Delocalized eta invariants, algebraicity, and K -theory of group C ∗ -algebras
aa r X i v : . [ m a t h . K T ] M a y DELOCALIZED ETA INVARIANTS, ALGEBRAICITY, AND K -THEORY OF GROUP C ∗ -ALGEBRAS ZHIZHANG XIE AND GUOLIANG YU
Abstract.
In this paper, we establish a precise connection between higher rho in-variants and delocalized eta invariants. Given an element in a discrete group, if itsconjugacy class has polynomial growth, then there is a natural trace map on the K -group of its group C ∗ -algebra. For each such trace map, we construct a determi-nant map on secondary higher invariants. We show that, under the evaluation of thisdeterminant map, the image of a higher rho invariant is precisely the correspondingdelocalized eta invariant of Lott. As a consequence, we show that if the Baum-Connesconjecture holds for a group, then Lott’s delocalized eta invariants take values in al-gebraic numbers. We also generalize Lott’s delocalized eta invariant to the case wherethe corresponding conjugacy class does not have polynomial growth, provided thatthe strong Novikov conjecture holds for the group. Introduction
Let X be a complete manifold of dimension n with a discrete group Γ acting on itproperly and cocompactly through isometries. Each Γ-equivariant elliptic differentialoperator D on X gives rise to a higher index class Ind Γ ( D ) ∈ K n ( C ∗ r (Γ)). This higherindex is an obstruction to the invertibility of D . It is a far-reaching generalizationof the classical Fredholm index and plays a fundamental role in the studies of manyproblems in geometry and topology such as the Novikov conjecture, the Baum-Connesconjecture and the Gromov-Lawson-Rosenberg conjecture. Higher index classes areoften referred to as primary invariants. When the higher index class of an operator istrivial and given a specific trivialization, a secondary index theoretic invariant naturallyarises. One such example is the associated Dirac operator e D on the universal covering f M of a closed spin manifold M , which is equipped with a positive scalar curvaturemetric g . In this case, it follows from the Lichnerowicz formula that the higher indexof the Dirac operator vanishes. And there is a natural secondary higher invariant of e D – introduced by Higson and Roe [12, 13, 14, 30] – called the higher rho invariant of e D (with respect to the metric g ), cf. Section 2 . e D above, Lott introduced the fol-lowing delocalized eta invariant η h h i ( e D ) [23]: η h h i ( e D ) := 2 √ π Z ∞ tr h ( e De − t e D ) dt, (1) The first author is partially supported by NSF 1500823, NSF 1800737.The second author is partially supported by NSF 1700021. under the condition that the conjugacy class h h i of h ∈ π M has polynomial growth.Here π M is the fundamental group of M , and tr h is the following trace map (seeSection 4 for more details): tr h ( A ) = X g ∈h h i Z F A ( x, gx ) dx on Γ-equivariant Schwartz kernels A ∈ C ∞ ( f M × f M ), where F is a fundamental domainof f M under the action of Γ.In this paper, we shall devise a conceptual K -theoretic approach to establish aprecise connection between Higson-Roe’s K -theoretic higher rho invariants and Lott’sdelocalized eta invariants. More precisely, we have the following theorem. Theorem 1.1.
Let M be a closed odd-dimensional spin manifold equipped with a pos-itive scalar curvature metric g . Suppose f M is the universal cover of M , ˜ g is theRiemannian metric on f M lifted from g , and e D is the associated Dirac operator on f M .Suppose the conjugacy class h h i of a non-identity element h ∈ π M has polynomialgrowth, then we have τ h ( ρ ( e D, e g )) = − η h h i ( e D ) , where ρ ( e D, e g ) is the K -theoretic higher rho invariant of e D with respect to the metric ˜ g , and τ h is a canonical determinant map associated to h h i . While the definition of Lott’s delocalized eta invariant requires certain growth con-ditions on π M (e.g. polynomial growth on a conjugacy class), the K -theoretic higherrho invariant can be defined in complete generality, without any growth conditions on π M . We shall show how to generalize Lott’s delocalized eta invariant without im-posing any growth conditions on π M , provided that the strong Novikov conjectureholds for π M . This is achieved by using the Novikov rho invariant introduced in [36,Section 7].As an application of Theorem 1 . Theorem 1.2.
With the same notation as above, if the rational Baum-Connes con-jecture holds for Γ , and the conjugacy class h h i of a non-identity element h ∈ Γ haspolynomial growth, then the delocalized eta invariant η h h i ( e D ) is an algebraic number.Moreover, if in addition h has infinite order, then η h h i ( e D ) vanishes. This theorem follows from the construction of the determinant map τ h and a L -Lefschetz fixed point theorem of B.-L. Wang and H. Wang [33, Theorem 5.10]. WhenΓ is torsion-free and satisfies the Baum-Connes conjecture, and the conjugacy class h h i of a non-identity element h ∈ Γ has polynomial growth, Piazza and Schick have provedthe vanishing of η h h i ( e D ) by a different method [25, Theorem 13.7].In light of this algebraicity result, we propose the following question. Question.
What values can delocalized eta invariants take in general? Are they alwaysalgebraic numbers?
ELOCALIZED ETA, ALGEBRAICITY AND K -THEORY 3 In particular, if a delocalized eta invariant is transcendental, then it will lead to acounterexample to the Baum-Connes conjecture [4, 5, 9]. Note that the above ques-tion is a reminiscent of Atiyah’s question concerning rationality of ℓ -Betti numbers[1]. Atiyah’s question was answered in negative by Austin, who showed that ℓ -Bettinumbers can be transcendental [2].As another application of Theorem 1 .
1, we give a K -theoretic proof of a version ofthe delocalized Atiyah-Patodi-Singer index theorem. See Proposition 5 . τ h on K ( C ∗ L, ( f M ) π M ) for each non-identity conjugacy class h h i with polynomial growth. The definition of K ( C ∗ L, ( f M ) π M )is reviewed in Section 2 . h on K ( C ∗ r ( π M )), and our construction is inspired by the workof de la Harpe and Skandalis [11] and Keswani [17]. In fact, combined with finite prop-agation speed of wave operators, our K -theoretic approach above can also be used togive a uniform treatment of various vanishing results and homotopy invariance resultsfor delocalized eta variants in [34, 17, 18, 25, 15, 7]. See a brief discussion in Remark3 .
10 below. We will present the details in a separate paper [32].One can use the same techniques developed in this paper to show that the analoguesof Theorem 1 . . . π M has polynomial growth or π M is hyperbolic.The paper is organized as follows. In Section 2, we recall some basic definitionsof certain geometric C ∗ -algebras. Given a discrete group, for each conjugacy classwith polynomial growth, we review how to extend a trace on the group algebra tothe Connes-Moscovici smooth dense subalgebra of the corresponding reduced group C ∗ -algebra. In Section 3, we then use this extended trace map to define an explicitdeterminant map on secondary higher invariants. In Section 4, we establish a preciseconnection between Higson-Roe’s K -theoretic higher rho invariants and Lott’s delo-calized eta invariants. We then apply it in Section 5 to prove an algebraicity resultconcerning the values of delocalized eta invariants and a version of delocalized Atiyah-Patodi-Singer index theorem.We would like to thank the referees for helpful comments. ZHIZHANG XIE AND GUOLIANG YU Conjugacy classes with polynomial growth and extension of traces
Let Γ be a discrete group and C Γ the corresponding group algebra. For each h ∈ Γ,there is a natural trace map on C Γ defined as follows:tr h ( a ) = X g ∈h h i a g where h h i is the conjugacy class of h and a = P g ∈ Γ a g g ∈ C Γ. In this section, wegive a brief construction on how to extend this trace to a smooth dense subalgebraof the reduced group C ∗ -algebra C ∗ r (Γ), provided that h h i has polynomial growth. Inparticular, such a trace map induces a map on K ( C ∗ r (Γ)). We shall use this inducedmap on K ( C ∗ r (Γ)) to define a determinant map on secondary higher invariants in thenext section.2.1. Roe algebras and localization algebras.
In this subsection, we briefly recallsome standard definitions of certain geometric C ∗ -algebras. We refer the reader to[29, 39] for more details. Let X be a proper metric space. That is, every closedball in X is compact. An X -module is a separable Hilbert space equipped with a ∗ -representation of C ( X ), the algebra of all continuous functions on X which vanishat infinity. An X -module is called nondegenerate if the ∗ -representation of C ( X ) isnondegenerate. An X -module is said to be standard if no nonzero function in C ( X )acts as a compact operator. Definition 2.1.
Let H X be a X -module and T a bounded linear operator acting on H X .(i) The propagation of T is defined to be sup { d ( x, y ) | ( x, y ) ∈ supp( T ) } , wheresupp( T ) is the complement (in X × X ) of the set of points ( x, y ) ∈ X × X forwhich there exist f, g ∈ C ( X ) such that gT f = 0 and f ( x ) = 0, g ( y ) = 0;(ii) T is said to be locally compact if f T and T f are compact for all f ∈ C ( X );(iii) T is said to be pseudo-local if [ T, f ] is compact for all f ∈ C ( X ). Definition 2.2.
Let H X be a standard nondegenerate X -module and B ( H X ) the setof all bounded linear operators on H X .(i) The Roe algebra of X , denoted by C ∗ ( X ), is the C ∗ -algebra generated by alllocally compact operators with finite propagations in B ( H X ).(ii) D ∗ ( X ) is the C ∗ -algebra generated by all pseudo-local operators with finite prop-agations in B ( H X ). In particular, D ∗ ( X ) is a subalgebra of the multiplier algebraof C ∗ ( X ).(iii) C ∗ L ( X ) (resp. D ∗ L ( X )) is the C ∗ -algebra generated by all bounded and uniformlynorm-continuous functions f : [0 , ∞ ) → C ∗ ( X ) (resp. f : [0 , ∞ ) → D ∗ ( X )) suchthat propagation of f ( t ) →
0, as t → ∞ .Again D ∗ L ( X ) is a subalgebra of the multiplier algebra of C ∗ L ( X ).(iv) C ∗ L, ( X ) is the kernel of the evaluation mapev : C ∗ L ( X ) → C ∗ ( X ) , ev( f ) = f (0) . In particular, C ∗ L, ( X ) is an ideal of C ∗ L ( X ). Similarly, we define D ∗ L, ( X ) as thekernel of the evaluation map from D ∗ L ( X ) to D ∗ ( X ). ELOCALIZED ETA, ALGEBRAICITY AND K -THEORY 5 Now in addition we assume that a discrete group Γ acts properly and cocompactlyon X by isometries. Let H X be a X -module equipped with a covariant unitary repre-sentation of Γ. If we denote the representation of C ( X ) by ϕ and the representationof Γ by π , this means π ( γ )( ϕ ( f ) v ) = ϕ ( f γ )( π ( γ ) v ) , where f ∈ C ( X ), γ ∈ Γ, v ∈ H X and f γ ( x ) = f ( γ − x ). In this case, we call ( H X , Γ , ϕ )a covariant system. Definition 2.3 ([41]) . A covariant system ( H X , Γ , ϕ ) is called admissible if(1) the Γ-action on X is proper and cocompact;(2) H X is a nondegenerate standard X -module;(3) for each x ∈ X , the stabilizer group Γ x acts on H X regularly in the sense that theaction is isomorphic to the action of Γ x on l (Γ x ) ⊗ H for some infinite dimensionalHilbert space H . Here Γ x acts on l (Γ x ) by translations and acts on H trivially.We remark that for each locally compact metric space X with a proper and cocom-pact isometric action of Γ, there exists an admissible covariant system ( H X , Γ , ϕ ). Also,we point out that the condition (3) above is automatically satisfied if Γ acts freely on X . If no confusion arises, we will denote an admissible covariant system ( H X , Γ , ϕ ) by H X and call it an admissible ( X, Γ)-module.
Definition 2.4.
Let X be a locally compact metric space X with a proper and cocom-pact isometric action of Γ. If H X is an admissible ( X, Γ)-module, we denote by C [ X ] Γ the ∗ -algebra of all Γ-invariant locally compact operators with finite propagations in B ( H X ). We define C ∗ ( X ) Γ to be the completion of C [ X ] Γ in B ( H X ).Since the action of Γ on X is cocompact, we have C ∗ ( X ) Γ ∼ = C ∗ r (Γ) ⊗ K , where C ∗ r (Γ)is the reduced group C ∗ -algebra of Γ and K is the algebra of all compact operators, cf.[30, Lemma 5.14].Similarly, we can also define D ∗ ( X ) Γ , C ∗ L ( X ) Γ , D ∗ L ( X ) Γ , C ∗ L, ( X ) Γ , D ∗ L, ( X ) Γ , C ∗ L ( Y ; X ) Γ and C ∗ L, ( Y ; X ) Γ . Remark . Up to isomorphism, C ∗ ( X ) = C ∗ ( X, H X ) does not depend on the choiceof the standard nondegenerate X -module H X . The same holds for D ∗ ( X ), C ∗ L ( X ), D ∗ L ( X ), C ∗ L, ( X ), D ∗ L, ( X ), C ∗ L ( Y ; X ), C ∗ L, ( Y ; X ) and their Γ-equivariant versions. Remark . Note that we can also define maximal versions of all the C ∗ -algebrasabove. For example, we define the maximal Γ-invariant Roe algebra C ∗ max ( X ) Γ to bethe completion of C [ X ] Γ under the maximal norm: k a k max = sup φ (cid:8) k φ ( a ) k | φ : C [ X ] Γ → B ( H ′ ) a ∗ -representation (cid:9) . Extension of traces to smooth dense subalgebras.
In this subsection, wegive a brief construction on how to extend the trace tr h : C Γ → C to be defined on asmooth dense subalgebra of the reduced group C ∗ -algebra C ∗ r (Γ), provided that h h i haspolynomial growth. Also see [31] for an alternative approach and relevant applications.Let M be a closed oriented Riemannian manifold. Let R be the algebra of smoothingoperators on M . Fix a basis of L ( M ), then R can be identified with the algebra ofmatrices ( a ij ) i,j ∈ N such thatsup i,j i k j l | a ij | < ∞ for all k, l ∈ N . ZHIZHANG XIE AND GUOLIANG YU
Let us recall the following smooth dense subalgebra of C ∗ r (Γ) ⊗ K , due to Connesand Moscovici [10]. Let ∆, resp. D , be the unbounded operator in ℓ ( N ), resp. ℓ (Γ),defined by ∆( δ j ) = jδ j for j ∈ N , resp. Dg = | g | · g for g ∈ Γ . Consider the unbounded derivations ∂ = [ D, · ] of B ( ℓ (Γ)) and e ∂ = [ D ⊗ I, · ] of B ( ℓ (Γ) ⊗ ℓ ( N )), and set : B ( f M ) Γ = { A ∈ C ∗ r (Γ) ⊗ K | e ∂ k ( A ) ◦ ( I ⊗ ∆) is bounded ∀ k ∈ N } . It follows from [10, Lemma 6.4] (and its proof) that B ( f M ) Γ contains C Γ ⊗ R and isclosed under holomorphic functional calculus.For each n ∈ N , define the following seminorm on B ( f M ) Γ : k A k n = n X k =0 k ! (cid:13)(cid:13) e ∂ k ( A ) ◦ ( I ⊗ ∆) (cid:13)(cid:13) op where k e ∂ k ( A ) ◦ ( I ⊗ ∆) (cid:13)(cid:13) op stands for the operator norm of e ∂ k ( A ) ◦ ( I ⊗ ∆) . Then B ( f M ) Γ is a Fr´echet algebra under this sequence of seminorms {k · k n : n ∈ N } .We associate to each element h ∈ Γ the following trace on C Γ ⊗ R :tr h ( γ ⊗ ω ) = ( trace( ω ) if γ ∈ h h i , γ / ∈ h h i , for γ ∈ Γ and ω ∈ R . Now suppose h ∈ Γ such that its conjugacy class h h i haspolynomial growth, that is, there exists C and d such that ♯ { g ∈ h h i : | g | ≤ n } ≤ C · n d . The following lemma shows that the trace tr h extends to a continuous trace on B ( f M ) Γ ,provided that h h i has polynomial growth. Lemma 2.7. If h h i has polynomial growth, then tr h extends to a continuous trace on B ( f M ) Γ .Proof. Our proof will follow closely the proof of [10, Lemma 6.4]. If A ∈ B ( f M ) Γ , A = ( a ij ) i,j ∈ N with a ij ∈ C ∗ r (Γ), we definetr h ( A ) = X j ∈ N X g ∈h h i a jj ( g ) . (2)We need to verify that the summation on the right side converges. Consider thefollowing inequality: X j ∈ N X g ∈h h i | a jj ( g ) | ≤ (cid:16) X j ∈ N X g ∈h h i j (1 + | g | ) k | a jj ( g ) | (cid:17) / (cid:16) X j ∈ N X g ∈h h i j − (1 + | g | ) − k (cid:17) / Since h h i has polynomial growth, the term P j ∈ N P g ∈h h i j − (1 + | g | ) − k converges bychoosing a sufficiently large k , for example, k > ( d + 1) / To be precise, the algebra B ( f M ) Γ defined here is slightly different from Connes-Moscovici’s algebra B in [10, Lemma 6.4]. Both of them are smooth dense subalgebras of C ∗ r (Γ) ⊗ K . In this paper, thealgebra B ( f M ) Γ works better for our purposes. ELOCALIZED ETA, ALGEBRAICITY AND K -THEORY 7 On the other hand, observe that (cid:16) e ∂ k ( A ) ◦ ( I ⊗ ∆) (cid:17) ( δ e ⊗ δ j ) = j X i ∈ N e ∂ k a ij ( δ e ) ⊗ δ i = j X ( g,i ) ∈ Γ × N | g | k a ij ( g ) δ g ⊗ δ i , which implies X ( g,i ) ∈ Γ × N j | g | k | a ij ( g ) | ≤ j − (cid:13)(cid:13) e ∂ k ( A ) ◦ ( I ⊗ ∆) (cid:13)(cid:13) op . It follows that there exists a fixed constant C k > X j ∈ N X g ∈h h i j (1 + | g | ) k | a jj ( g ) | ≤ C k · k A k k . Therefore, tr h extends to a continuous linear map on B ( f M ) Γ .Now let us verify that tr h is a trace on B ( f M ) Γ , that is, tr h ( AB ) = tr h ( BA ) for A, B ∈ B ( f M ) Γ . First, assume that A, B ∈ C Γ ⊗ R . In this case, a straightforwardcalculation shows that tr h ( AB ) = tr h ( BA ). Now the general case follows, since tr h iscontinuous and C Γ ⊗ R is dense in B ( f M ) Γ . This finishes the proof. (cid:3) Since B ( f M ) Γ is a dense subalgebra of C ∗ ( f M ) Γ ∼ = C ∗ r (Γ) ⊗ K and is closed underholomorphic functional calculus, we see that the trace tr h induces a homomorphism:tr h : K ( B ( f M ) Γ ) = K ( C ∗ r (Γ) ⊗ K ) → C . Secondary higher invariants and determinant maps
In this section, we will use the trace maps from the previous section to constructcertain determinant maps on secondary higher invariants. More precisely, for each h = e ∈ Γ and tr h : K ( B ( f M ) Γ ) = K ( C ∗ r (Γ) ⊗ K ) → C , we will construct a linear map τ h : K ( C ∗ L, ( f M ) Γ ) → C . Let us first introduce somenotation. Definition 3.1.
We define B L ( f M ) Γ to be the dense subalgebra of C ∗ L ( f M ) Γ consistingof elements f ∈ C ∗ L ( f M ) Γ such that f ( t ) ∈ B ( f M ) Γ for all t ∈ [0 , ∞ ) and f is piecewisesmooth with respect to the Fr´echet topology of B ( f M ) Γ . B L ( f M ) Γ is a dense subalgebra of C ∗ L ( f M ) Γ and is closed under holomorphic functionalcalculus. Similarly, we define B L, ( f M ) Γ to be the kernel of the evaluation mapev : B L ( f M ) Γ → B ( f M ) Γ defined by f f (0) . The following lemma is an immediate consequence of the above definitions.
Lemma 3.2.
The inclusion maps B L ( f M ) Γ ֒ → C ∗ L ( f M ) Γ and B L, ( f M ) Γ ֒ → C ∗ L, ( f M ) Γ induce natural isomorphisms: K j ( B L ( f M ) Γ ) ∼ = K j ( C ∗ L ( f M ) Γ ) and K j ( B L, ( f M ) Γ ) ∼ = K j ( C ∗ L, ( f M ) Γ ) . ZHIZHANG XIE AND GUOLIANG YU
Now for each non-identity element h ∈ Γ such that the conjugacy class h h i haspolynomial growth, we shall construct a determinant map τ h : K ( B L, ( f M ) Γ ) → C , which can be equivalently viewed as a map τ h : K ( C ∗ L, ( f M ) Γ ) → C . Roughly speaking,the explicit formula for τ h is given by τ h ( u ) := 12 πi Z ∞ tr h (cid:0) ˙ u ( t ) u − ( t ) (cid:1) dt (3)for each [ u ] ∈ K ( B L, ( f M ) Γ ), where ˙ u is the derivative of u . In order to justify thevalidity of this integral, we need the following technical results. Definition 3.3.
Let SC ∗ ( f M ) Γ be the suspension of C ∗ ( f M ) Γ , and ϕ ∈ be an invertibleelement in SC ∗ ( f M ) Γ , that is, a loop ϕ : S = [0 , / { , } → ( C ∗ ( f M ) Γ ) + of invertibleelements such that ϕ (1) = 1, where ( C ∗ ( f M ) Γ ) + is the unitization C ∗ ( f M ) Γ . We say ϕ is local if it is the image of an invertible element ψ ∈ SC ∗ L ( f M ) Γ under the evaluationmap SC ∗ L ( f M ) Γ → SC ∗ ( f M ) Γ . Similarly, an invertible element ϕ ∈ S B ( f M ) Γ is calledlocal if it is the image of an invertible element ψ ∈ S B L ( f M ) Γ under the evaluationmap.Local loops of invertible elements have the following property. Lemma 3.4. If ϕ is a local invertible element in SC ∗ ( f M ) Γ ( resp. S B ( f M ) Γ ) , thenfor ∀ ε > , there exists an idempotent p in C ∗ ( f M ) Γ ( resp. B ( f M ) Γ ) such that thepropagation of p is ≤ ε and ϕ is equivalent to the invertible element e πiθ p + (1 − p ) in SC ∗ ( f M ) Γ ( resp. S B ( f M ) Γ ) .Proof. By the Bott periodicity map β : K ( C ∗ L ( f M ) Γ ) ∼ = −−→ K ( SC ∗ L ( f M ) Γ ) , P e πiθ P + (1 − P ) , every invertible element in SC ∗ L ( e X ) Γ is equivalent to an invertible element of the form e πiθ P + (1 − P ). It follows from the Baum-Douglas geometric description of K -homology [6, Section 11] that P can be chosen to be a family of idempotents such thatthe propagation of P ( t ) goes to zero as t goes to infinity. Indeed, since P representsa K -homology class, it can be chosen to be the local index (cf. [39, Section 3]) of atwisted Dirac operator over a spin c manifold. The standard construction of K -theoretic(local) index classes shows that the propagation of the idempotent P ( t ) can be madefinite and goes to zero as t goes to infinity.Since ϕ is the image of an invertible element in SC ∗ L ( e X ) Γ under the evaluation map,it follows immediately that for ∀ ε >
0, there exists an idempotent p ∈ C ∗ ( f M ) Γ suchthat the propagation of p is ≤ ε and ϕ is homotopic to the loop e πiθ p + (1 − p ) througha family of loops of invertible elements. It is important that we use idempotents instead of projections here. In general, the constructionof K -theoretic index classes does not produce a projection (an idempotent that is self-adjoint) withfinite propagation, but it does produce an idempotent with arbitrary small propagation. On the otherhand, if one insists on having both self-adjointness and finite propagation, one possibility is to usequasi-projections, cf. [40]. ELOCALIZED ETA, ALGEBRAICITY AND K -THEORY 9 By applying Lemma 3 .
2, the case of a local invertible element in S B ( f M ) Γ alsofollows. (cid:3) In order to rigorously define the determinant map τ h : K ( B L, ( f M ) Γ ) → C , we shallprove that every element of K ( B L, ( f M ) Γ ) has a nice representative with certain reg-ularities. The main motivation for choosing such nice representatives is to guaranteethe convergence of the integral in line (3). Moreover, we show that for a given elementof K ( B L, ( f M ) Γ ), two different such regularized representatives can be connected bya family of representatives of the same kind. This allows us to show that the integralin line (3) is independent of the choice of such representatives. Proposition 3.5.
Every element [ u ] ∈ K ( B L, ( f M ) Γ ) has a representative w : [0 , ∞ ) → ( B ( f M ) Γ ) + such that w ( t ) = u ( t ) if 0 ≤ t ≤ h ( t ) if 1 ≤ t ≤ e πi F ( t − if t ≥ where h is a piecewise smooth path of invertible elements connecting u (1) and e πi F (1)+12 ,and F is a piecewise smooth map F : [1 , ∞ ) → D ∗ ( f M ) Γ satisfying (1) F ( t ) − ∈ B ( f M ) Γ and F ∗ ( t ) = F ( t ) , (2) its derivative F ′ ( t ) ∈ B ( f M ) Γ , (3) and propagation of F ( t ) goes to , as t → ∞ .Moreover, if v is another such representative, then there exists a piecewise smoothfamily of invertibles u s ∈ B L, ( f M ) Γ and piecewise smooth maps F s : [1 , ∞ ) → D ∗ ( f M ) Γ satisfying conditions (1) , (2) and (3) above, with s ∈ [0 , , such that ( i ) u = w , ( ii ) u s ( t ) = e πi Fs ( t − for all t ≥ iii ) u v − ( t ) = 1 for all t / ∈ (1 , and u v − : [1 , → B ( f M ) Γ is a local loop ofinvertible elements.Remark . For simplicity, we shall call a representative as in the proposition a regu-larized representative.
Proof.
View the invertible element u ∈ B L, ( f M ) Γ as an invertible element in B L ( f M ) Γ .Consider the element ˆ u = u : [1 , ∞ ) → ( B ( f M ) Γ ) + in K ( B L ( f M ) Γ ). Since the K -theory of B L ( f M ) Γ is the K -homology of M , it follows from the Baum-Douglas geo-metric description of K -homology [6, Section 11] that ˆ u can be represented by a twistedDirac operator over a spin c manifold. In particular, it follows that there exists a piece-wise smooth map F : [1 , ∞ ) → D ∗ ( f M ) Γ satisfying(1) F ( t ) − ∈ B ( f M ) Γ and F ∗ ( t ) = F ( t ),(2) its derivative F ′ ( t ) ∈ B ( f M ) Γ , (3) propagation of F ( t ) goes to 0, as t → ∞ ; Note that ˆ u starts at t = 1 instead of t = 0. and ˆ u ( t ) is homotopic to the path e πi F ( t )+12 with t ∈ [1 , ∞ ) . In particular, there is apath of invertible elements, denoted by h , connecting u (1) and e πi F (1)+12 . Then u ishomotopic to the invertible element w defined by w ( t ) = u ( t ) if 0 ≤ t ≤ h ( t ) if 1 ≤ t ≤ e πi F ( t − if t ≥ u ( t ) h ( t ) e πi F ( t − h ( t ) u ( t ) u ( t ) Figure 1. homotopy between u and w .Now suppose v is another representative of [ u ] such that v ( t ) = u ( t ) if 0 ≤ t ≤ g ( t ) if 1 ≤ t ≤ e πi G ( t − if t ≥ g is a path of invertible elements connecting u (1) and e πi G (1)+12 , and G is apiecewise smooth map G : [1 , ∞ ) → D ∗ ( f M ) Γ satisfying that G ( t ) − ∈ B ( f M ) Γ and G ∗ ( t ) = G ( t ); its derivative G ′ ( t ) ∈ B ( f M ) Γ ; and propagation of G ( t ) goes to 0, as t → ∞ .By [16, Theorem 3.8], there exists a piecewise smooth family F s : [1 , ∞ ) → D ∗ ( f M ) Γ with s ∈ [0 ,
1] such that F = F and F = G ; F s ( t ) − ∈ B ( f M ) Γ and F ∗ s ( t ) = F s ( t );its derivative ∂∂t F s ( t ) ∈ B ( f M ) Γ ; and propagation of F s ( t ) goes to 0, as t → ∞ .Let ̟ : [0 , ∞ ) → ( B ( f M ) Γ ) + be the path of invertibles defined as ̟ ( t ) = u ( t ) if 0 ≤ t ≤ h ( t ) if 1 ≤ t ≤ e πi Fs (1)+12 if 2 ≤ t = s + 2 ≤ e πi G ( t − if t ≥ w is homotopic to ̟ . On the other hand, after a re-parametrization, it is notdifficult to see that ̟ differs from v by the loop f : [0 , → ( B ( f M ) Γ ) + with f ( t ) = g ( t ) − h (2 t ) if 0 ≤ t ≤ / g ( t ) − e πi F t − if 1 / ≤ t ≤ ELOCALIZED ETA, ALGEBRAICITY AND K -THEORY 11 Moreover, f is a local loop in the sense of Definition 3 . u ( t ) h ( t ) e πi F ( t − h ( t ) g ( t ) e πi Fs (1)+12 u ( t ) g ( t ) e πi G ( t − Figure 2. ̟ and v differs by a local loop. The picture should be viewedas 3-dimensional. The circular sectors form an element in S B L ( f M ) Γ ,which shows that the circular sector on the left is local in the sense ofthe Definition 3 . (cid:3) As before, suppose that h is a non-identity element in Γ such that its conjugacy class h h i has polynomial growth. Definition 3.7.
For each [ u ] ∈ K ( B L, ( f M ) Γ ), let w be a regularized representativeof u as in Proposition 3 .
5. We define τ h ( u ) := 12 πi Z ∞ tr h (cid:0) ˙ w ( t ) w − ( t ) (cid:1) dt. (4)where ˙ w is the derivative of w .Let us show that the above formula (4) gives a well-defined map τ h : K ( B L, ( f M ) Γ ) → C . In particular, we shall prove that the integral in the formula (4) converges and isindependent of the choice of regularized representative. Proposition 3.8.
If the conjugacy class h h i of a non-identity element h ∈ Γ haspolynomial growth, then the map τ h : K ( C ∗ L, ( f M ) Γ ) → C is well-defined.Proof. Let [ u ] ∈ K ( C ∗ L, ( f M ) Γ ) ∼ = K ( B L, ( f M ) Γ ). Let w be a regularized representativeof [ u ] as in Proposition 3 .
5. First, we shall show that the integral in the formula (4)converges. Indeed, we have12 πi Z ∞ tr h (cid:0) ˙ w ( t ) w − ( t ) (cid:1) dt = 12 πi Z tr h (cid:0) ˙ w ( t ) w − ( t ) (cid:1) dt + 12 πi Z ∞ tr h ( ˙ F ( t )) dt The first integral on the right is clearly well-defined. Observe that there exists ε > f M ) such that tr h ( ˙ F ( t )) = 0 as long as the propagation of ˙ F ( t ) is less than ε . Since the propagation of ˙ F ( t ) goes 0 as t goes to ∞ , it follows that thesecond integral on the right is well-defined.Now let us show that τ h ([ u ]) is independent of the choice of regularized representa-tives. suppose v is another regularized representative of [ u ]. By Proposition 3 .
5, thereexists a piecewise smooth family of invertibles u s ∈ B L, ( f M ) Γ ) + with the stated prop-erties ( i ) − ( iii ) as in Proposition 3 .
5. A key consequence of these properties is that itguarantees the convergence of each integral in the following transgression formula: ∂∂s τ h ( u s ) = − Z ∞ ∂ s tr h (cid:0) ( ∂ t u ) u − (cid:1) dt (5)= − Z ∞ tr h (cid:0) ( ∂ s ∂ t u ) u − (cid:1) dt + − Z ∞ tr h (cid:0) ( ∂ t u ) ∂ s ( u − ) (cid:1) dt = − Z ∞ tr h (cid:0) ( ∂ s ∂ t u ) u − (cid:1) dt − − Z ∞ tr h (cid:0) ( ∂ t u ) u − ( ∂ s u ) u − (cid:1) dt = − Z ∞ tr h (cid:0) ( ∂ s ∂ t u ) u − (cid:1) dt + − Z ∞ tr h (cid:0) ∂ t ( u − )( ∂ s u ) (cid:1) dt = − Z ∞ ∂ t tr h (cid:0) ( ∂ s u ) u − (cid:1) dt = tr h (cid:0) ( ∂ s F s )( n ) (cid:1) − tr h (cid:0) ( ∂ s u s )(0) u − s (0) (cid:1) for n sufficiently large= 0 , where − R stands for πi R . It follows that τ h ( w ) = τ h ( u ) = τ h ( u ). On the other hand, v and u differ by a local loop ϕ . By Lemma 3 .
4, a local loop ϕ : S → ( B ( f M ) Γ ) + ishomotopic to a loop e πiθ p + (1 − p ) with the propagation of the idempotent p beingsufficiently small. It follows that12 πi Z tr h ( ˙ ϕ ( θ ) ϕ − ( θ )) dθ = Z tr h ( p ) dθ = 0 , since the propagation of p is sufficiently small. Therefore, we have τ h ( v ) = τ h ( u ) = τ h ( w ). This finishes the proof. (cid:3) The determinant map τ h : K ( C ∗ L, ( f M ) Γ ) → C is related to the trace maptr h : K ( C ∗ r (Γ)) → C as follows. Lemma 3.9.
With the same notation as above, if the conjugacy class h h i of a non-identity element h ∈ Γ has polynomial growth, then the following diagram commutes: K ( C ∗ r (Γ)) − tr h (cid:15) (cid:15) ∂ / / K ( C ∗ L, ( f M ) Γ ) τ h (cid:15) (cid:15) C / / C where ∂ : K ( C ∗ r (Γ)) → K ( C ∗ L, ( f M ) Γ ) is the connecting map in the six-term K -theorylong exact sequence for the short exact sequence: → C ∗ L, ( f M ) Γ → C ∗ L ( f M ) Γ → C ∗ ( f M ) Γ → . ELOCALIZED ETA, ALGEBRAICITY AND K -THEORY 13 Proof.
For each [ p ] ∈ K ( C ∗ r (Γ)), recall that ∂ [ p ] is defined as follows: let { a ( t ) } t ∈ [0 , ∞ ) be a lift of p in B L ( f M ) Γ such that a ( t ) = 0 for all t ≥
1, in particular, a (0) = p , then ∂p := u with u ( t ) = e πia ( t ) for t ∈ [0 , ∞ ) . It follows that τ h ( ∂p ) = 12 πi Z ∞ tr h (cid:0) ˙ u ( t ) u − ( t ) (cid:1) dt = Z ∞ tr h ( ˙ a ( t )) dt = − tr h ( p ) . More precisely, by our construction of the map τ h , we need to choose a regularizedrepresentative of u as in Proposition 3 .
5. Since u ( t ) = 1 for all t ≥
1, a regularizedrepresentative of u can tautologically be chosen to be itself. (cid:3) Remark . The same method in this section can be also applied to the following(relative) traces defined on C ∗ r (Γ) or C ∗ max (Γ). Throughout this remark, we do notassume any growth conditions on the group Γ.(1) Let σ : Γ → U ( n ) and σ : Γ → U ( n ) be two unitary representations of Γ of thesame dimension. They induces traces tr σ i on C ∗ max (Γ) by γ tr( σ i ( γ )) . By using regularized representatives as in Proposition 3 .
5, the relative trace tr σ − tr σ induces a homomorphism τ σ ,σ : K ( C ∗ L, ( f M ) Γmax ) → C by τ σ ,σ ( u ) = 12 πi Z ∞ (tr σ − tr σ ) (cid:0) ˙ u ( t ) u − ( t ) (cid:1) dt. A key observation here is again that there exists ε > (tr σ − tr σ )( a ) = 0if the propagation of a ∈ C ∗ max ( f M ) Γ is less than ε . Combined with the finitepropagation speed of wave operators, this provides a conceptual approach to someresults of Keswani [17], Piazza and Schick [25] and Higson and Roe [15]. We shallpresent the details in a separate paper [32].(2) Let ν be the L -trace on the group von Neumann algebra N Γ of Γ. It induces atrace, still denoted by ν , on C ∗ max (Γ) by the natural map C ∗ max (Γ) → C ∗ r (Γ) → N Γ.Now suppose λ : C ∗ max (Γ) → C is the trivial representation. Then the formula ρ (2) ( u ) := 12 πi Z ∞ ( ν − λ ) (cid:0) ˙ u ( t ) u − ( t ) (cid:1) dt defines a homomorphism ρ (2) : K ( C ∗ L, ( f M ) Γmax ) → C , which is precisely the L - ρ -invariant of Cheeger and Gromov [8]. This provides a more conceptual approachto some results of Keswani [18] and Benameur and Roy [7]. Again, the details willbe given in [32]. For example, the local index of the Dirac operator on the empty set gives a constant path ofinvertible elements w ( t ) = 1. To be precise, since C ∗ max ( f M ) Γ ∼ = C ∗ max (Γ) ⊗ K , one needs to pass to an appropriate smoothdense subalgebra of C ∗ max (Γ) ⊗ K on which the traces tr σ and tr σ are defined. Such smooth densesubalgebras always exist. Higher rho invariants and delocalized eta invariants
In this section, we shall establish a precise connection between higher rho invariantsand delocalized eta invariants. More precisely, let M be an odd dimensional closedspin manifold equipped with a positive scalar curvature metric. Denote its fundamentalgroup π M by Γ. Suppose f M is the universal cover of M . We denote the Riemannnianmetric lifted to f M by ˜ g . Then the Dirac operator e D on f M with respect to ˜ g naturallydefines a higher rho invariant ρ ( e D, ˜ g ) ∈ K ( C ∗ L, ( f M ) Γ ). We shall show that if theconjugacy class h h i of a non-identity element h ∈ Γ has polynomial growth, then τ h ( ρ ( e D, ˜ g )) is equal to the delocalized eta invariant of Lott.Let us briefly recall the construction of ρ ( e D, ˜ g ) ∈ K ( C ∗ L, ( f M ) Γ ). Recall that e D = ∇ ∗ ∇ + κ , where ∇ : C ∞ ( f M , S ) → C ∞ ( f M , T ∗ f M ⊗ S ) is the connection on the spinor bundle S over f M , ∇ ∗ is the adjoint of ∇ , and κ is the scalar curvature of the metric ˜ g . Byassumption, κ > ε for some ε >
0, it follows immediately that e D is invertible in thiscase. We define F = e D | e D | − . Now for each n ∈ N , let { U n,j } be a Γ-invariant locally finite open cover of f M withdiameter( U n,j ) < /n and { φ n,j } a Γ-invariant partition of unity subordinate to { U n,j } .We define F ( t ) = X j (1 − ( t − n )) φ / n,j F φ / n,j + ( t − n ) φ / n +1 ,j F φ / n +1 ,j (6)for t ∈ [ n, n + 1]. Form the path of unitaries u ( t ) = e πi F ( t )+12 , ≤ t < ∞ . Note that F +12 is a genuine projection, hence u (0) = 1. So the path u ( t ) , ≤ t < ∞ , defines a class in K ( C ∗ L, ( M ) Γ ). Definition 4.1.
The higher rho invariant ρ ( e D, ˜ g ) is defined to be the K -theory class[ u ] ∈ K ( C ∗ L, ( M ) Γ ) . Now let us also recall the definition of delocalized eta invariants due to Lott.
Definition 4.2 ([23, Definition 7]) . With the above notation, the delocalized eta in-variant of e D at h h i is defined to be η h h i ( e D ) := 2 √ π Z ∞ tr h ( e De − t e D ) dt. (7)Here the convergence of the integral does not hold in general, and relies on the growthrate of the conjugacy class h h i , cf. [26, Section 3] for a more thorough discussion. Asufficient condition for the convergence of the integral is that the conjugacy class h h i has polynomial growth. The even dimensional case is completely parallel. For simplicity, we will only discuss the odddimensional case here. If n = 0, we choose the open cover to be { f M } consisting of a single open set f M itself. ELOCALIZED ETA, ALGEBRAICITY AND K -THEORY 15 We have the following main result of this section.
Theorem 4.3.
Let M be a closed odd-dimensional spin manifold equipped with a pos-itive scalar curvature metric g . Suppose f M is the universal cover of M , ˜ g is the Rie-mannnian metric on f M lifted from g , and e D the associated Dirac operator. Supposethe conjugacy class h h i of a non-identity element h ∈ π ( M ) has polynomial growth.Then we have τ h ( ρ ( e D, e g )) = − η h h i ( e D ) . Before we prove the theorem, let us point out that the definition of higher rhoinvariant ρ ( e D, ˜ g ) does not require any growth condition on h h i or π M . In fact, ifthe strong Novikov conjecture holds for Γ = π ( M ), then we can generalize Lott’sdelocalized eta invariant without any growth conditions of the conjugacy class of h .This can be achieved by using the Novikov rho invariant introduced in [36, Section 7].Let us briefly recall the construction below, and refer the reader to [36, Section 7] formore details. Consider the following commutative diagram: K Γ1 ( E Γ , f M ) / / Λ (cid:15) (cid:15) K Γ0 ( f M ) / / ∼ = (cid:15) (cid:15) K Γ0 ( E Γ) / / µ ∗ (cid:15) (cid:15) K Γ0 ( E Γ , f M ) Λ (cid:15) (cid:15) K ( C ∗ L, ( f M ) Γ ) / / K ( C ∗ L ( f M ) Γ ) / / K ( C ∗ r (Γ)) ∂ / / K ( C ∗ L, ( f M ) Γ ) (8)where E Γ is the universal space for proper Γ-actions and K Γ i ( E Γ , f M ) is the Γ-equivariantrelative K -homology group for the pair of Γ-spaces ( E Γ , f M ). Let us assume that µ ∗ : K Γ i ( E Γ) → K i ( C ∗ r (Γ)) is a split injection . In this case, let us denote the splittingmap by α : K ( C ∗ r (Γ)) → K Γ0 ( E Γ) which induces a direct sum decomposition: K ( C ∗ r (Γ)) ∼ = K Γ0 ( E Γ) ⊕ E . A routine diagram chase shows that(1) the homomorphism Λ : K Γ0 ( E Γ , f M ) → K ( C ∗ L, ( f M ) Γ ) is also an injection;(2) and ∂ ( E ) ∩ ∂ ( K Γ0 ( E Γ)) = 0.It follows that we have the following commutative diagram: K Γ0 ( f M ) / / ∼ = (cid:15) (cid:15) K Γ0 ( E Γ) / / µ ∗ (cid:15) (cid:15) K Γ0 ( E Γ , f M ) Λ (cid:15) (cid:15) / / K Γ1 ( f M ) ∼ = (cid:15) (cid:15) K ( C ∗ L ( f M ) Γ ) / / = (cid:15) (cid:15) K Γ0 ( E Γ) ⊕ E ∂ / / α (cid:15) (cid:15) K ( C ∗ L, ( f M ) Γ ) q (cid:15) (cid:15) / / K ( C ∗ L ( f M ) Γ ) = (cid:15) (cid:15) K ( C ∗ L ( f M ) Γ ) / / K Γ0 ( E Γ) ∂ / / K ( C ∗ L, ( f M ) Γ ) /∂ ( E ) / / K ( C ∗ L ( f M ) Γ ) (9)where q is the quotient map q : K ( C ∗ L, ( f M ) Γ ) → K ( C ∗ L, ( f M ) Γ ) /∂ ( E ) . So far, in all known cases where the strong Novikov conjecture holds, the split injectivity of theBaum-Connes assembly map is known to be true as well.
Note that the last row in diagram (9) is also a long exact sequence. By the five lemma,it follows that the composition q ◦ Λ : K Γ0 ( E Γ , f M ) ∼ = −−→ K ( C ∗ L, ( f M ) Γ ) /∂ ( E )is an isomorphism. Now we define β := ( q ◦ Λ) − ◦ q : K ( C ∗ L, ( f M ) Γ ) → K Γ0 ( E Γ , f M ) . Let E Γ be the universal space for free and proper Γ-actions. By Composing β withthe natural morphism K Γ0 ( E Γ , f M ) → K Γ0 ( E Γ , E Γ)induced by the inclusion ( E Γ , f M ) ֒ → ( E Γ , E Γ) and the Chern character map K Γ0 ( E Γ , E Γ) → M k ∈ Z H Γ2 k ( E Γ , E Γ) ⊗ C , we get a morphism Θ : K ( C ∗ L, ( f M ) Γ ) → M k ∈ Z H Γ2 k ( E Γ , E Γ) ⊗ C . Recall that (cf. [3], [5, Section 7]) M k ∈ Z H k ( E Γ , E Γ) ⊗ C ∼ = M h γ i γ finite order and γ = e M k ∈ Z H k ( Z γ ; C ) , where e is the identity element of Γ, h γ i runs through all conjugacy classes of finiteorder elements γ with γ = e , and Z γ is the centralizer group of γ in Γ. Definition 4.4.
If the Baum-Connes assembly map for π ( M ) is a split injection ,then we define the generalized delocalized eta invariant e D at h h i to be the complexnumber in the H ( Z h ; C )-component of ρ ( e D, ˜ g ) under the map Θ. In particular, if h has infinite order, then the generalized delocalized eta invariant e D at h h i is defined tobe zero.Note that if the conjugacy class h h i has polynomial growth and in addition the Baum-Connes assembly map is an isomorphism , then the above generalized delocalized etainvariant coincides with the delocalized eta invariant of Lott. Indeed, in this case, themap Θ : K ( C ∗ L, ( E Γ) Γ ) ⊗ C → M k ∈ Z H Γ2 k ( E Γ , E Γ) ⊗ C is an isomorphism. It follows from Lemma 3 . . .
3. In order to make the exposition moretransparent, we shall work with the following alternative smooth dense subalgebra of C ∗ ( f M ) Γ = C ∗ r (Γ) ⊗ K .Let S ( f M ) Γ be the convolution algebra of all elements A ∈ C ∞ ( f M × f M ) satisfying(1) A is Γ-invariant, that is, A ( gx, gy ) = A ( x, y ) for all g ∈ Γ, We remark that this definition of generalized delocalized eta invariants depends on the choice ofthe split injection in general.
ELOCALIZED ETA, ALGEBRAICITY AND K -THEORY 17 (2) A has finite propagation, that is, there exists R > A ( x, y ) = 0 for all x, y ∈ f M with d ( x, y ) ≥ R .The algebra S ( f M ) Γ acts on L ( f M ) by( Af )( x ) = Z f M A ( x, y ) f ( y ) dy, for A ∈ S ( f M ) Γ and f ∈ L ( f M ).Fix a point x ∈ f M and let σ : f M → R be the distance function σ ( x ) = d ( x, x ) on f M . In fact, we shall choose a smooth approximation σ of σ such that | σ ( x ) − σ ( x ) | < k dσ ( x ) k ≤ x ∈ f M . For notational simplicity, we shall continue to denotethis modified distance function by σ . Multiplication by the function σ acts as anunbounded operator on L ( f M ). Taking commutator with σ defines a derivation on S ( f M ) Γ : e ∂ = [ σ, · ] : S ( f M ) Γ → S ( f M ) Γ . Now let ∆ be the Laplace operator on f M and r an integer > dim M . We define A ( f M ) Γ = { A ∈ C ∗ ( f M ) Γ | e ∂ k ( A ) ◦ ( ∆ + 1) r is bounded for ∀ k ∈ N } . (10)The same proof from [10, Lemma 6.4] shows that A ( f M ) Γ contains S ( f M ) Γ and isclosed under holomorphic functional calculus.We associate to each element h ∈ Γ the following trace on S ( f M ) Γ :tr h ( A ) = X g ∈h h i Z F A ( x, gx ) dx where F is a fundamental domain of f M under the action of Γ. Here we have identified L ( f M ) with L ( F ) ⊗ ℓ (Γ) through the mapping f → ˆ f by the formula ˆ f ( x, α ) = f ( αx )for x ∈ F and α ∈ Γ. In particular, each element A ∈ S ( f M ) Γ becomes a finite sum P g ∈ Γ ( A g ) R g , where A g ( x, y ) = A ( x, gy ) for x, y ∈ F and R denotes the right regularrepresentation of Γ.The following lemma and its proof are essentially the same as Lemma 2 .
7. We shallbe brief.
Lemma 4.5. If h h i has polynomial growth, then tr h extends to a continuous trace on A ( f M ) Γ .Proof. Let A be an element in A ( f M ) Γ . By assumption, e ∂ k ( A ) ◦ ( ∆ + 1) r is boundedfor all k ∈ N . It follows from Sobolev embedding theorem that the Schwartz kernel e ∂ k ( A )( x, y ) of e ∂ k ( A ) is a uniformly bounded continuous function on f M × f M for each k ∈ N . We define tr h ( A ) = X g ∈h h i Z F A ( x, gx ) dx (11)We need to verify that the summation on the right side converges. Observe that e ∂ k ( A )( x, gx ) = (cid:0) σ ( x ) − σ ( gx ) (cid:1) k A ( x, gx ) , and furthermore | σ ( x ) − σ ( gx ) | ≥ | g | − diam( F ) for all x ∈ F , where diam( F ) is thediameter of F . It follows that there exists a fixed constant C k > | g | ) k Z F | A ( x, gx ) | dx ≤ C k Z F | e ∂ k ( A )( x, gx ) | dx. (12)Now for all k ∈ N , we have | tr h ( A ) | ≤ X g ∈h h i Z F | A ( x, gx ) | dx ≤ (cid:16) X g ∈h h i (1 + | g | ) k Z F | A ( x, gx ) | dx (cid:17) / (cid:16) X g ∈h h i (1 + | g | ) − k (cid:17) / ≤ C / k (cid:16) X g ∈h h i (1 + | g | ) − k Z F | e ∂ k ( A )( x, gx ) | dx (cid:17) / (cid:16) X g ∈h h i (1 + | g | ) − k (cid:17) / , which is finite for k sufficiently large, since h h i has polynomial growth and e ∂ k A ( x, gx )is uniformly bounded for all g . Now a similar argument as in Lemma 2 . h is a continuous map and tr h ( AB ) = tr h ( BA ) for A, B ∈ A ( f M ) Γ . (cid:3) Let S be the associated spinor bundle on f M and S ∗ its dual bundle. Consider thebundle End( S ) = p ∗ ( S ) ⊗ p ∗ ( S ∗ ) on f M × f M , where p i : f M × f M → f M is the projectiononto the first and second component respectively. There is a natural diagonal actionof Γ on End( S ). Define C ∞ ( f M × f M ,
End( S )) to be the set of all smooth sections ofthe bundle End( S ) over f M × f M . Let S ( f M , S ) Γ be the convolution algebra of all Γ-invariant finite propagation elements in C ∞ ( f M × f M ,
End( S )). The algebra S ( f M , S ) Γ acts on L ( f M , S ) by ( Af )( x ) = Z f M A ( x, y ) f ( y ) dy, for A ∈ S ( f M , S ) Γ and f ∈ L ( f M , S ), where L ( f M , S ) is the space of L -sections of S over f M .Recall that the Γ-equivariant Roe algebra C ∗ ( f M ) Γ is (up to isomorphism) indepen-dent of the choice of admissible ( f M ,
Γ)-modules. We shall still denote the Γ-equivariantRoe algebra obtained from the ( f M ,
Γ)-module L ( f M , S ) by C ∗ ( f M ) Γ . Similar to line(10), we define A ( f M , S ) Γ = { A ∈ C ∗ ( f M ) Γ | e ∂ k ( A ) ◦ ( e D n + 1) is bounded for ∀ k ∈ N } . where e D is the Dirac operator on f M and n is a fixed integer > dim M . As before, wecan similarly define the algebras A L ( f M , S ) Γ and A L, ( f M , S ) Γ as follows Definition 4.6.
We define A L ( f M , S ) Γ to be the dense subalgebra of C ∗ L ( f M ) Γ consistingof elements f ∈ C ∗ L ( f M ) Γ such that f is piecewise smooth and f ( t ) ∈ A ( f M , S ) Γ for all t ∈ [0 , ∞ ). Also, we define A L, ( f M , S ) Γ to be the kernel of the evaluation mapev : A L ( f M , S ) Γ → A ( f M , S ) Γ defined by f f (0) . Note that there exists a fixed constant Λ j such that the supremum norm of the continuousfunction e ∂ j ( A )( x, gy ) is ≤ Λ j · k e ∂ j ( A ) ◦ ( ∆ + 1) r k for all A ∈ A ( f M ) Γ . ELOCALIZED ETA, ALGEBRAICITY AND K -THEORY 19 Recall that a smooth function ϕ on R is called a Schwartz function if the function x k ϕ ( j ) ( x ) is bounded on R for each k, j ∈ N , where ϕ ( j ) is the j -th derivative of ϕ . Proposition 4.7.
Suppose ϕ is a Schwartz function, then ϕ ( e D ) is an element in A ( f M , S ) Γ .Proof. Recall that ϕ ( e D ) = 12 π Z ∞−∞ ˆ ϕ ( s ) e is e D ds, where ˆ ϕ is the Fourier transform of ϕ , which is also a Schwartz function, since ϕ is.Let us first verify that [ ϕ ( e D ) , σ ] is a bounded operator. Consider f ( s ) = e is e D Ae − is e D − A. Then f ′ ( s ) = e is e D i e DAe − is e D − e is e D A ( i e D ) e − isD = ie is e D [ e D, A ] e − is e D and f (0) = 0, which implies that f ( s ) = e is e D Ae − is e D − A = i Z s e it e D [ e D, A ] e − it e D dt, or equivalently, [ e is e D , A ] = i Z s e it e D [ e D, A ] e i ( s − t ) e D dt. It follows that [ ϕ ( e D ) , σ ] = i π Z ∞−∞ ˆ ϕ ( s ) Z s e it e D [ e D, σ ] e i ( s − t ) e D dtds. Note that k e it e D [ e D, σ ] e i ( s − t ) e D k ≤ (cid:13)(cid:13) [ e D, σ ] (cid:13)(cid:13) for all s, t ∈ R . We see that (cid:13)(cid:13)(cid:13)(cid:13)Z s e it e D [ e D, σ ] e i ( s − t ) e D dt (cid:13)(cid:13)(cid:13)(cid:13) ≤ s · (cid:13)(cid:13) [ e D, σ ] (cid:13)(cid:13) . This implies that [ ϕ ( e D ) , σ ] has finite operator norm, since the Fourier transform ˆ ϕ isa Schwartz function. Now a straightforward inductive argument shows that e ∂ k ( ϕ ( e D ))is a bounded operator for each k ∈ N .Now let us show that e ∂ k ( ϕ ( e D )) ◦ e D n is bounded. Note that[ σ, ϕ ( e D )] e D n = [ σ, ϕ ( e D ) e D n ] − ϕ ( e D )[ σ, e D n ] . By the same argument as above, the operator [ σ, ϕ ( e D ) e D n ] is bounded, since ϕ ( e D ) e D n = ψ ( e D ), where ψ ( x ) = ϕ ( x ) x n is a Schwartz function. Also, observe that ϕ ( e D )[ σ, e D n ] = n − X k =0 ϕ ( e D ) e D k [ σ, e D ] e D n − k − = n − X k =0 (cid:16) ϕ ( e D ) e D k ( e D n + 1) (cid:17) ( e D n + 1) − [ σ, e D ] e D n − k − . Note that ( e D n + 1) − [ σ, e D ] e D n − k − is bounded for all 0 ≤ k ≤ n −
1. It follows that ϕ ( e D )[ σ, e D n ] is bounded. This finishes the proof. (cid:3) Now let us prove Theorem 4 . Proof of Theorem . . Recall that a smooth normalizing function φ : R → [ − ,
1] is anodd function such that φ ( x ) > x >
0, and φ ( x ) → ± as x → ±∞ . Considerthe normalizing function ϕ ( x ) = 2 √ π Z x e − s ds. Define F ( t ) = ϕ ( t − e D ) for t ∈ (0 , ∞ ). Since the scalar curvature of e g is uniformlybounded below by a positive number, it follows that e D is invertible. In particular,there is a spectral gap near 0 in the spectrum of e D . This implies that F ( t ) convergesto sign( e D ) = e D | e D | − in operator norm, as t →
0. Define F (0) = sign( e D ). The path u ( t ) = e πi F ( t )+12 with t ∈ [0 , ∞ )defines an element in K ( C ∗ L, ( f M ) Γ ). Indeed, by construction, we have u (0) = 1.Moreover, we have ϕ ( e D ) = 12 π Z ∞−∞ ˆ ϕ ( s ) e is e D ds, where ˆ ϕ is the Fourier transform of ϕ . Since every smooth normalizing function canbe approximated (in supremum norm) by smooth normalizing functions whose distri-butional Fourier transform has compact support. It follows from functional calculusand finite propagation of the wave operator e is e D that the path u can be uniformlyapproximated by paths of invertible elements with finite propagation. Observe that,for each smooth normalizing function ψ , the operator ψ ( e D ) − ψ ( e D )+12 is a projection modulo locally compact operators. This implies that e πi ψ ( e D )+12 ≡ u defines an invertible element in ( C ∗ L, ( f M ) Γ ) + . Moreover, it is not difficult tosee that 1 − exp(2 πi ϕ +12 ) is a Schwartz function. By Proposition 4 .
7, it follows that u ( t ) ∈ ( A ( f M , S ) Γ ) + for each t ∈ [0 , ∞ ). To show that u is an invertible element in( A L, ( f M , S ) Γ ) + , it suffices to show that u − A ( f M , S ) Γ .For each N ∈ N , we define the algebra A N = { A ∈ C ∗ ( f M ) Γ | e ∂ k ( A ) ◦ ( e D n + 1) is bounded for ∀ k ≤ N } and a norm on A N by k A k A N = N X k =0 k ! (cid:13)(cid:13) e ∂ k ( A ) ◦ ( e D n + 1) (cid:13)(cid:13) op where k e ∂ k ( A ) ◦ ( e D n + 1) (cid:13)(cid:13) op stands for the operator norm of e ∂ k ( A ) ◦ ( e D n + 1). Bydefinition, we have A ( f M , S ) Γ = T N ∈ N A N , which becomes a Fr´echet algebra underthis sequence of norms {k · k A N : N ∈ N } .Since e ∂ is a derivation, it is straightforward to see that A N is closed under holo-morphic functional calculus. Therefore, it suffices to show that u − k · k A N for each N ∈ N . ELOCALIZED ETA, ALGEBRAICITY AND K -THEORY 21 Recall that e D is invertible. Let σ > e D at zero, that is,the spectrum of e D is disjoint from the interval ( − σ, σ ). Since A N is closed underholomorphic functional calculus, the spectral radius of e − e D in A N is e − σ . Now applythe spectral radius formula lim n →∞ (cid:13)(cid:13) ( e − e D ) n (cid:13)(cid:13) n A N = e − σ . It follows that there exists C > (cid:13)(cid:13) e − s e D (cid:13)(cid:13) A N C · e − sσ / for all sufficiently large s ≫
0. Therefore, there exists
C > k πi ˙ F ( t ) k A N = (cid:13)(cid:13) − i √ πt − e De − e D /t (cid:13)(cid:13) A N t − (cid:13)(cid:13) √ π e De − e D (cid:13)(cid:13) A N · (cid:13)(cid:13) e − (1 /t − e D (cid:13)(cid:13) A N Ct − e − σ /t , (13)where ˙ F ( t ) is the derivative of F ( t ) with respect to t . Note that we have u ( t ) − (cid:18) πi Z t ˙ F ( r ) dr (cid:19) − . It follows that, for sufficiently small t > k u ( t ) − k A N = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X n =1 n ! (cid:18)Z t πi ˙ F ( r ) dr (cid:19) n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) A N ∞ X n =1 n ! (cid:16) Ct − e − σ /t (cid:17) n = exp (cid:16) Ct − e − σ /t (cid:17) − , (14)where the last term goes to 0 as t →
0. This implies that u − k · k A N . Now take the derivatives of u −
1, and a similar argumentalso shows that the derivatives of u − k · k A N .Therefore, u − k · k A N . This proves that u is aninvertible element in ( A L, ( f M , S ) Γ ) + .Observe that 12 πi Z ∞ tr h ( ˙ u ( t ) u − ( t )) dt = − √ π Z ∞ tr h ( e De − t e D ) dt. It follows from the work of Lott [23, Section 4, Page 20] that the above integral con-verges absolutely.On the other hand, our definition of τ h ( ρ ( e D, e g )) is expressed in terms of regularizedrepresentatives of ρ ( e D, e g ) (cf. Proposition 3 . . χ be a smooth normalizingfunction such that the support of its distributional Fourier transform ˆ χ has compactsupport. If we denote G ( t ) = χ ( t − e D ), then the path w defined by w ( t ) = u ( t ) 0 t e πi (2 − t ) F (1)+( t − G (1)+12 t e πi G ( t − t > is a regularized representative of [ u ] in the sense of Proposition 3 .
5. Now a similarcalculation as the transgression formula (5) in Proposition 3 . πi Z ∞ tr h ( ˙ w t w − t ) dt = 12 πi Z ∞ tr h ( ˙ u t u − t ) dt. Here let us briefly comment on the convergence of various terms that appear in thetransgression formula. Observe that there exists ε > h ( a ) = 0 for a ∈ A ( f M , S ) Γ , as long as the propagation of a is less than ε . Now the convergence ofvarious terms that appear in the transgression formula follows from this observationand the work of Lott [23, Section 4, Page 20]. To summarize, we have proved that τ h ( ρ ( e D, e g )) = − η h h i ( e D ) . This finishes the proof. (cid:3) Baum-Connes conjecture and algebraicity of delocalized etainvariants
In this section, we prove an algebraicity result concerning the values of delocalizedeta invariants. We also give a K -theoretic proof of a version of delocalized Atiyah-Patodi-Singer index theorem. Definition 5.1.
Given a discrete group Γ, let Q Γ be the field extension of Q by thefollowing set of roots of unity: { e πi/n | there exists α ∈ Γ such that the order of α is n } . We have the following algebraicity result concerning the values of delocalized etainvariants.
Theorem 5.2.
Assume the same notation as in Theorem . . If the rational Baum-Connes conjecture holds for Γ , that is, the assembly map µ : K Γ i ( E Γ) → K i ( C ∗ r (Γ)) isa rational isomorphism, and the conjugacy class h h i of a non-identity element h ∈ Γ has polynomial growth, then η h h i ( e D ) is an element in Q Γ . If in addition h has infiniteorder, then η h h i ( e D ) = 0 .Proof. Consider the following long exact sequence: K ( C ∗ L, ( E Γ) Γ ) ⊗ Q / / K ( C ∗ L ( E Γ) Γ ) ⊗ Q µ / / K ( C ∗ r (Γ)) ⊗ Q ∂ (cid:15) (cid:15) K ( C ∗ r (Γ)) ⊗ Q O O K ( C ∗ L ( E Γ) Γ ) ⊗ Q o o K ( C ∗ L, ( E Γ) Γ ) ⊗ Q o o Recall that the morphism K i ( C ∗ L ( E Γ) Γ ) → K i ( C ∗ L ( E Γ) Γ ) induced by the natural inclu-sion from E Γ to E Γ is rationally injective (cf. [5, Section 7]). It follows that, if the ratio-nal Baum-Connes conjecture holds for Γ, that is, the assembly map µ : K i ( C ∗ L ( E Γ) Γ ) → K i ( C ∗ r (Γ)) is rationally isomorphic, then the map µ is injective and the map ∂ is sur-jective. In particular, rationally every element u ∈ K ( C ∗ L, ( E Γ) Γ ) is the image of some p ∈ K ( C ∗ r (Γ)) under the map ∂ . By Lemma 3 .
9, we have τ h ( u ) = − tr h ( p ) . ELOCALIZED ETA, ALGEBRAICITY AND K -THEORY 23 Moreover, the map τ h : K ( C ∗ L, ( f M ) Γ ) → C factors through K ( C ∗ L, ( E Γ) Γ ). Therefore,the image of the map τ h : K ( C ∗ L, ( f M ) Γ ) → C is (up to multiplication by rationalnumbers) equal to the image of the map tr h : K ( C ∗ r (Γ)) → C .By Theorem 4 .
3, it suffices to show that image of the map tr h : K ( C ∗ r (Γ)) → C iscontained in Q Γ . By using the Baum-Douglas model of K -homology, K ( C ∗ L ( E Γ) Γ ) isgenerated by ( M, E, ϕ ), where M is a complete spin c manifold equipped with a properand cocompact Γ-action, E is a Γ-equivariant bundle over M , and ϕ : M → E Γ is aΓ-equivariant map. In this case, the Baum-Connes assembly map takes (
M, E, ϕ ) toits higher index Ind Γ ( D E ), where D E is the associated Dirac operator on M twisted by E . Now by Corollary A. h (Ind Γ ( D E )) is an algebraicnumber in Q Γ , and if in addition h has infinite order, then tr h (Ind Γ ( D E )) = 0. Thisfinishes the proof. (cid:3) In light of the above theorem, we propose the following question.
Question 1.
What values can delocalized eta invariants take in general? Are theyalways algebraic numbers?
In particular, if a delocalized eta invariant is transcendental, then it would lead toa counterexample to the Baum-Connes conjecture. Note that the above question is areminiscent of Atiyah’s question concerning rationality of ℓ -Betti numbers [1]. Atiyah’squestion was answered in negative by Austin, who showed that ℓ -Betti numbers canbe transcendental [2].Now let us turn to a delocalized Atiyah-Patodi-Singer index theorem. Let W bea compact n -dimensional spin manifold with boundary ∂W . Suppose W is equippedwith a Riemannian metric g W which has product structure near ∂W and in additionhas positive scalar curvature on ∂W . Let f W be the universal covering of W and g f W the Riemannian metric on f W lifted from g W . Denote π ( W ) by Γ. With respect tothe metric g f W , the associated Dirac operator e D on f W naturally defines a higher index,denoted by Ind Γ ( e D, g f W ), in K n ( C ∗ ( f W ) Γ ) = K n ( C ∗ r (Γ)), cf. [37, Section 3]. Let ˜ g ∂ be the restriction of g f W on ∂ f W . As we have seen above, with respect to the metric˜ g ∂ , the associated Dirac operator e D ∂ on ∂ f W naturally defines a higher rho invariant ρ ( e D ∂ , ˜ g ∂ ) in K n − ( C ∗ L, ( ∂ f W ) Γ ). If no confusion is likely to arise, the image of ρ ( e D ∂ , ˜ g ∂ )in K n − ( C ∗ L, ( f W ) Γ ) under the natural morphism K n − ( C ∗ L, ( ∂ f W ) Γ ) → K n − ( C ∗ L, ( f W ) Γ )will still be denoted by ρ ( e D ∂ , ˜ g ∂ ).We denote by ∂ : K n ( C ∗ ( f W ) Γ ) → K n − ( C ∗ L, ( f W ) Γ ) the connecting map in the K-theory long exact sequence induced by the short exact sequence of C ∗ -algebras:0 → C ∗ L, ( f W ) Γ → C ∗ L ( f W ) Γ → C ∗ ( f W ) Γ → . Then we have ∂ (Ind Γ ( e D, g f W )) = ρ ( e D ∂ , ˜ g ∂ ) in K n − ( C ∗ L, ( f W ) Γ ) , cf. [27, Theorem 1.14], [37, Theorem A].We have the following version of the delocalized Atiyah-Patodi-Singer index theorem. Proposition 5.3.
Let W be a compact even-dimensional spin manifold with boundary ∂W . Suppose W is equipped with a Riemannian metric g W which has product structurenear ∂W and in addition has positive scalar curvature on ∂W . If the conjugacy class h h i of a non-identity element h ∈ Γ = π ( W ) has polynomial growth, then tr h (Ind Γ ( e D, g f W )) = η h h i ( e D ∂ )2 . Proof.
Since ∂ (Ind Γ ( e D, g f W )) = ρ ( e D ∂ , ˜ g ∂ ) in K ( C ∗ L, ( f W ) Γ ) , the statement follows immediately from Theorem 4 . . (cid:3) Remark . When Γ is virtually nilpotent, a similar result has been proved at thelevel of noncommutative de Rham homology by Leichtnam and Piazza [20, Theorem14.1].
Appendix A. L -Lefschetz fixed point theorem In this appendix, we shall briefly review a modified version of the L -Lefschetz fixedpoint theorem of B.-L. Wang and H. Wang for proper actions of discrete groups oncomplete manifolds [33, Theorem 5.10].Let X be a complete spin c manifold of dimension n , and E a Hermitian vectorbundle over X . Suppose a discrete group Γ acts properly and cocompactly on X through isometries. Moreover, assume this action lifts to actions on the associated spin c bundle S and the bundle E over X through isometric bundle morphisms.For each h ∈ Γ, let X h = { x ∈ X | h · x = x } the fixed point set of h . Denote thenormal bundle of X h by N . The bundle N admits a decomposition N = N ( π ) ⊕ M <θ<π N ( θ )where the differential d h of the map h acts on N ( π ) by multiplication by −
1, and foreach 0 < θ < π , N ( θ ) is a complex bundle in which d h acts by multiplication by e iθ .Since h is orientation preserving, the bundle N ( π ) is an oriented even-dimensional realbundle.The L -Lefschetz fixed point theorem will be expressed in terms of characteristicclasses as follows. We refer to [19, Chapter III, Section 14] for more details. If V is acomplex vector bundle with formal splitting V = ℓ ⊕ · · · ⊕ ℓ k into line bundles withthe corresponding first Chern class denoted by c ( ℓ j ) = x j , then for each 0 < θ < π ,we define ˆ A θ ( V ) = 2 − k k Y j =1 ( x j + iθ ) = k Y j =1 e ( x j + iθ ) e ( x j + iθ ) − θ = π , we define a characteristic class ˆ A π ( V ) for any oriented real 2 k -dimensionalbundle as follows. Let V = V ⊕ · · · ⊕ V k be a formal splitting into oriented 2-planebundles, and set x j = χ ( V j ) the Euler class of V j . We defineˆ A π ( V ) = 2 − k k Y j =1 ( x j + iπ ) = (2 i ) − k k Y j =1 x j / . ELOCALIZED ETA, ALGEBRAICITY AND K -THEORY 25 Let ℓ be the associated line bundle for the spin c -structure of X , and c = c ( ℓ ) itsfirst Chern class. Suppose d h acts on ℓ by multiplication by e iβ .Furthermore, X h consists of Let D E be the associated Dirac operator twisted by E on X and denote its higher index by Ind Γ ( D E ) ∈ K n ( C ∗ r (Γ)). We have the followingmodified version of L -Lefschetz fixed point theorem of B.-L. Wang and H. Wang [33,Theorem 5.10 & Theorem 6.1]. Theorem A.1.
With the same notation as above, if the conjugacy class h h i of h ∈ Γ has polynomial growth, then tr h (Ind Γ ( D E )) = Z F (cid:16) Y <θ ≤ π ˆ A θ ( N ( θ )) · ˆ A ( X h ) · e c iβ · ch( E ) (cid:17) . (15) Here tr h (Ind Γ ( D E )) stands for the evaluation of the linear map tr h : K ( C ∗ r (Γ)) → C on the higher index class Ind Γ ( D E ) , and F is a fundamental domain of X h under theaction of Z h , where Z h is the centralizer group of h in Γ . Although B.-L. Wang and H. Wang made the assumption that tr h extends to a tracemap on C ∗ r (Γ) in [33, Theorem 5.10], we point out that it suffices to have the lessrestrictive assumption that the conjugacy class h h i of h ∈ Γ has polynomial growth.Indeed, the higher index class Ind Γ ( D E ) can be represented by elements in terms of theheat kernel operator e − ( e D E ) (see for example [10, Page 356]). Such a representative liesin A ( e X, S ⊗ E ) Γ (see Proposition 4 . Remark
A.2 . The assumption that the conjugacy class h h i has polynomial growthis only used to guarantee that the trace map tr h : C Γ → C induces a linear maptr h : K ( C ∗ r (Γ)) → C . On the other hand, the L -Lefschetz fixed point theorem con-tinues to hold in complete generality, without any growth conditions on h h i . Indeed,observe that Ind Γ ( D E ) can be represented by elements with finite propagation. Nowthe local index formula can be calculated by using finite propagation speed methodssuch as those employed in [24, Theorem 3.4].As an immediate consequence of the theorem above, we have the following corollary. Definition A.3.
Let Q Γ be the field extension of Q by the following set of roots ofunity: { e πi/n | there exists α ∈ Γ such that the order of α is n } . Corollary A.4.
With the same notation as above, tr h (Ind Γ ( D E )) is an algebraic num-ber in Q Γ . If in addition h has infinite order, then tr h (Ind Γ ( D E )) = 0 .Proof. Suppose h has finite order n , then the possible values for θ and β that appearin the formula (15) are (2 kπ/n ) for 0 ≤ k < [ n/ θ/ β/ kπ/n ) for 0 ≤ k < [ n/ h has infinite order, then the fixed point set of h is empty, since the action of Γ on X isproper. It follows that tr h (Ind Γ ( D E )) = 0. (cid:3) References [1] M. F. Atiyah. Elliptic operators, discrete groups and von Neumann algebras. In
Colloque “Analyseet Topologie” en l’Honneur de Henri Cartan (Orsay, 1974) , pages 43–72. Ast´erisque, No. 32–33.Soc. Math. France, Paris, 1976. [2] Tim Austin. Rational group ring elements with kernels having irrational dimension.
Proc. Lond.Math. Soc. (3) , 107(6):1424–1448, 2013.[3] Paul Baum and Alain Connes. Chern character for discrete groups. In
A fˆete of topology , pages163–232. Academic Press, Boston, MA, 1988.[4] Paul Baum and Alain Connes. K -theory for discrete groups. In Operator algebras and applications,Vol. 1 , volume 135 of
London Math. Soc. Lecture Note Ser. , pages 1–20. Cambridge Univ. Press,Cambridge, 1988.[5] Paul Baum, Alain Connes, and Nigel Higson. Classifying space for proper actions and K -theory ofgroup C ∗ -algebras. In C ∗ -algebras: 1943–1993 (San Antonio, TX, 1993) , volume 167 of Contemp.Math. , pages 240–291. Amer. Math. Soc., Providence, RI, 1994.[6] Paul Baum and Ronald G. Douglas. K homology and index theory. In Operator algebras andapplications, Part I (Kingston, Ont., 1980) , volume 38 of
Proc. Sympos. Pure Math. , pages117–173. Amer. Math. Soc., Providence, R.I., 1982.[7] Moulay-Tahar Benameur and Indrava Roy. The Higson-Roe exact sequence and ℓ eta invariants. J. Funct. Anal. , 268(4):974–1031, 2015.[8] Jeff Cheeger and Mikhael Gromov. Bounds on the von Neumann dimension of L -cohomologyand the Gauss-Bonnet theorem for open manifolds. J. Differential Geom. , 21(1):1–34, 1985.[9] Alain Connes.
Noncommutative geometry . Academic Press Inc., San Diego, CA, 1994.[10] Alain Connes and Henri Moscovici. Cyclic cohomology, the Novikov conjecture and hyperbolicgroups.
Topology , 29(3):345–388, 1990.[11] P. de la Harpe and G. Skandalis. D´eterminant associ´e `a une trace sur une alg´ebre de Banach.
Ann. Inst. Fourier (Grenoble) , 34(1):241–260, 1984.[12] Nigel Higson and John Roe. Mapping surgery to analysis. I. Analytic signatures. K -Theory ,33(4):277–299, 2005.[13] Nigel Higson and John Roe. Mapping surgery to analysis. II. Geometric signatures. K -Theory ,33(4):301–324, 2005.[14] Nigel Higson and John Roe. Mapping surgery to analysis. III. Exact sequences. K -Theory ,33(4):325–346, 2005.[15] Nigel Higson and John Roe. K -homology, assembly and rigidity theorems for relative eta invari-ants. Pure Appl. Math. Q. , 6(2, Special Issue: In honor of Michael Atiyah and Isadore Singer):555–601, 2010.[16] Navin Keswani. Geometric K -homology and controlled paths. New York J. Math. , 5:53–81 (elec-tronic), 1999.[17] Navin Keswani. Relative eta-invariants and C ∗ -algebra K -theory. Topology , 39(5):957–983, 2000.[18] Navin Keswani. Von Neumann eta-invariants and C ∗ -algebra K -theory. J. London Math. Soc.(2) , 62(3):771–783, 2000.[19] H. Blaine Lawson, Jr. and Marie-Louise Michelsohn.
Spin geometry , volume 38 of
PrincetonMathematical Series . Princeton University Press, Princeton, NJ, 1989.[20] Eric Leichtnam and Paolo Piazza. The b -pseudodifferential calculus on Galois coverings and ahigher Atiyah-Patodi-Singer index theorem. M´em. Soc. Math. Fr. (N.S.) , (68):iv+121, 1997.[21] Eric Leichtnam and Paolo Piazza. On higher eta-invariants and metrics of positive scalar curva-ture. K -Theory , 24(4):341–359, 2001.[22] John Lott. Higher eta-invariants. K -Theory , 6(3):191–233, 1992.[23] John Lott. Delocalized L -invariants. J. Funct. Anal. , 169(1):1–31, 1999.[24] H. Moscovici and F.-B. Wu. Localization of topological Pontryagin classes via finite propagationspeed.
Geom. Funct. Anal. , 4(1):52–92, 1994.[25] Paolo Piazza and Thomas Schick. Bordism, rho-invariants and the Baum-Connes conjecture.
J.Noncommut. Geom. , 1(1):27–111, 2007.[26] Paolo Piazza and Thomas Schick. Groups with torsion, bordism and rho invariants.
Pacific J.Math. , 232(2):355–378, 2007.[27] Paolo Piazza and Thomas Schick. Rho-classes, index theory and Stolz’ positive scalar curvaturesequence.
J. Topol. , 7(4):965–1004, 2014.[28] Michael Puschnigg. New holomorphically closed subalgebras of C ∗ -algebras of hyperbolic groups. Geom. Funct. Anal. , 20(1):243–259, 2010.
ELOCALIZED ETA, ALGEBRAICITY AND K -THEORY 27 [29] John Roe. Coarse cohomology and index theory on complete Riemannian manifolds. Mem. Amer.Math. Soc. , 104(497):x+90, 1993.[30] John Roe.
Index theory, coarse geometry, and topology of manifolds , volume 90 of
CBMS RegionalConference Series in Mathematics . Published for the Conference Board of the MathematicalSciences, Washington, DC, 1996.[31] S¨uleyman Ka˘gan Samurka¸s. Bounds for the rank of the finite part of operator.
J. Noncommut.Geom. , 2017. arXiv:1705.07378v2.[32] Xiang Tang, Yi-Jun Yao, Zhizhang Xie, and Guoliang Yu. Higher rho invariants, wave operatorand rigidity theorems of rho invariants. under preparation, 2018.[33] Bai-Ling Wang and Hang Wang. Localized index and L -Lefschetz fixed-point formula for orb-ifolds. J. Differential Geom. , 102(2):285–349, 2016.[34] Shmuel Weinberger. Homotopy invariance of η -invariants. Proc. Nat. Acad. Sci. U.S.A. ,85(15):5362–5363, 1988.[35] Shmuel Weinberger. Higher ρ -invariants. In Tel Aviv Topology Conference: Rothenberg Festschrift(1998) , volume 231 of
Contemp. Math. , pages 315–320. Amer. Math. Soc., Providence, RI, 1999.[36] Shmuel Weinberger, Zhizhang Xie, and Guoliang Yu. Additivity of higher rho invariants andnonrigidity of topological manifolds. submitted, 2016.[37] Zhizhang Xie and Guoliang Yu. Positive scalar curvature, higher rho invariants and localizationalgebras.
Adv. Math. , 262:823–866, 2014.[38] Zhizhang Xie and Guoliang Yu. Higher rho invariants and the moduli space of positive scalarcurvature metrics.
Adv. Math. , 307:1046–1069, 2017.[39] Guoliang Yu. Localization algebras and the coarse Baum-Connes conjecture. K -Theory ,11(4):307–318, 1997.[40] Guoliang Yu. The Novikov conjecture for groups with finite asymptotic dimension. Ann. of Math.(2) , 147(2):325–355, 1998.[41] Guoliang Yu. A characterization of the image of the Baum-Connes map. In
Quanta of Maths ,volume 11 of
Clay Math. Proc. , pages 649–657. Amer. Math. Soc., Providence, RI, 2010.(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 ::