∗ max ( M ) Γ , such that:(i) π restricts to a ∗ -homomorphism C b ( R ) → M ; (ii) If | f ( t ) | ≤ | g ( t ) | for all t ∈ R , then dom( g ( D )) ⊆ dom( f ( D )) ;(iii) If ( f n ) n ∈ N is a sequence in C ( R ) for which there exists f ′ ∈ C ( R ) suchthat | 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 ( D ) x → f ( D ) x foreach x ∈ dom( f ( D )) ;(iv) Id( D ) = D . The wave operator.
We now discuss the relationship between the wave operator formed using the func-tional calculus from Theorem 4.2 and the classical wave operator on L . Both ofthese operators can be viewed as bounded multipliers of the maximal Roe algebra C ∗ max ( M ) Γ , and we will see that: Proposition 4.3.
For each t ∈ R , we have e itDL = e itD ∈ M ( C ∗ max ( M ) Γ ) . Let us begin by explaining both sides of the equation, starting with the right-handside. For each t ∈ R , the functional calculus from Theorem 4.2 allows one to form abounded adjointable operator e itD on the Hilbert module C ∗ max ( M ) Γ . The resultinggroup of operators { e itD } t ∈ R is strongly continuous in the sense of Theorem 4.2 (iii)and uniquely solves the wave equation on C ∗ max ( M ) Γ : Lemma 4.4.
For any κ ∈ S Γ , u ( t ) = e itD κ is the unique solution of the problem dudt = iDu, u (0) = κ, (4.3) with u : R → C ∗ max ( M ) Γ a differentiable map taking values in dom( D ) .Proof. For each t ∈ R , the function s e its is a unitary in C b ( R ). Hence e itD isbounded adjointable and unitary. Let h n be a sequence of positive real numbersconverging to 0 as n → ∞ . Then for each t ∈ R , the sequence of functions f n ( s ) := e i ( t + h n ) s − e its h n converges to f ( s ) := ise its uniformly on compact subsets of R in the limit n → ∞ .Also, each f n is bounded above by | s | . By Theorem 4.2 (iii), this implies (4.3).For the uniqueness claim, let v be another solution of (4.3) with v (0) = κ . Forany fixed s ∈ R and 0 ≤ t ≤ s , set w ( t ) = e itD w ( s − t ). Then we have dwdt = iDe itD v ( s − t ) − ie itD Dv ( s − t ) = 0 . It follows that w ( t ) is constant for all t , hence v ( s ) = w (0) = w ( s ) = e isD κ = u ( s ) . (cid:3) UNCTORIALITY FOR HIGHER RHO INVARIANTS OF ELLIPTIC OPERATORS 21
On the other hand, we may apply the functional calculus on L ( E ) to the essen-tially self-adjoint operator D : L ( E ) → L ( E ) (4.4)to form the classical wave operator e itDL , not to be confused with with the boundedadjointable operator e itD . The resulting operator group { e itDL } t ∈ R is strongly con-tinuous in B ( L ( E )) and uniquely solves the wave equation on L ( E ): for every v ∈ C ∞ c ( E ), we have ddt ( e itDL ( v )) = iDe itDL ( v ) . Via composition, e itDL defines a map e itDL : S Γ → S Γ . (4.5)Indeed, since the quotient M/ Γ is compact, a standard argument involving theSobolev embedding theorem shows that for any T κ ∈ S Γ , the kernel of e itDL ◦ T κ is smooth, in addition to having finite propagation and being Γ-equivariant. Thus e itDL ◦ T κ ∈ S Γ . By Remark 2.9, (4.5) extends uniquely to a bounded multiplier of C ∗ max ( M ) Γ , e itDL : C ∗ max ( M ) Γ → C ∗ max ( M ) Γ . (4.6)This is the operator on the left-hand side of Proposition 4.3.The proof of Proposition 4.3, involves a more general form of the following obser-vation: if k is a smooth, compactly supported Schwartz kernel on M × M , then wehave the pointwise equality (cid:16) ddt e itDL k (cid:17) ( x, y ) = ( iDe itDL k )( x, y ) , (4.7)where e itDL k denotes the smooth Schwartz kernel of the composition e itDL ◦ T k . Wewill need the following more general form of this observation for kernels in S Γ : Lemma 4.5.
Let κ ∈ S Γ , and let e itDL κ denote the smooth Schwartz kernel of e itDL ◦ T κ . Then for every t ∈ R and x, y ∈ M , we have the pointwise equality (cid:16) ddt e itDL κ (cid:17) ( x, y ) = ( iDe itDL κ )( x, y ) . (4.8) Moreover, the path t e itDL κ is continuous with respect to the operator norm, and lim h → (cid:13)(cid:13)(cid:13)(cid:13) e i ( t + h ) DL ◦ T κ − e itDL ◦ T κ h − iDe itDL ◦ T κ (cid:13)(cid:13)(cid:13)(cid:13) B ( L ( E )) = 0 . (4.9) Proof.
Fix t ∈ R , ǫ >
0, and x , y ∈ M . Take φ ∈ C ∞ c ( M ) such that φ ( x ) = 1if x ∈ B t ( x ), and φ ( x ) = 0 if x ∈ M \ B t ( x ). Since κ has finite propagation, thesmooth kernel φκ is compactly supported in M × M , so by (4.7) we have (cid:16) ddt e itDL φκ (cid:17) ( x, y ) = ( iDe itDL φκ )( x, y ) . (4.10) The fact that (1 − φ ) κ ( z, y ) = 0 for all z ∈ B t ( x ) implies that for all w ∈ B t ( x ) wehave ( e itDL (1 − φ ) κ )( w, y ) = 0, so that( e itDL κ )( w, y ) = ( e itDL φκ )( w, y ) . (4.11)Combined with (4.10), this gives (4.8).To establish norm continuity, it suffices to show that t e itDL κ is norm-continuousat t = 0. For each x , y , and t in an interval [ − t , t ], we have (cid:12)(cid:12)(cid:12) e itDL κ ( x, y ) − κ ( x, y ) (cid:12)(cid:12)(cid:12) ≤ | t | · sup s ∈ [ − t ,t ] (cid:12)(cid:12)(cid:12) iDe isDL κ ( x, y ) (cid:12)(cid:12)(cid:12) (4.12)by the mean value theorem applied to (4.8). Since the operator iDe isDL κ is Γ-equivariant with finite propagation, and M is cocompact, there exists a compactsubset K ⊆ M such thatsup x,y ∈ M,s ∈ [ − t ,t ] (cid:12)(cid:12)(cid:12) iDe isDL κ ( x, y ) (cid:12)(cid:12)(cid:12) = sup x,y ∈ K,s ∈ [ − t ,t ] (cid:12)(cid:12)(cid:12) iDe isDL κ ( x, y ) (cid:12)(cid:12)(cid:12) ≤ C t (4.13)for some constant C t . Now since e itDL κ − κ has finite propagation, its operator normcan be estimated using and (4.12) and (4.13): k e itDL κ − κ k B ( L ( E )) ≤ C · sup x,y ∈ M (cid:12)(cid:12)(cid:12) e itDL κ ( x, y ) − κ ( x, y ) (cid:12)(cid:12)(cid:12) ≤ CC t | t | , for some constant C . We obtain norm continuity by taking a limit t →
0. The proofof (4.9) is a straightforward adaptation of this argument. (cid:3)
With these preparations, we now prove Proposition 4.3.
Proof of Proposition 4.3.
Fix t ∈ R . Let T κ ∈ S Γ with smooth Schwartz kernel κ .We claim that the kernel e itDL κ satisfies the wave equation in C ∗ max ( M ) Γ . To seethis, first note that by equation (4.9) we havelim h → (cid:13)(cid:13)(cid:13)(cid:13) e i ( t + h ) DL ◦ T κ − e itDL ◦ T κ h − iDe itDL ◦ T κ (cid:13)(cid:13)(cid:13)(cid:13) B ( L ( E )) = 0 . Now since the kernels e i ( t + h ) DL κ and De itDL κ each have propagation at most r := prop( κ ) + | t + h | , by Remark 2.9 there exists a constant C r such that the norm of e i ( t + h ) DL κ − e itDL κh − iDe itDL ( κ )in C ∗ max ( M ) Γ is bounded above by C r times its norm in B ( L ( E )). Thuslim h → (cid:13)(cid:13)(cid:13)(cid:13) e i ( t + h ) DL κ − e itDL κh − iDe itDL ( κ ) (cid:13)(cid:13)(cid:13)(cid:13) max = 0 , UNCTORIALITY FOR HIGHER RHO INVARIANTS OF ELLIPTIC OPERATORS 23 so the operator group { e itDL } t ∈ R solves the wave equation (4.3) in C ∗ max ( M ) Γ . Itfollows from the uniqueness property in Lemma 4.4 that e itDL and e itD coincide on S Γ , and hence are equal as elements of M . (cid:3) Functoriality for the higher index
In this section we give a new proof of Theorem 1.1, using properties of the foldingmap Ψ and the wave operator from sections 3 and 4. The techniques we develophere will be crucial to proving our main result, Theorem 1.2, in section 6.5.1.
Higher index.
We begin by recalling the definition of the maximal higher index of an equivariantelliptic operator. Let Γ, M , and D be as in section 4.Let Q := M /C ∗ max ( M ) Γ . Consider the short exact sequence of C ∗ -algebras0 → C ∗ max ( M ) Γ → M → Q → , where M is shorthand for the multiplier algebra M (cid:0) C ∗ max ( M ) Γ (cid:1) . This induces thefollowing six-term exact sequence in K -theory: K ( C ∗ max ( M ) Γ ) K ( M ) K ( Q ) K ( Q ) K ( M ) K ( C ∗ max ( M ) Γ ) , ∂ ∂ where the connecting maps ∂ and ∂ , known as index maps , are defined as follows. Definition 5.1. (i) ∂ : let u be an invertible matrix over Q representing a class in K ( Q ). Let v be the inverse of u . Let U and V be lifts of u and v to a matrix algebraover M . Then the matrix W = (cid:18) U (cid:19) (cid:18) − V (cid:19) (cid:18) U (cid:19) is invertible, and P = W (cid:18) (cid:19) W − is an idempotent. We define ∂ [ u ] := [ P ] − (cid:20) (cid:21) ∈ K (cid:0) C ∗ max ( M ) Γ (cid:1) . (5.1)(ii) ∂ : let q be an idempotent matrix over Q representing a class in K ( Q ). Let Q be a lift of q to a matrix algebra over M . Then e πiQ is a unitary in theunitized algebra, and we define ∂ [ q ] := (cid:2) e πiQ (cid:3) ∈ K ( C ∗ max ( M ) Γ ) . (5.2) This construction is applied to the operator D via the functional calculus fromTheorem 4.2, as follows. Let χ : R → R be a continuous, odd function such thatlim x → + ∞ χ ( x ) = 1 , known as a χ a normalizing function . From Theorem 4.2 (i) we obtain an element χ ( D ) ∈ M . We now have: Lemma 5.2.
The class of χ ( D ) in M /C ∗ max ( M ) Γ is invertible and independent ofthe choice of normalizing function χ .Proof. Let S ( R ) denote the Schwartz space of functions R → C . Then for every f ∈ S ( R ) with compactly supported Fourier transform b f , the operator f ( D ) isgiven by a smooth kernel [20, Proposition 2.10]. Since every f ∈ C ( R ) function is auniform limit of such functions, the first part of Theorem 4.2 implies that for everysuch f we have f ( D ) ∈ C ∗ max ( M ) Γ .Now if χ is a normalizing function, then χ − ∈ C ( R ). Hence the class of χ ( D ) in M /C ∗ max ( M ) Γ is invertible. Since any two normalizing functions differ byan element of C ( R ), this class is independent of the choice of χ . (cid:3) Using this lemma, one computes that χ ( D ) + 12is an idempotent modulo C ∗ max ( M ) Γ and so defines element of K (cid:0) M /C ∗ max ( M ) Γ (cid:1) .This leads us to the definition of the maximal higher index of D : Definition 5.3.
For i = 1 ,
2, let ∂ i be the connecting maps from Definition 5.1.The maximal higher index of D is the elementInd Γ , max D := ∂ [ χ ( D )] ∈ K (cid:0) C ∗ max ( M ) Γ (cid:1) , if dim M is even ,∂ h χ ( D )+12 i ∈ K (cid:0) C ∗ max ( M ) Γ (cid:1) , if dim M is odd . Functoriality.
In this subsection, we return to the geometric setup described in subsection 2.3 andgive a new proof of Theorem 1.1.A key idea is to use the local nature of the wave operator to prove:
Proposition 5.4.
For all t ∈ R , we have Ψ( e itD ) = e itD .Proof. For j = 1 ,
2, let e itD j L be the wave operator on L ( E j ). By (4.6), this operatorextends uniquely to a bounded multiplier of C ∗ max ( M j ) Γ j . By Proposition 4.3, wehave e itD j L = e itD j ∈ M j . Thus to prove this proposition, it suffices to show thatΨ( e itD L ) = e itD L , as elements of B fp ( L ( E j )) Γ j . UNCTORIALITY FOR HIGHER RHO INVARIANTS OF ELLIPTIC OPERATORS 25
As a notational convenience, we will write e itD j for e itD j L ∈ B fp ( L ( E j )) Γ j . Letthe open covers U M and U M be as in (3.3) and (3.4), so that by definition, thereexists some ǫ > ǫ in M is evenly covered by M .Let { V k } be another open cover of M such that each V k has diameter at most ǫ and such that any compact subset of M intersects only finitely many of the V k .Let { ρ k } be a partition of unity subordinate to { V k } .Choose a positive integer n such that tn < ǫ . Now since e itD = (cid:0) e i tn D (cid:1) n , and Ψ is a ∗ -homomorphism, it suffices to show that Ψ( e i tn D ) = e i tn D . Noting thatany section u ∈ C c ( E ) can be written as a finite sum u = P k ρ k u, we haveΨ( e i tn D ) u = X k Ψ( e i tn D )( ρ k u ) . We first claim that for each k we haveΨ( e i tn D )( ρ k u ) = e i tn D ( ρ k u ) . (5.3)To see this, note that the ball of radius ǫ around supp( ρ k u ) has diameter at most ǫ and so is evenly covered by M . Since the definition of Ψ is independent of the choiceof compatible partitions of unity by Proposition 3.4, we may work with a partitionof unity { φ [ g ] j } for M subordinate to a cover U M , such that B ǫ (supp( ρ k u )) ⊆ U j for some open set U [ s ] j ∈ U M and φ [ s ] j ≡ B ǫ (supp( ρ k u )), for some s ∈ S and j ∈ I . By the definition of the folding map (3.6) applied to this choice of opencover, we have Ψ( e i tn D )( ρ k u ) = X g ∈ Γ ,s ∈ S X i,j ∈ I φ [ g ] i (cid:0) e i tn D φ sj ( ρ k u ) (cid:1) = φ [ s ] j (cid:0) e i tn D (cid:0) π | ∗ U s j ( ρ k u ) (cid:1)(cid:1) = π ∗ (cid:0) e i tn D (cid:0) π | ∗ U s j ( ρ k u ) (cid:1)(cid:1) . Now the wave equation on M reads ∂∂t (cid:0) e i tn D (cid:0) π | ∗ U s j ( ρ k u ) (cid:1)(cid:1) = iD (cid:0) e i tn D (cid:0) π | ∗ U s j ( ρ k u ) (cid:1)(cid:1) . Applying π ∗ to both sides of this equation and using that D is the lift of D , weobtain ∂∂t (cid:0) Ψ (cid:0) e i tn D (cid:1) ( ρ k u ) (cid:1) = iD (cid:0) Ψ (cid:0) e i tn D (cid:1) ( ρ k u ) (cid:1) , whence (5.3) follows from uniqueness of the solution to the wave equation on M . Taking a sum over k now yieldsΨ( e i tn D ) u = X k Ψ( e i tn D )( ρ k u ) = X k e i tn D ( ρ k u ) = e i tn D u. Thus Ψ( e i tn D ) = e i tn D , as bounded operators on L ( E ). By our previous remarks,this means that Ψ( e itD ) = e itD . (cid:3) Applying Fourier inversion together with Proposition 5.4 leads to:
Proposition 5.5.
For any f ∈ C ( R ) we have Ψ( f ( D )) = f ( D ) ∈ M . Proof.
Suppose first that f ∈ S ( R ) with compactly supported Fourier transform.By the Fourier inversion formula, we have f ( D j ) = 12 π Z R b f ( t ) e itD j dt, (5.4)where the integral converges strongly in M j . Now for any κ ∈ C ∗ max ( M ) Γ , Propo-sition 3.6 implies that there exists κ ∈ C ∗ max ( M ) Γ such that Ψ( κ ) = κ . Since Ψis a ∗ -homomorphism,Ψ( f ( D ))( κ ) = Ψ( f ( D ) κ ) = 12 π Z R b f ( t )Ψ( e itD κ ) dt. By Proposition 5.4, this equals12 π Z R b f ( t ) e itD κ dt = f ( D ) κ. This proves the claim for f ∈ S ( R ). The general claim now follows from density of S ( R ) in C ( R ). (cid:3) Proof of Theorem 1.1.
The expressions (5.1) and (5.2) show that, for j = 1 ,
2, thehigher index of D j is represented by a matrix A j = A j ( χ ) whose entries are operatorsformed using functional calculus of D j . More precisely, if the initial operator D N on N is ungraded, as is typically the case for M odd-dimensional, then A j = e πi ( χ +1) ( D j ) . (5.5)When D j is odd-graded with respect to a Z -grading on the bundle E j = E + j ⊕ E − j ,as typically occurs when dim N is even, we have a direct sum decomposition χ ( D j ) = χ ( D j ) + ⊕ χ ( D j ) − . In this case, the index element is represented explicitly by thematrix A j = (cid:18) (1 − χ ( D j ) − χ ( D j ) + ) χ ( D j ) − (1 − χ ( D j ) + χ ( D j ) − ) χ ( D j ) + (2 − χ ( D j ) − χ ( D j ) + )(1 − χ ( D j ) − χ ( D j ) + ) χ ( D j ) + χ ( D j ) − (2 − χ ( D j ) + χ ( D j ) − ) − (cid:19) (5.6) UNCTORIALITY FOR HIGHER RHO INVARIANTS OF ELLIPTIC OPERATORS 27
Observe that in either case, each entry of A j is an operator of the form f ( D j ), forsome f ∈ C ( R ) (modulo grading and the identity operator). By Proposition 5.5,Ψ maps each entry of A to the corresponding entry of A . Hence Ψ ∗ [ A ] = [ A ],which proves the claim. (cid:3) Remark 5.6.
When Γ = π N and Γ is the trivial group, Theorem 1.1 reduces tothe maximal version of Atiyah’s L -index theorem mentioned in section 1.Atiyah’s original L -index theorem, which uses the von Neumann trace τ insteadof the folding map, can be proved by an argument along lines similar to the proofof Theorem 1.1.6. Functoriality for the higher rho invariant
The higher index is a primary obstruction to the existence of positive scalarcurvature metrics on a manifold. When the manifold is spin with positive scalarcurvature, so that the higher index of the Dirac operator vanishes, one can define asecondary invariant called the higher rho invariant , introduced in [21, 15]. This isan obstruction to the inverse of the Dirac operator being local [4]. In this section weshow that the higher rho invariant behaves functorially under the map Ψ L, fromDefinition 3.9.6.1. Higher rho invariant.
Consider the geometric situation in subsection 2.3, with the additional condition thatthe Riemannian manifold N is spin with positive scalar curvature. The operator D N is then the Dirac operator acting on the spinor bundle E N .As the definition of the higher rho invariant is the same for either M or M , wewill simply write M , Γ to mean either M , Γ or M , Γ . Similarly, D will refer toeither of the lifted operators D or D acting on the equivariant spinor bundles E or E lifted from E N .Let κ be the scalar curvature function of the lifted metric on M , which is uniformlypositive. Let ∇ : C ∞ ( E ) → C ∞ ( T ∗ M ⊗ E ) be the connection on E induced by theLevi-Civita connection on M . Recall that by the Lichnerowicz formula, D = ∇ ∗ ∇ + κ . Since κ is uniformly positive, D is strictly positive as an unbounded operator on theHilbert module C ∗ max ( M ) Γ . Thus we may use the functional calculus from Theorem4.2 to form the operator F ( D ) := D | D | , an element of the multiplier algebra M = M ( C ∗ max ( M ) Γ ). Observe that F ( D )+12 isa projection in M . Since D is invertible, there exists ǫ > D is containedin R \ ( − ǫ, ǫ ). Let { F t } t ∈ R + be a set of normalizing functions satisfying the followingconditions: • F t has compactly supported distributional Fourier transform for each t ; • diam(supp b F t ) → t → ∞ ; • F t → x | x | uniformly on R \ ( − ǫ, ǫ ) in the limit t → t → ∞ , the propagations of F t ( D ) tend to 0. By Theorem 4.2 (i), as t →
0, the operators F t ( D ) converge to F ( D ) in the norm of M .Define a path R ≥ → ( C ∗ max ( M ) Γ ) + given by R D : t A ( F t ) , (6.1)where the matrix A ( F t ) is defined by (5.5) or (5.6) depending upon the dimensionof N . Then R D is a projection matrix with entries in ( C ∗ L, max ( M ) Γ ) + when dim N is even, and a unitary in ( C ∗ L, max ( M ) Γ ) + when dim N is odd. Further, by notingthat R D (0) = A ( F ) = (cid:16) (cid:17) if dim N is even , N is odd , one sees that R D is a matrix with entries in (cid:0) C ∗ L, , max ( M ) Γ (cid:1) + . Definition 6.1.
The higher rho invariant of D on the Riemannian manifold M is ρ max ( D ) = [ R D ] ∈ K • ( C ∗ L, , max ( M ) Γ ) , where • = dim M (mod 2).6.2. Functoriality.
We are now ready to complete the proof of our main result, Theorem 1.2, using thetools we developed in sections 3, 4, and 5.Recall from Definition 3.9 that we have a folding map at level of obstructionsalgebras, Ψ L, : C ∗ L, , max ( M ) Γ → C ∗ L, , max ( M ) Γ . This map is well-defined because the folding map Ψ at the level of maximal equi-variant Roe algebras preserves small propagation of operators, by Proposition 3.5.The induced map on K -theory,(Ψ L, ) ∗ : K • (cid:0) C ∗ L, , max ( M ) Γ (cid:1) → K • (cid:0) C ∗ L, , max ( M ) Γ (cid:1) , implements functoriality of the maximal higher rho invariant. Proof of Theorem 1.2.
For j = 1 ,
2, let the higher rho invariants of D j be denotedby ρ max ( D j ), as in Definition 6.1. By (6.1), this class is represented the path R D j : t A j ( F t ) , UNCTORIALITY FOR HIGHER RHO INVARIANTS OF ELLIPTIC OPERATORS 29 where the matrix A j is as in (5.5) and (5.6). By Definition 3.9, the map (Ψ L ) ∗ takesthe class [ R D ] ∈ K • (cid:0) C ∗ L, , max ( M ) Γ (cid:1) to the class of the composed pathΨ ◦ R D : t Ψ( A ( F t ))in K • (cid:0) C ∗ L, , max ( M ) Γ (cid:1) . Since each entry of A j is an operator of the form f ( D j ), forsome f ∈ C ( R ) (up to grading and the identity operator), Proposition 5.5 impliesthat for each t ≥
0, we haveΨ ◦ R D ( t ) = Ψ( A ( F t )) = A ( F t ) = R D ( t ) . It follows that (Ψ L, ) ∗ ( ρ max ( D ) = ρ max ( D ) ∈ K • ( C ∗ L, , max ( M ) Γ ). (cid:3) A generalization to the non-cocompact setting
The methods in this paper can be used to establish analogous results in moregeneral geometric settings. In this final section, we give one such generalization ofTheorem 1.1 to the case of operators that are invertible at infinity on non-cocompactmanifolds.In the following, we will work with the non-cocompact analogue of the geometricsetup in subsection 2.3, so that the manifold N is no longer assumed to be compact.In place of the finite partition of unity (3.2) used to define the folding map Ψ, wetake a locally finite partition of unity U N whose elements are evenly covered by both M and M , and with the property that any compact subset of N intersects onlyfinitely many elements of U N . The equivariant partitions of unity U M and U M of M and M are defined in the same way according to (3.3) and (3.4).The local nature of the wave operator e itD allows us to obtain the followinggeneralization of Proposition 5.4: Proposition 7.1.
Let N , M , and M be as in this section, with N not necessarilycompact. Then for all t ∈ R , we have Ψ( e itD ) = e itD .Proof. We adapt the proof of Proposition 5.4, indicating only what needs to bechanged. Let the open covers U M and U M be as above. The difference now isthat since N may be non-compact, we cannot assume the existence of a uniformlypositive covering diameter ǫ as we did in the proof of Proposition 5.4.Instead, the key point is to observe that for any fixed t ∈ R and u ∈ C c ( E ),the section e itD u is supported within the compact subset B t (supp u ). Thus we canfind ǫ > x ∈ B t (supp u ), the ball B ǫ ( x ) is evenly covered by M .From here, the proof proceeds exactly as in the argument that Ψ( e itD ) u = e itD u proceeds exactly as in the proof of Proposition 5.4, with M replaced by B t (supp u ).Since t and u are arbitrary, we conclude. (cid:3) To ensure that the operator D N is Fredholm, we suppose that it is invertibleat infinity , meaning that there exists some compact subset Z N ⊆ N on whosecomplement we have D N ≥ a for some a >
0. An important case is when D N is theDirac operator, and the metric g N has uniformly positive scalar curvature outsideof Z N .For j = 1 ,
2, the lifted operator D j then satisfies the analogous relation D j ≥ a on the complement of a cocompact , Γ j -invariant subset Z j ⊆ M j . We can define aversion of the higher index of D j , localized around the set Z j , as follows. (For moredetails of this construction, see [22] and [12, section 3].)For each R >
0, let C ∗ max ( B R ( Z j )) Γ j be the maximal equivariant Roe algebra ofthe R -neighborhood of Z j . Since B R ( Z j ) is cocompact, this algebra is isomorphicto C ∗ max (Γ j ) ⊗ K by Remark 2.9. One can then show that for any f ∈ C c ( − a, a ), wehave f ( D j ) ∈ lim R →∞ C ∗ max ( B R ( Z j )) Γ j , where lim R →∞ C ∗ max ( B R ( Z j )) Γ j is the direct limit of these C ∗ -algebras. Indeed, thislimit algebra is isomorphic to C ∗ max (Γ) ⊗ K , where K denotes the compact operatorson a (possibly non-compact) fundamental domain of the Γ-action. The constructionfrom subsection 5.1 then allows one to define an index elementInd Γ j , max D j ∈ K • ( C ∗ max (Γ j )) . Similar to the cocompact case, we can use Proposition 7.1 to obtain the followingversion of Theorem 1.1 for operators invertible at infinity:
Theorem 7.2.
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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
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