An equivariant Atiyah-Patodi-Singer index theorem for proper actions II: the K -theoretic index
aa r X i v : . [ m a t h . K T ] J un An equivariant Atiyah–Patodi–Singer indextheorem for proper actions II: the K -theoretic index Peter Hochs, ∗ Bai-Ling Wang † and Hang Wang ‡ June 16, 2020
Abstract
Consider a proper, isometric action by a unimodular locally com-pact group G on a Riemannian manifold M with boundary, such that M/G is compact. Then an equivariant Dirac-type operator D on M un-der a suitable boundary condition has an equivariant index index G ( D )in the K -theory of the reduced group C ∗ -algebra C ∗ r G of G . This isa common generalisation of the Baum–Connes analytic assembly mapand the (equivariant) Atiyah–Patodi–Singer index. In part I of thisseries, a numerical index index g ( D ) was defined for an element g ∈ G ,in terms of a parametrix of D and a trace associated to g . An Atiyah–Patodi–Singer type index formula was obtained for this index. In thispaper, we show that, under certain conditions, τ g (index G ( D )) = index g ( D ) , for a trace τ g defined by the orbital integral over the conjugacy class of g . This implies that the index theorem from part I yields informationabout the K -theoretic index index G ( D ). It also shows that index g ( D )is a homotopy-invariant quantity. Contents ∗ University of Adelaide, Radboud University, [email protected] † Australian National University, [email protected] ‡ East China Normal University, [email protected] Preliminaries and results 6 g -trace 15 S and S . . . . . . . . . . . . . . . . . . . . . 163.3 Properties of the g -trace . . . . . . . . . . . . . . . . . . . . . 173.4 G -integrable kernels . . . . . . . . . . . . . . . . . . . . . . . 183.5 Basic estimates for heat operators . . . . . . . . . . . . . . . 20 S and S are g -trace class 21 t . . . . . . . . . . . . . 224.2 Convergence of an integral for large t . . . . . . . . . . . . . . 234.3 Proof of Proposition 4.1 . . . . . . . . . . . . . . . . . . . . . 27 g TR . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.3 Two maps from Roe algebras to C ∗ r G ⊗ K . . . . . . . . . . . 335.4 Proof of Proposition 5.6 . . . . . . . . . . . . . . . . . . . . . 355.5 Proof of Proposition 5.1 . . . . . . . . . . . . . . . . . . . . . 385.6 Proof of Theorem 2.7 . . . . . . . . . . . . . . . . . . . . . . . 39 D N This paper is about a K -theoretic index defined for Dirac operators on man-ifolds with boundary, equivariant with respect to proper, cocompact actionsby locally compact groups. It is a companion paper to part I [21] of thisseries of two papers, in which numerical indices were defined for such opera-tors, and an index formula was proved for those indices. The main result inthis paper is Theorem 2.7, stating that, under certain conditions, numericalinvariants extracted from the K -theoretic index via orbital integral traces2qual the indices from [21]. In that way, the index formula from [21] appliesto K -theoretic index as well.Consider a unimodular, locally compact group G acting properly andisometrically on a Riemannian manifold M , with boundary N , such that M/G is compact. Let D be a G -equivariant Dirac-type operator on a G -equivariant, Z -graded Hermitian vector bundle E = E + ⊕ E − → M . Sup-pose that all structures have a product form near N . In particular, supposethat near N , the restriction of D to sections of E + equals σ (cid:16) − ∂∂u + D N (cid:17) , (1.1)where σ : E + | N → E − | N is an equivariant vector bundle isomorphism, u isthe coordinate in (0 ,
1] in a neighbourhood of N equivariantly isometric to N × (0 , D N is a Dirac operator on E + | N .We initially assume D N to be invertible, and later show how to weakenthis assumption to 0 being isolated in the spectrum of D N . If D N is invert-ible, then we use the construction of an indexindex G ( D ) ∈ K ( C ∗ r G ) (1.2)from [16], where C ∗ r G is the reduced group C ∗ -algebra of G . This indexwas defined in [16] in a more general setting, and applied to, for example,Callias-type operators and positive scalar curvature [17] and the quantisationcommutes with reduction problem [18].To extract relevant numbers from this K -theoretic index, we apply tracesdefined by orbital integrals . Let g ∈ G , let Z g be its centraliser, and supposethat G/Z g has a G -invariant measure d ( hZ g ). Then the orbital integral withrespect to g of a function f ∈ C c ( G ) is the number τ g ( f ) := Z G/Z g f ( hgh − ) d ( hZ g ) . (1.3)If the integral on the right hand side converges absolutely for all f in adense subalgebra A ⊂ C ∗ r G , closed under holomorphic functional calculus,then this defines a trace τ g on A . That trace induces τ g : K ( C ∗ r G ) = K ( A ) → C . (1.4)Orbital integrals for semisimple Lie groups are fundamental to Harish-Chandra’s development of harmonic analysis on such groups. They also playan important role in Bismut’s work on hypo-elliptic Laplacians [6]. The map31.4) on K -theory is given by evaluating characters at g if G is compact. Onecan also use (1.4) to recover the values at elliptic elements g of charactersof discrete series representation of semisimple groups [22]. This was usedto link index theory to representation theory in [22]. Higher cyclic cocyclesgeneralising orbital integrals and capturing all information about classes in K ∗ ( C ∗ r G ) were developed by Song and Tang [35].For discrete groups, where they are sums over conjugacy classes, orbitalintegrals and the map (1.4) have found various applications to geometry andtopology in recent years, see for example [24, 36, 37, 39].Applying (1.4) to (1.2) yields the number τ g (index G ( D )) , (1.5)which is the main object of interest in this paper. The index (1.2) and thenumber (1.5) generalise various earlier indices. • If N = ∅ , then (1.2) is the image of D under the Baum–Connes analyticassembly map [4], see Corollary 4.3 in [16]. That is the most natu-ral and widely-used generalisation of the classical equivariant index toproper, cocompact actions. It has been applied to various problemsin geometry and topology, such as questions about positive scalar cur-vature and the Novikov conjecture. In this context, the number (1.5)was shown to be relevant to representation theory, orbifold geometryand trace formulas [20, 22, 23, 36]. • If M and G are compact, then (1.2) becomes the equivariant APSindex used in [12], and (1.5) is the evaluation of that index at g . (SeeLemma 2.9 in [21].) If G is trivial, then this index reduces to the usualAPS index. • In the case where
M/G is a compact manifold with boundary, M is its universal cover, and G is its fundamental group, the number τ e (index G ( D )) is the index used by Ramachandran in [33], see Remark2.13. In this setting, the index (1.2) was introduced in Section 3 of[38]. Indices with values in K ∗ ( C ∗ r G ) in this setting were also definedin [25, 26, 27, 28], via operators on Hilbert C ∗ r G -modules and in [32]in terms of Roe algebras. We expect these to be special cases of (1.2),because they generalise the case of manifolds without boundary [30], aspecial case of the Baum–Connes assembly map; see also for exampleProposition 2.4 in [32]. 4hese special cases suggest that the index (1.2) and the number (1.5) arenatural objects to study. They generalise to the case where 0 is isolated inthe spectrum of D N , as discussed in Section 6.In [21], the notion of a g -Fredholm operator was introduced. Such op-erators have a numerical g -index , defined in terms of a parametrix of theoperator and a trace related to τ g . It was shown that for several classes ofgroups and actions, the Dirac operator D on the manifold with boundary M is g -Fredholm, and hence has a g -index, denoted by index g ( D ). An in-dex formula was proved for this index. In the case where D is a twistedSpin c -Dirac operator, this index formula takes the formindex g ( D ) = Z M g χ g ˆ A ( M g ) e c ( L | Mg ) / tr( ge − R V | Mg / πi )det(1 − ge − R N / πi ) / − η g ( D N ) . The first term in the right hand side is a direct generalisation of the righthand side of the Atiyah–Segal–Singer fixed point formula [2, 3, 5]. Thenumber η g ( D N ) is a delocalised η -invariant . These were first constructed byLott [28, 29].The main result in this paper, Theorem 2.7, states that, under certainconditions, index g ( D ) = τ g (index G ( D )) . This links the index index g ( D ) to K -theory, and allows us to apply the indexformula from [21] to the number (1.5). This generalises the index theoremsin [1, 12, 33], for example. Furthermore, homotopy invariance of index G implies homotopy invariance of index g in these cases. Outline of this paper
The index (1.2), and the Roe algebras needed to define it, are introduced inSection 2. There we also recall the definition of the index index g from [21],and state the main result, Theorem 2.7.We prepare for the proof of Theorem 2.7 in Section 3, by introducinga parametrix for the operator D , and discussing some properties of the g -trace and of heat kernels. Then we prove the two main steps in the proofof Theorem 2.7 in Sections 4 and 5, Propositions 4.1 and 5.1. Combiningthese with a last extra step, Proposition 5.12, we obtain a proof of Theorem2.7. In Section 6, we show how to weaken the assumption that the boundaryDirac operator D N in (1.1) is invertible, to the assumption that 0 is isolatedin its spectrum. 5 cknowledgements HW was supported by the Australian Research Council, through DiscoveryEarly Career Researcher Award DE160100525, by the Shanghai Rising-StarProgram grant 19QA1403200, and by NSFC grant 11801178. PH thanksEast China Normal University for funding a visit there in 2018.
For a proper, cocompact action by a general locally compact group G , themost widely-used equivariant index of equivariant elliptic operators is theBaum–Connes analytic assembly map [4]. (Here an action is called co-compact if its quotient is compact.) This is a generalisation of the usualequivariant index in the compact case, and takes values in K ∗ ( C ∗ r G ), the K -theory of the reduced group C ∗ -algebra of G . In [16], a generalisation ofthe assembly map was constructed and studied, which applies to possiblynon-cocompact actions, as long as the operator it is applied to is invertibleoutside a cocompact set in the appropriate sense. This index also generalisesthe Gromov–Lawson index [14], an equivariant index of Callias-type opera-tors [15], the (equivariant) APS index on manifolds with boundary [1, 11],and the index used by Ramachandran for manifolds with boundary [33].This index is an equivariant version of the localised coarse index of Roe [34],for actions by arbitrary locally compact groups. For actions by fundamentalgroups of manifolds on their universal covers, this index was constructed in[38].We briefly review the construction of the index in [16] in Subsection2.2, in the case we need here. This involves localised Roe algebras, whichwe discuss in Subsection 2.1. The index takes values in the K -theory ofthe reduced C ∗ -algebra of the group acting. Using traces on subalgebrasof this algebra defined by orbital integrals, defined in Subsection 2.3, weextract numbers from that index. The main result in this paper is Theorem2.7, which states that, under certain conditions, those numbers equal thenumbers for which an index formula was proved in [21]. Let (
X, d ) be a metric space in which all closed balls are compact. Let G be a locally compact, unimodular group acting properly and isometricallyon X . Let Z ⊂ X be a nonempty, closed, G -invariant subset such that Z/G is compact. Fix a G -invariant Borel measure on X for which every open6et has positive measure. Let E → X be a G -equivariant Hermitian vectorbundle.The Hilbert space L ( E ) of square-integrable sections of E has a naturalunitary representation of G , and an action by C ( X ) given by pointwisemultiplication of sections by functions. In this sense, it is a G -equivariant C ( X ) -module . We will not define the various types of such modules here,but always work with concrete examples. Apart from L ( E ), we will also usethe module L ( E ) ⊗ L ( G ), where G acts diagonally (acting on L ( G ) viathe left regular representation), and where C ( X ) acts on the factor L ( E )via pointwise multiplication. If X/G is compact, then L ( E ) ⊗ L ( G ) is an admissible equivariant C ( X )-module, under the non-essential assumptionthat either X/G or G/K , for a maximal compact subgroup
K < G , isinfinite. See Theorem 2.7 in [16]. This type of C ( X )-module is central tothe constructions in [16].We denote the algebra of G -equivariant bounded operators on a Hilbertspace H with a unitary representation of G by B ( H ) G . Definition 2.1.
Let T ∈ B ( L ( E ) ⊗ L ( G )). Then T is locally compact ifthe operators T f and f T are compact for all f ∈ C ( X ). The operator T has finite propagation if there is a number r > f , f ∈ C ( X )whose supports are further than r apart, we have f T f = 0. Finally, T is supported near Z if there is an r ′ > f ∈ C ( X ) whosesupport is further than r ′ away from Z , the operators T f and f T are zero.The localised equivariant Roe algebra of X is the closure in B ( L ( E ) ⊗ L ( G )) of the algebra of locally compact operators in B ( L ( E ) ⊗ L ( G )) G with finite propagation, supported near Z . It is denoted by C ∗ ( X ) G loc .The algebra C ∗ ( X ) G loc is independent of Z . And, assuming either Z/G or G/K is an infinite set, C ∗ ( X ) G loc ∼ = C ∗ r G ⊗ K , (2.1)where C ∗ r G is the reduced group C ∗ -algebra of G , and K is the algebra ofcompact operators on a separable, infinite-dimensional Hilbert space. See(5) in [16]. (If Z/G and
G/K are both finite, then (2.1) still holds with K replaced by a matrix algebra.) This equality implies that C ∗ ( X ) G loc is alsoindependent of E . (In fact, it is independent of the choice of a more generalkind of admissible module.) Remark . There is no reason a priori to assume that
Z/G is compact. Theresulting localised Roe algebra will then depend on Z . We always assumethat Z/G is compact, so that we have the isomorphism (2.1), and we canapply the traces of Subsection 2.3 to classes in the K -theory of C ∗ ( X ) G loc .7e will also use a version of the localised equivariant Roe algebra definedwith respect to the C ( X )-module L ( E ), instead of L ( E ) ⊗ L ( G ). Thisis defined exactly as in Definition 2.1, with L ( E ) ⊗ L ( G ) replaced by L ( E ) everywhere. The resulting algebra is denoted by C ∗ ( X ; L ( E )) G loc .This algebra is less canonical than C ∗ ( X ) G loc , and is not stably isomorphicto C ∗ r G in general. If X/G itself is compact, then we omit the subscript loc,since being supported near Z then becomes a vacuous condition. Suppose, from now on, that X = M is a complete Riemannian manifold,and E is a smooth, Z -graded, G -equivariant, Hermitian vector bundle. Let D be an elliptic, odd-graded, essentially self-adjoint, first order differentialoperator on E . Suppose that D ≥ c (2.2)on M \ Z , for a positive constant c . Let b ∈ C ( R ) be an odd functionsuch that b ( x ) = 1 for all x ≥ c . Lemma 2.3 in [34] states that b ( D ) − ∈ C ∗ ( X ; L ( E )) G loc . By Lemma 2.1 in [34], the operator b ( D ) lies inthe multiplier algebra M ( C ∗ ( X ; L ( E )) G loc ) of C ∗ ( X ; L ( E )) G loc . Hence therestriction of b ( D ) to even-graded sections defines a class[ b ( D )] ∈ K (cid:0) M ( C ∗ ( X ; L ( E )) G loc ) /C ∗ ( X ; L ( E )) G loc (cid:1) . Let ∂ : K (cid:0) M ( C ∗ ( X ; L ( E )) G loc ) /C ∗ ( X ; L ( E )) G loc (cid:1) → K ( C ∗ ( X ; L ( E )) G loc )be the boundary map in the six-term exact sequence associated to the ideal C ∗ ( X ; L ( E )) G loc of M ( C ∗ ( X ; L ( E )) G loc ). We setindex L ( E ) G ( D ) := ∂ [ b ( D )] ∈ K ( C ∗ ( X ; L ( E )) G loc ) . (2.3)To obtain an index in K ( C ∗ r G ), let χ ∈ C ∞ ( M ) be a cutoff function, inthe sense that it is nonnegative, its support has compact intersections withall G -orbits, and that for all m ∈ M , Z G χ ( gm ) dg = 1 . (2.4)The map j : L ( E ) → L ( E ) ⊗ L ( G ) , (2.5)8iven by ( j ( s ))( m, g ) = χ ( g − m ) s ( m ) , for s ∈ L ( E ), m ∈ M and g ∈ G , is a G -equivariant, isometric embedding.Let ⊕ C ∗ ( X ; L ( E )) G loc → C ∗ ( X ) G loc (2.6)be given by mapping operators on L ( E ) to operators on j ( L ( E )) by con-jugation with j , and extending them by zero on the orthogonal complementof j ( L ( E )). We denote the map on K -theory induced by ⊕ Definition 2.3.
The localised equivariant coarse index of D isindex G ( D ) := index L ( E ) G ( D ) ⊕ ∈ K ( C ∗ r G ) . Remark . In [16], the localised equivariant coarse index is defined slightlydifferently from Definition 2.3, but also in terms of j . The two definitionsagree by (13) in [16]. In that paper, a version for ungraded vector bundles,with values in odd K -theory, is also defined. An illustration of how (repre-sentation theoretic) information that may not be encoded by index L ( E ) G ( D )is recovered through the map ⊕ M/G iscompact, then it reduces to the analytic assembly map from the Baum–Connes conjecture [4]. See Section 3.5 in [16] for other special cases. In thispaper, we apply the index to manifolds with boundary, to generalise theAPS index and its generalisations in [1, 11, 33].
Fix an element g ∈ G . Let Z g < G be its centraliser. Suppose that G/Z g has a G -invariant measure d ( hZ g ) such that for all f ∈ C c ( G ), Z G f ( h ) dh = Z G/Z g Z Z g f ( hz ) dz d ( hZ g ) , for fixed Haar measures dh on G and dz on Z g . (This is the case, for example,if G is discrete, or if G is real semisimple and g is a semisimple element.)The orbital integral of a function f ∈ C c ( G ) is τ g ( f ) := Z G/Z g f ( hgh − ) d ( hZ g ) .
9e assume that there is a dense subalgebra
A ⊂ C ∗ r G , closed under holo-morphic functional calculus, such that τ g extends to a continuous linearfunctional on A . Then it defines a trace on A . Existence of A is a nontrivialquestion; positive answers for discrete and semisimple groups were given in[24] and [22], respectively. The trace τ g on A defines a map τ g : K ( C ∗ r G ) = K ( A ) → C . Consider the setting of Subsection 2.2. Then we have the number τ g (index G ( D )) . In part I [21], we used a trace related to τ g to define the notion of a g -Fredholm operator, and the g -index of such operators. We briefly recallthe definitions here.Let χ ∈ C ∞ ( M ) be a cutoff function for the action, as in (2.4). Considerthe bundle End( E ) := E ⊠ E ∗ → M × M. Definition 2.5.
A section κ ∈ Γ ∞ (End( E )) G is g -trace class if the integral Z G/Z g Z M χ ( hgh − m ) tr( hgh − κ ( hg − h − m, m )) dm d ( hZ g ) (2.7)converges absolutely. Then the value of this integral is the g -trace of κ ,denoted by Tr g ( κ ). If T is a bounded, G -equivariant operator on L ( E ),with a g -trace class Schwartz kernel κ , then we say that T is g -trace class,and define Tr g ( T ) := Tr g ( κ ). Definition 2.6.
Let D be a G -equivariant, elliptic differential operator on E , odd with respect to a Z -grading on E . Let D + be its restriction toeven-graded sections. Then D is g -Fredholm if D + has a parametrix R suchthat the operators S := 1 − RD + ; S := 1 − D + R ; (2.8)are g -trace class.The g -index of a g -Fredholm operator D is the numberindex g ( D ) := Tr g ( S ) − Tr g ( S ) , (2.9)with S and S as in (2.8).The g -index is independent of the parametrix R by Lemma 2.4 in [21].10 .4 Manifolds with boundary We now specialise to the case we are interested in in this paper. The settingis the same as in Subsection 2.2 in [21].Slightly changing notation from the previous subsections, we let M bea Riemannian manifold with boundary N . We still suppose that G actsproperly and isometrically on M , preserving N , such that M/G is compact.We assume that a G -invariant neighbourhood U of N is G -equivariantlyisometric to a product N × (0 , δ ], for a δ >
0. To simplify notation, weassume that δ = 1; the case for general δ is entirely analogous.As before, let E = E + ⊕ E − → M be a Z -graded G -equivariant, Her-mitian vector bundle. We assume that E is a Clifford module , in the sensethat there is a G -equivariant vector bundle homomorphism, the Clifford ac-tion, from the Clifford bundle of T M to the endomorphism bundle of E ,mapping odd-graded elements of the Clifford bundle to odd-graded endo-morphisms. We also assume that there is a G -equivariant isomorphism ofClifford modules E | U ∼ = E | N × (0 , D be a Dirac-type operator on E ; i.e. the principal symbol of D isgiven by the Clifford action. Let D + be the restriction of D to sections of E + . Suppose that D + | U = σ (cid:16) − ∂∂u + D N (cid:17) , (2.10)where σ : E + | N → E − | N is a G -equivariant vector bundle isomorphism, u isthe coordinate in the factor (0 ,
1] in U = N × (0 , D N is an (ungraded)Dirac-type operator on E + | N . We initially assume that D N is invertible , andshow how to remove this assumption in Section 6.Consider the cylinder C := N × [0 , ∞ ), equipped with the product of themertic on M restricted to N , and the Euclidean metric. Because the metric,group action, Clifford module and Dirac operator have a product form on U , all these structures extend to C . We denote the extension of D to C by D C . We form the complete manifoldˆ M := ( M ⊔ C ) / ∼ , where m ∼ ( n, u ) if m = ( n, u ) ∈ U = N × (0 , E → ˆ M and ˆ D be theextensions of E and D to ˆ M , respectively, obtained by gluing the relevantobjects on M and C together along U .Since D N is invertible, there is a c > D N ≥ c. (2.11)11ence also D C ≥ c . In other words, ˆ D ≥ c outside the cocompact set M ,so that Definition 2.3 applies to ˆ D . This gives us the localised coarse indexindex G ( ˆ D ) ∈ K ( C ∗ r G ) , (2.12)which is the main object of study in this paper. Our goal is to give atopological expression for the number τ g (index G ( ˆ D )).The index (2.12) and the number τ g (index G ( ˆ D )) simultaneously gen-eralise several widely-used indices, as mentioned in the introduction. Theindex generalises to a case where D N is not invertible, as discussed in Section6. We assume that two heat kernels associated to D satisfy standard Gaussiandecay properties. Let ˜ D be the extension of D to the double ˜ M of M . Let κ t be the smooth kernel of either e − t ˜ D , ˜ De − t ˜ D , e − tD C or D C e − tD C . weassume that for all t >
0, there are b , b , b > t ∈ (0 , t ]and all m, m ′ in the relevant manifold ˜ M or N × R , k κ t ( m, m ′ ) k ≤ b t − b e − b d ( m,m ′ ) /t , (2.13)where d is the Riemannian distance. Estimates of this type were proved inmany places. A classical result is the one by Chen–Li–Yau [9] for the scalarLaplacian. Note that any bounded geometry-type conditions are automati-cally satisfied in our setting, because N/G and ˜
M /G are compact.In the case where G = Γ is discrete and finitely generated, let l be a wordlength function on Γ with respect to a fixed, finite, symmetric, generatingset. Because Γ is finitely generated, there are C, k > n ∈ N , { γ ∈ Γ; l ( γ ) = n } ≤ Ce kn . (2.14)Fix m ∈ M . By the Svarc–Milnor lemma, there are a , a > γ ∈ Γ, d ( γm , m ) ≥ a l ( γ ) − a . (2.15)Let c be as in (2.11). Theorem 2.7.
Suppose that ˆ D is g -Fredholm, and that the heat kernel decay (2.13) holds for the operators mentioned. Suppose that an algebra A as inSubsection 2.3 exists. If either(a) G/Z g is compact; or b) G = Γ is discrete and finitely generated, and (2.14) holds for a k < a √ c ,then τ g (index G ( ˆ D )) = index g ( ˆ D ) . Conditions for ˆ D to be g -Fredholm were given in Theorem 2.11 andCorollaries 2.13, 2.16 and 2.18 in [21]. Remark . The growth condition on Γ in part (b) of Theorem 2.7 holds inparticular if Γ has slower than exponential growth. In general, the conditiondepends on D , Γ and the group action. The factor 2 / a √ c may be increased to any number smaller than 1. This can be achieved ifwe replace the factors 1 / / g ( ˆ D ).This follows by homotopy invariance of index G ( ˆ D ). Corollary 2.9.
In the setting of Theorem 2.7, the number index g ( ˆ D ) is ahomotopy-invariant property of ˆ D . Combining Theorem 2.7 with Corollaries 2.13 and 2.16 in [21], we obtainan index formula for τ g (index G ( ˆ D )). Corollary 2.10.
Let D be a twisted Spin c -Dirac operator. Suppose that theheat kernel decay (2.13) holds for the operators mentioned. Suppose thateither • g = e , or • G = Γ is discrete and finitely generated, (2.14) holds for a k < a √ c ,and ( g ) has polynomial growth.Then τ g (index G ( ˆ D )) = Z M g χ g ˆ A ( M g ) e c ( L | Mg ) / tr( ge − R V | Mg / πi )det(1 − ge − R N / πi ) / − η g ( D N ) . (2.16)Notation is as in [21]; the integrand on the right hand side is the Atiyah–Segal–Singer integrand [2, 3, 5] times a cutoff function χ g , and η g ( D N ) isthe delocalised η -invariant of D N , as in [28, 29] and Subsection 2.3 of [21].Theorem 2.7, combined with results from [8], also implies a version ofProposition 5.3 in [39] and Theorem 1.4 in [8] in the case of fundamentalgroups of compact manifolds with boundary acting on their universal covers.13 orollary 2.11. Suppose that X is a compact Riemannian Spin c -manifoldwith boundary, with a product structure near the boundary. Let M be theuniversal cover of X , and let N = ∂M as before. Let G = Γ = π ( X ) . Let D be the lift to M of a twisted Spin c -Dirac operator on X . Let g ∈ Γ bedifferent from the identity element. If the constant c such that D N ≥ c islarge enough, then τ g (index G ( ˆ D )) = − η g ( D N ) . Proof.
If the constant c is large enough, then the delocalised η -invariant η g ( D N ) converges by Theorem 1.1 in [8]. Furthermore, condition (b) inTheorem 2.7 also holds if c is large enough. Finally, index Γ ( ˆ D ) ∈ K ( C ∗ r Γ)may be replaced by an index in K ( l (Γ)); see for example Remark A.2in [39]. The trace τ g converges on l (Γ) without growth conditions on theconjugacy class of g . So Theorem 2.7 applies in this setting, and so doesthe index formula in Corollary 2.17 in [21]. The interior contribution nowequals zero, because a nontrivial group element has no fixed points becausethe action is free. Remark . The index theorem in [21] also applies to semisimple Liegroups. So it is a natural question if a version of Theorem 2.7 appliesin that setting. We expect the techniques needed to prove this (particularlyProposition 4.1) to be very different from the discrete case. We have notlooked into the details so far.
Remark . The case of Theorem 2.7 where g = e , combined with Lemma2.6 in [21], shows that τ e (index G ( ˆ D )) generalises the index used by Ra-machandran in [33], and that Corollary 2.10 generalises Ramachandran’sindex theorem for manifolds with boundary. Remark . Consider the setting of Corollary 2.11. Let D X be the twistedSpin c -Dirac operator on X that lifts to the operator D on M . As a conse-quence of Theorem 3.9 in [18], where reduced group C ∗ -algebras and Roealgebras are replaced by maximal ones (one can also use l (Γ)), we have X ( g ) τ g (index Γ ( ˆ D )) = index( D X ) , (2.17)where the sum runs over all conjugacy classes ( g ) in Γ, and the index onthe right hand side is the APS index of D X . Since Γ acts freely on M ,Corollaries 2.10 and 2.11 imply that the left hand side of (2.17) equals Z M χ e ˆ A ( ˜ M ) e c ( L ) / tr( ge − R ˜ V / πi ) − X ( g ) η g ( D N ) . D X . We conclude that η ( D Y ) = X ( g ) η g ( D N ) , where D Y is the Dirac operator on the boundary Y = N/ Γ of X corre-sponding to D N . In other words, the delocalised η -invariants of D N arerefinements of the η -invariant of D Y . This remains true in a case where D N is not invertible, but there is a large enough gap in the spectrum of D N around zero. See Section 6. (See (I.6) in [11] for the case where G is finite.)We expect Corollary 2.10, and its extension to non-invertible D N , torefine Farsi’s orbifold APS index theorem (Theorem 4.1 in [13]) in a similarway. g -trace We prepare for the proof of Theorem 2.7 by introducing a specific parametrixfor ˆ D , and discussing some properties of the g -trace and of heat operators.We will use these things in Section 4 to prove that for the parametrix chosen,the squares of the remainder terms S j as in (2.8) are g -trace class in thesetting of Theorem 2.7. We will use a parametrix of ˆ D introduced in Subsection 5.1 of [21]. Considerthe setting of Subsection 2.4. As before, let ˜ M be the double of M , and let˜ E = ˜ E + ⊕ ˜ E − and ˜ D be the extensions of E and D to ˜ M , respectively. Moreexplicitly, as on page 55 of [1], ˜ M is obtained from M by gluing together acopy of M and a copy of M with reversed orientation, while ˜ E is obtainedby gluing together a copy of E and a copy of E with reversed grading. Toglue these copies of E together along N , we use the isomorphism σ . Let ˜ D ± be the restrictions of ˜ D to the sections of ˜ E ± .Let ψ : (0 , ∞ ) → [0 ,
1] be a smooth function such that ψ equals 1on (0 , ε ) and 0 on (1 − ε, ∞ ), for some ε ∈ (0 , / ψ := 1 − ψ .Let ϕ , ϕ : (0 , ∞ ) → [0 ,
1] be smooth functions such that ϕ equals 1 on(0 , − ε/
2) and 0 on (1 , ∞ ), while ϕ equals 0 on (0 , ε/
4) and 1 on ( ε/ , ∞ ).Then ϕ j ψ j = ψ j for j = 1 ,
2, and ϕ ′ j and ψ j have disjoint supports.We pull back the functions ϕ j and ψ j to C along the projection onto(0 , ∞ ), and extend these functions smoothly to ˆ M by setting ψ and ϕ M \ U , and ψ and ϕ equal to 0 on M \ U . We denotethe resulting functions by the same symbols ψ j and ϕ j . (No confusion ispossible in what follows, because we will always use these symbols to denotethe functions on ˆ M .) We denote the derivatives of these functions in the(0 , ∞ ) directions by ϕ ′ j and ψ ′ j , respectively. These derivatives are onlydefined and used on N × (0 , ∞ ) ⊂ ˆ M .Fix t >
0, and consider the parametrix˜ Q := 1 − e − t ˜ D − ˜ D + ˜ D − ˜ D + ˜ D − (3.1)of ˜ D + . (The part without the last factor ˜ D − is formed via functional calcu-lus, by an application of the function x − e − tx x to ˜ D − ˜ D + ; this does notrequire invertibility of ˜ D − ˜ D + .)The extension of D C to the complete manifold N × R is essentially self-adjoint and positive. Hence its self-adjoint closure is invertible. Let Q C bethe restriction to sections of E − of the inverse of that closure. We define R := ϕ ˜ Qψ + ϕ Q C ψ . Note that the operator ˜ Q is well-defined on the supports of ϕ and ψ , andthat Q C is well-defined on the supports of ϕ and ψ . The following twooperators play key roles in this paper. S := 1 − R ˆ D + ; S := 1 − ˆ D + R. (3.2) S and S Consider the setting of Subsection 2.5. In addition to the parametrix R andthe remainder terms S and S , we will also use the remainders˜ S := 1 − ˜ Q ˜ D + = e − t ˜ D − ˜ D + ;˜ S := 1 − ˜ D + ˜ Q = e − t ˜ D + ˜ D − . (3.3)We recall Lemmas 5.1 and 5.2 from [21]. Lemma 3.1.
We have S = ϕ ˜ S ψ + ϕ ˜ Qσψ ′ + ϕ Q C σψ ′ ; S = ϕ ˜ S ψ − ϕ ′ σ ˜ Qψ − ϕ ′ σQ C ψ . (3.4)16 emma 3.2. The operators S and S have smooth kernels. Lemma 3.3.
The operators S and S lie in C ∗ ( ˆ M ; L ( ˆ E )) G loc .Proof. The operators S and S have smooth kernels by Lemma 3.2. Thisimplies that these operators are locally compact.The operator Q C equals b ( D C ), where b ∈ C ( R ) satisfies b ( x ) = 1 /x for all x ∈ spec( D N )
0. Hence, by Lemma 2.1 in [34], Q C is a norm-limit of a sequence ( Q C,j ) ∞ j =1 operators with finite propagation. Similarly,˜ Q is a norm-limit of operators with finite propagation. So S and S arenorm-limits of operators with finite propagation.Since ϕ ′ and ψ ′ are supported near M and Q C,j has finite propagation,the operators ϕ ′ σQ C,j ψ and ϕ Q C,j σψ ′ are supported near M . Hence ϕ ′ σQ C ψ and ϕ Q C σψ ′ are norm-limits of operators that are supportednear M . The other terms on the right hand sides of (3.4) are supportednear M because ϕ and ψ are. So S and S are norm-limits of operatorsthat are supported near M . g -trace We consider a general setting, where E → M is an equivariant, Hermitianvector bundle over a complete Riemannian metric with a proper, isometricaction by G . In Subsection 4.3, we return to the setting of Subsection 2.4.This trace property is Lemma 3.4 in [21]. Lemma 3.4.
Let S and T are G -equivariant operators on Γ ∞ ( E ) . Supposethat S has a distributional kernel supported on the diagonal, and T has asmooth kernel in Γ ∞ (End( E )) G . If ST and T S are g -trace class, then theyhave the same g -trace. Lemma 3.5.
A section κ ∈ Γ ∞ (End( E )) G is g -trace class if and only if theintegral Z G/Z g Z M χ ( m ) | tr( hgh − κ ( hg − h − m, m )) | dm d ( hZ g ) (3.5) converges.Proof. In (2.7), substituting m ′ = hgh − m , using G -invariance of κ and the17race property shows that (2.7) equals Z G/Z g Z M χ ( m ′ ) | tr( hgh − κ ( hg − h − m ′ , hg − h − m ′ )) | dm ′ d ( hZ g )= Z G/Z g Z M χ ( m ′ ) | tr( κ ( hg − h − m ′ , m ′ ) hgh − ) | dm ′ d ( hZ g )= Z G/Z g Z M χ ( m ′ ) | tr( hgh − κ ( hg − h − m ′ , m ′ )) | dm ′ d ( hZ g ) . Lemma 3.6.
Let κ ∈ Γ ∞ (End( E )) G be such that there exists a cocompactlysupported ϕ ∈ C ∞ ( M ) G such that either κ = ( ϕ ⊗ κ or κ = (1 ⊗ ϕ ) κ .Suppose that G/Z g is compact. Then κ is g -trace class.Proof. We prove the case where κ = ( ϕ ⊗ κ , the other case is analogous.The integral (3.5) then equals Z G/Z g Z M χ ( m ) ϕ ( m ) | tr( hgh − κ ( hg − h − m, m )) | dm d ( hZ g ) . Because
G/Z g and the support of χ ϕ are compact, this integral converges.In the setting of Lemma 3.6, if κ is well-defined, then it has the sameproperty as κ , so that it is also g -trace class. G -integrable kernels The composition of two g -trace class operators need not be g -trace class.The notion of G -integrability (or Γ-summability for discrete groups Γ) canbe used to prove that such compositions are g -trace class under certainconditions. Definition 3.7.
A section κ ∈ Γ ∞ (End( E )) G is G -integrable if for all ϕ, ψ ∈ C ∞ c ( M ), the integral Z G (cid:18)Z M × M ϕ ( m ) ψ ( m ′ ) k xκ ( x − m, m ′ ) k dm dm ′ (cid:19) / dx converges. 18 emma 3.8. Let κ, λ ∈ Γ ∞ (End( E )) G be G -integrable, and such that thereexist cocompactly supported ϕ, ψ ∈ C ∞ ( M ) G such that either κ = ( ϕ ⊗ κ and λ = ( ψ ⊗ λ or κ = (1 ⊗ ϕ ) κ and λ = (1 ⊗ ψ ) λ . Suppose thatthe composition κλ is a well-defined element of Γ ∞ (End( E )) G . Then theintegral Z G Z M χ ( m ) | tr( x ( κλ )( x − m, m )) | dm dx (3.6) converges.Proof. We prove the case where κ = ( ϕ ⊗ κ and λ = ( ψ ⊗ λ , the othercase is analogous. In this situation, the integral (3.6) equals Z G Z M χ ( m ) (cid:12)(cid:12)(cid:12)(cid:12)Z M ϕ ( m ) ψ ( m ′ ) tr( xκ ( x − m, m ′ ) λ ( m ′ , m )) dm ′ (cid:12)(cid:12)(cid:12)(cid:12) dm dx. Inserting a factor 1 = R G χ ( ym ′ ) dy and substituting m ′′ = ym ′ , we findthat this integral equals at most Z G Z M Z M Z G χ ( m ) χ ( ym ′ ) (cid:12)(cid:12) ϕ ( m ) ψ ( m ′ ) tr( xκ ( x − m, m ′ ) λ ( m ′ , m )) (cid:12)(cid:12) dy dm ′ dm dx = Z G Z M Z M Z G χ ( m ) χ ( m ′′ ) (cid:12)(cid:12) ϕ ( m ) ψ ( m ′′ ) tr( xκ ( x − m, y − m ′′ ) λ ( y − m ′′ , m )) (cid:12)(cid:12) dy dm ′′ dm dx. By Fubini’s theorem, the integral on the right converges if and only if Z G Z G Z M Z M χ ( m ) χ ( m ′′ ) (cid:12)(cid:12) ϕ ( m ) ψ ( m ′′ ) tr( xκ ( x − m, y − m ′′ ) λ ( y − m ′′ , m )) (cid:12)(cid:12) dm ′′ dm dx dy converges. It is enough to consider the case where χ , ϕ and ψ are nonnega-tive. Then the latter integral is at most equal to Z G Z G Z M Z M χ ( m ) ϕ ( m ) χ ( m ′′ ) ψ ( m ′′ ) k xκ ( x − m, y − m ′′ ) λ ( y − m ′′ , m )) k dm ′′ dm dx dy. Using G -invariance of κ , subtituting z = xy − for x and applying theCauchy–Schwartz inequality, we see that this integral equals Z M Z M χ ( m ) ϕ ( m ) χ ( m ′′ ) ψ ( m ′′ ) Z G Z G k xy − κ ( yx − m, m ′′ ) yλ ( y − m ′′ , m )) k dx dy dm ′′ dm ≤ Z G (cid:18)Z M × M χ ( m ) ϕ ( m ) χ ( m ′′ ) ψ ( m ′′ ) k zκ ( z − m, m ′′ ) k (cid:19) / dz · Z G (cid:18)Z M × M χ ( m ) ϕ ( m ) χ ( m ′′ ) ψ ( m ′′ ) k yλ ( y − m, m ′′ ) k (cid:19) / dy. (3.7)The right hand side converges by G -integrability of κ and λ .19f G = Γ is discrete, then we will call a G -integrable smooth kernelΓ -summable . Lemma 3.9.
Suppose G = Γ is discrete. Let κ, λ ∈ Γ ∞ (End( E )) Γ be Γ -summable, and such that there exist cocompactly supported ϕ, ψ ∈ C ∞ ( M ) Γ such that either κ = ( ϕ ⊗ κ and λ = ( ψ ⊗ λ or κ = (1 ⊗ ϕ ) κ and λ = (1 ⊗ ψ ) λ . Suppose that the composition κλ is a well-defined element of Γ ∞ (End( E )) Γ . Then κλ is γ -trace class for all γ ∈ Γ .Proof. By Lemma 3.8, X γ ′ ∈ Γ Z M χ ( m ) | tr( γ ′ ( κλ )( γ ′− m, m )) | dm converges. So the sum over the conjugacy class of γ also converges, whichis (3.5) in this case. Let D be a Dirac operator on E → M . Lemma 3.10.
Let f ∈ S ( R ) . Let r ≥ . Consider bounded endomorphisms Φ and Ψ of E whose supports are at least a distance r apart. Then k Φ f ( D )Ψ k B ( L ( E )) ≤ π k Φ kk Ψ k Z R \ [ − r,r ] | ˆ f ( ξ ) | dξ. Proof.
For D = √− ∆, with ∆ the scalar Laplacian and f even, this isProposition 1.1 in [7]. The arguments apply directly to D : the claim followsfrom the decomposition f ( D ) = 12 π Z R ˆ f ( λ ) e iλD dλ and the fact that e iλD has propagation at most | λ | . See Propositions 10.3.5and 10.3.1 in [19], respectively. Corollary 3.11.
In the setting of Lemma 3.10, for all t > , k Φ e − tD Ψ k B ( L ( E )) ≤ √ π k Φ kk Ψ k e − r t k Φ De − tD Ψ k B ( L ( E )) ≤ √ πt k Φ kk Ψ k e − r t . roof. Applying Lemma 3.10 with f ( x ) = e − tx , we obtain k Φ e − tD Ψ k B ( L ( E )) ≤ √ πt k Φ kk Ψ k Z ∞ r e − λ t dλ = 2 √ π k Φ kk Ψ k erfc (cid:18) r √ t (cid:19) . The first inequality now follows form the inequality erfc( x ) ≤ e − x for all x > f ( x ) = xe − tx . Then Lemma 3.10yields k Φ De − tD Ψ k B ( L ( E )) ≤ √ πt / k Φ kk Ψ k Z ∞ r λe − λ t dλ = 1 √ πt k Φ kk Ψ k e − r t . Lemma 3.12.
Suppose that M has bounded geometry, and that the kernelsof e − tD and e − tD D satisfy bounds of the type (2.13) . The operators e − tD ϕ and e − tD Dϕ are Hilbert–Schmidt operators for all t > and ϕ ∈ C ∞ c ( M ) .Proof. Let κ be the Schwartz kernel of either e − tD D or e − tD . The bound(2.13) means that κϕ can be bounded by a Gaussian function. Since M hasbounded geometry, volumes of balls in M are bounded by an exponentialfunction of their radii. This implies that a Gaussian function is square-integrable. S and S are g -trace class Let S and S be as in (3.2). Our main goal in this section is to prove thefollowing proposition. Proposition 4.1.
Under the conditions in Theorem 2.7, the operators S and S are g -trace class. In [21], it is shown that S and S are g -trace class in a general setting.An important subtlety is that this is true for the notion of g -trace classoperators in Definition 2.5, which is relatively weak. For example, it doesnot reduce to the usual notion of trace class operators if G is trivial, and itis not preserved by composition with bounded, or even other g -trace class21perators. For this reason, Proposition 4.1 does not follow directly from thefact that S and S are g -trace class, and the arguments in this section areneeded to prove it. t In this subsection and the next, we consider a general setting, where E → M is an equivariant, Hermitian vector bundle over a complete Riemannianmetric with a proper, isometric action by G .Let D be a Dirac operator on E , assuming a Clifford action is given.Let t >
0. In this subsection and the next, we suppose that the kernels of e − tD and e − tD D satisfy bounds of the type (2.13), for t ∈ (0 , t ].We will use some calculus. Lemma 4.2.
Let a, b > , and t ∈ (0 , b/a ] . Then Z t t − a e − b/t dt ≤ t min( t , t ) − a e − b/t . (4.1) Proof.
The function t t − a e − b/t is increasing on (0 , b/a ], hence on (0 , t ].So Z t t − a e − b/t dt ≤ t t − a e − b/t ≤ t − a e − b/t , (4.2)and a similar estimate holds for the integral from 0 to t if t ≤ t . If t ≥ t ,then Z t t t − a e − b/s ds ≤ ( t − t ) t − a e − b/t . (4.3)The claim (4.1) follows from a combination of (4.2) and (4.3). Lemma 4.3.
Let κ t be the Schwartz kernel of either e − tD or e − tD D . Let ϕ, ψ ∈ C ∞ ( M ) Γ have supports separated by a positive distance ε , and let ˜ ϕ, ˜ ψ ∈ C ∞ c ( M ) . Then the integral X γ ∈ Γ (cid:18)Z M × M ˜ ϕ ( m ) ϕ ( m ) ˜ ψ ( m ′ ) ψ ( m ′ ) Z t k γκ t ( γ − m, m ′ ) k dt dm dm ′ (cid:19) / (4.4) converges.Proof. For γ ∈ Γ, set r ( γ ) := d ( γ supp( ˜ ϕϕ ) , supp( ˜ ψψ )) . κ t implies that for all γ ∈ Γ and t ∈ (0 , t ], Z M × M ˜ ϕ ( m ) ϕ ( m ) ˜ ψ ( m ′ ) ψ ( m ′ ) k γκ t ( γ − m, m ′ ) k dm dm ′ = Z M × M ˜ ϕ ( γm ) ϕ ( γm ) ˜ ψ ( m ′ ) ψ ( m ′ ) k κ t ( m, m ′ ) k dm dm ′ ≤ b t − b e − b r ( γ ) /t k ˜ ϕϕ k L k ˜ ψψ k L . So (cid:18)Z M × M ˜ ϕ ( m ) ϕ ( m ) ˜ ψ ( m ′ ) ψ ( m ′ ) Z t k γκ t ( γ − m, m ′ ) k dt dm dm ′ (cid:19) / ≤ b k ˜ ϕϕ k / L k ˜ ψψ k / L (cid:18)Z t t − b e − b r ( γ ) /t (cid:19) / . The assumptions on ϕ and ψ imply that r ( γ ) ≥ ε for all γ ∈ Γ. Set t := b ε /b . Then by Lemma 4.2, (cid:18)Z t t − b e − b r ( γ ) /t (cid:19) / ≤ t / min( t , t ) − b e − b r ( γ ) /t . The Svarc–Milnor lemma and compactness of the supports of ˜ ϕ and ˜ ψ imply that there are a, b > γ ∈ Γ, r ( γ ) ≥ al ( γ ) − b ,where l denotes the word length with respect to a fixed, finite, symmetric,generating set. So there are α, β > γ ∈ Γ, e − b r ( γ ) /t ≤ e − b ( al ( γ ) − b ) /t ≤ αe − βl ( γ ) /t . The sum of the right hand side over γ ∈ Γ converges, because of (2.14). t We still consider a Dirac operator D , and now assume that D ≥ c > l be a word length function on Γ with respect to a fixed,finite, symmetric, generating set. Because Γ is finitely generated, there are C, k > n ∈ N . Let ϕ, ψ ∈ C ∞ c ( M ), and fix m ∈ supp( ψ ). Let a and a be as in (2.15). Proposition 4.4.
Suppose that M has bounded geometry. Suppose that (2.14) holds for a k < a √ c . Then for all t > , the expression X γ ∈ Γ (cid:18)Z M × M Z ∞ t ϕ ( m ) ψ ( m ′ ) k γe − sD D ( γ − m, m ′ ) k ds dm dm ′ (cid:19) / (4.5)23 onverges. By Lemma 3.12, the operators e − tD ϕ and e − tD Dϕ are Hilbert–Schmidtfor all t > Lemma 4.5.
For all ϕ ∈ C ∞ c ( M ) , and all t > , there exists an a > such that for all t > t , k e − tD ϕ k HS ≤ ae − ct ; k e − tD Dϕ k HS ≤ ae − ct . Proof.
For t >
0, let A t be either the operator e − tD or e − tD D . Then forall t > t >
0, and all s ∈ L ( E ), k A t ϕs k = k e − ( t − t ) D A t ϕs k = (cid:0) e − t − t ) D A t ϕs, A t ϕs (cid:1) ≤ e − c ( t − t ) k A t ϕs k . Let { e j } ∞ j =1 be an orthonormal basis of L ( E ). Then by the above estimate, k A t ϕ k = ∞ X j =1 k A t ϕe j k ≤ e − c ( t − t ) k A t ϕ k . Let ϕ, ψ ∈ C ∞ c ( M ), and suppose for simplicity that these functions takevalues in [0 , γ ∈ Γ, set r ( γ ) := d ( γ supp( ϕ ) , supp( ψ )) . (Here we note that r ( γ ) may be zero.) Fix γ ∈ Γ and t >
0. Let ζ ∈ C ∞ c ( M )be a function with values in [0 ,
1] such that d (supp( ψ ) , supp(1 − ζ )) ≥ r ( γ ) / d ( γ supp( ϕ ) , ζ ) ≥ r ( γ ) / . (4.6)Write ( γ · ϕ ) e − tD Dψ = A ( γ ) + B ( γ ) , where A ( γ ) := ( γ · ϕ ) e − tD / ζe − tD / Dψ ; B ( γ ) := ( γ · ϕ ) e − tD / (1 − ζ ) e − tD / Dψ. emma 4.6. The operator A ( γ ) is Hilbert–Schmidt, and there is a b > ,independent of γ , such that for all t ≥ t , k A ( γ ) k HS ≤ be − r ( γ ) / t − ct . Proof.
For all s ∈ L ( E ) and γ ∈ Γ, k A ( γ ) s k ≤ k ( γ · ϕ ) e − tD / ζ / k B ( L ( E )) k ζ / e − tD / Dψs k . By Corollary 3.11 and the second inequality in (4.6), k ( γ · ϕ ) e − tD / ζ / k B ( L ( E )) ≤ √ π e − r ( γ )218 t . So, if { e j } ∞ j =1 is an orthonormal basis of L ( E ), k A ( γ ) k ≤ π e − r ( γ )29 t ∞ X j =1 k ζ / e − tD / Dψe j k ≤ π e − r ( γ )29 t k e − tD / Dψ k . The claim now follows by Lemma 4.5.
Lemma 4.7.
The operator B ( γ ) is Hilbert–Schmidt, and there is a b > ,independent of γ , such that for all t ≥ t , k B ( γ ) k HS ≤ be − r ( γ ) / t − ct . Proof.
The operator B ( γ ) is Hilbert–Schmidt if and only its adjoint is, andthen these operators have the same Hilbert–Schmidt norm. Now B ( γ ) ∗ = ψe − tD / D (1 − ζ ) e − tD / ( γ · ϕ )= ψe − tD / (1 − ζ ) De − tD / ( γ · ϕ ) − ϕe − tD / c ( dζ ) e − tD / ( γ · ϕ ) . The distance between the supports of ϕ and 1 − ζ is at least r ( γ ) /
3. Thesupport of dζ lies inside the support of 1 − ζ , so the distance between thesupports of ϕ and dζ is at least r ( γ ) / k ψe − tD / (1 − ζ ) k ≤ √ π e − r ( γ )218 t ; k ψe − tD / c ( dζ ) k ≤ k dζ k ∞ √ π e − r ( γ )218 t . e − tD / ( γ · ϕ ) = γe − tD / ϕγ − and De − tD / ( γ · ϕ ) = γDe − tD / ϕγ − are independent of γ . So a similar argument to the proof of Lemma 4.6applies to show that there is a b > t ≥ t , k B ( γ ) k HS = k B ( γ ) ∗ k HS ≤ be − r ( γ ) / t − ct . Lemma 4.8.
Let
C, k, α , α , α , t > , and suppose that (2.14) holds forall n ∈ N . Suppose that k < α α . Then X γ ∈ Γ Z ∞ t e − α l ( γ ) − α s − α s ds (4.7) converges.Proof. The sum (4.7) equals ∞ X n =0 X γ ∈ Γ; l ( γ )= n Z ∞ t e − α n − α s − α s ds ≤ C ∞ X n =0 Z ∞ t e − α n − α s − α s + kn ds = Ce kα Z ∞ t e (cid:16) k α − α (cid:17) s ∞ X n =0 e − α s (cid:16) n − α − ks α (cid:17) ! ds. (4.8)(Because all terms and integrands are positive, convergence does not dependon the order of summation and integration.) Convergence of the right handside of (4.8) is equivalent to convergence of the double integral Z ∞ t e (cid:16) k α − α (cid:17) s (cid:18)Z ∞ e − α s (cid:16) x − α − ks α (cid:17) dx (cid:19) ds. (4.9)And for all s > Z ∞ e − α s (cid:16) x − α − ks α (cid:17) dx ≤ Z R e − α s (cid:16) x − α − ks α (cid:17) dx = r πsα . We find that a sufficient condition for the convergence of (4.9) is convergenceof Z ∞ t e (cid:16) k α − α (cid:17) s r πsα ds. This is equivalent to the condition k < α α .26 roof of Proposition 4.4. The integral (4.5) equals X γ ∈ Γ (cid:13)(cid:13)(cid:13)(cid:13)Z ∞ t ϕ ◦ γ ◦ e − sD D ◦ ψ ds (cid:13)(cid:13)(cid:13)(cid:13) HS ≤ X γ ∈ Γ Z ∞ t k ϕ ◦ γ ◦ e − sD D ◦ ψ k HS ds. (4.10)By Lemmas 4.6 and 4.7, there is a b > t ≥ t and all γ ∈ Γ, k ϕ ◦ γ ◦ e − sD D ◦ ψ k HS = k ( γ · ϕ ) ◦ e − sD D ◦ ψ k HS ≤ be − r ( γ ) / s − cs . The condition (2.15) and compactness of supp( ϕ ) and supp( ψ ) implythat there is a > γ ∈ Γ, r ( γ ) ≥ a l ( γ ) − a . So k ϕ ◦ γ ◦ e − sD D ◦ ψ k HS ≤ be − ( a l ( γ ) − a s − cs . So the right hand side of (4.10) is at most equal to b X γ ∈ Γ Z ∞ t e − ( a l ( γ ) − a s − cs ds. By Lemma 4.8, this converges if Γ satisfies (2.14) for some
C, k > k < a c . We return to the setting of Subsection 2.4, where M is a manifold withboundary N , on which G acts cocompactly, and ˆ M is obtained from M byattaching a cylinder N × [0 , ∞ ).The operators ˜ Q and Q C as in Subsection 3.1 do not have smooth kernels,but if ϕ, ψ ∈ C ∞ ( M ) have disjoint supports, then ϕ ˜ Qψ and ϕQ C ψ do. Lemma 4.9.
Suppose that the kernel of the operator ˜ De − t ˜ D satisfies abound of the type (2.13) . If ϕ, ψ ∈ C ∞ ( M ) Γ have supports separated by apostive distance, then ϕ ˜ Qψ is Γ -summable.Proof. We have ˜ Q = − Z t e − s ˜ D + ˜ D − ˜ D − ds. So the claim follows from Lemma 4.3.
Proposition 4.10.
Consider the setting of Theorem 2.7(b). If ϕ, ψ ∈ C ∞ ( M ) Γ have supports separated by a postive distance, then ϕQ C ψ is Γ -summable. roof. We have Q C = Z ∞ e − s ( D C ) + ( D C ) − ( D C ) − ds. The operator ϕ Z e − s ( D C ) + ( D C ) − ( D C ) − ds ψ is Γ-summable by Lemma 4.3, and the operator ϕ Z ∞ e − s ( D C ) + ( D C ) − ( D C ) − ds ψ is Γ-summable by Proposition 4.4. The coefficient that appears a in (2.15)and in the growth condition on Γ is independent of the choice of m ∈ supp( ψ ) by compactness of M/ Γ and Γ-invariance of the distance on M .Let the functions ϕ j and ψ j , and the operator S be as in Subsection3.1, and let and ˜ S be as in (3.3). Proposition 4.11.
Consider the setting of Theorem 2.7(b). The operator ( ϕ ˜ Q − ϕ Q C ) ψ ′ has a smooth kernel, and is Γ -summable.Proof. The operator S has a smooth kernel by Lemma 3.2, and ϕ ˜ S ψ hasa smooth kernel as well. Hence so does( ϕ ˜ Q − ϕ Q C ) ψ ′ = S − ϕ ˜ S ψ . As in the proof of Proposition 5.7 in [21],( ϕ ˜ Q − ϕ Q C ) ψ ′ = ( ϕ ˜ Q − ϕ Q ′ C ) ψ ′ − ϕ ( Q C − Q ′ C ) ψ ′ = ( ϕ ˜ Q − ϕ Q ′ C ) ψ ′ − ϕ e − D C, + D C, − Q C ψ ′ − Z t (cid:0) ϕ e − s ˜ D + ˜ D − − ϕ e − sD C, + D C, − (cid:1) D − ds σψ ′ − ϕ Z ∞ t e − sD C, − D C, + D C, − ds σψ ′ (4.11)The second term on the right hand side is Γ-summable by Proposition 4.4,in which it is not assumed that the functions ϕ and ψ have disjoint supports.Here we again use the fact that the coefficient a that appears in (2.15) andin the growth condition on Γ is independent of the choice of m ∈ supp( ψ )by compactness of M/ Γ and Γ-invariance of the distance on M .We now focus on the first term on the right hand side of (4.11). As inthe proof of Lemma 5.5 in [21], let ϕ ∈ C ∞ ( ˆ M ) be such that for j = 1 , ϕ ψ ′ j , and zero outside the support of 1 − ϕ j . Since(1 − ϕ ) and ψ ′ have supports separated by a positive distance, Lemma 4.3implies that Z t (1 − ϕ ) (cid:0) ϕ e − s ˜ D + ˜ D − − ϕ e − s ( D C, + D C, − ) (cid:1) D − σψ ′ ds is Γ-summable.Let ˜ ϕ, ˜ ψ ∈ C ∞ c ( M ). Then as in Lemma 5.4 in [21], for all m, m ′ ∈ M (cid:0) ˜ ϕϕ (cid:0) ϕ e − s ˜ D + ˜ D − − ϕ e − s ( D C, + D C, − ) (cid:1) D − σψ ′ ˜ ψ (cid:1) ( m, m ′ ) = 1(2 πs ) dim( M ) / e − d ( m,m ′ ) / s F ( s, m, m ′ ) , where F ( s, m, m ′ ) vanishes to all orders in s as s ↓
0, uniformly in m, m ′ incompact sets. This implies that ˜ ϕϕ (cid:0) ϕ e − s ˜ D + ˜ D − − ϕ e − s ( D C, + D C, − ) (cid:1) D − σψ ′ is Γ-summable via a simpler version of the proof of Lemma 4.3. Proof of Proposition 4.1.
First suppose that
G/Z g is compact. Because thefunctions ϕ and ϕ ′ are cocompactly supported, Lemma 3.1 implies thatthere is a cocompactly supported function ϕ ∈ C ∞ ( M ) G such that ϕS = S . So S is g -trace class by Lemma 3.6 and the comment below it. Andbecause ψ and ψ ′ are cocompactly supported, Lemma 3.1 implies that thereis a cocompactly supported function ϕ ∈ C ∞ ( M ) G such that S ϕ = S . So S is g -trace class, again by Lemma 3.6. Part (a) follows.For case (b) in Theorem 2.7, suppose that G = Γ is discrete. Theoperator ˜ S is Γ-summable, so Lemma 4.9 and Proposition 4.10 imply thatthe three terms in the expression for S in Lemma 3.1 are all Γ-summable.As in the proof of part (a), there is a cocompactly supported function ϕ ∈ C ∞ ( M ) G such that ϕS = S . So S is g -trace class by Lemma 3.9.The operator ˜ S is Γ-summable, so Proposition 4.11 and Lemma 3.1imply that S = ϕ ˜ S ψ + ( ϕ ˜ Q − ϕ Q C ) ψ ′ is Γ-summable as well. As in the proof of part (a), there is a cocompactlysupported function ϕ ∈ C ∞ ( M ) G such that S ϕ = S . So S is g -trace classby Lemma 3.9. Our main goal in this section is to prove the following part of Theorem 2.7.29 roposition 5.1. If S and S are g -trace class, then τ g (index G ( ˆ D )) = Tr g ( S ) − Tr g ( S ) . (5.1)Together with Proposition 4.1, this is the main part of the proof ofTheorem 2.7. Let C ∞ ( ˆ M ; L ( ˆ E )) G loc be the subalgebra of elements of C ∗ ( ˆ M ; L ( ˆ E )) G loc withsmooth kernels. Because ˆ D is a multiplier of C ∞ ( ˆ M ; L ( ˆ E )) G loc , Lemmas 3.2and 3.3 imply that e := (cid:18) S S (1 + S ) RS ˆ D + − S (cid:19) (5.2)is an idempotent in C ∞ ( ˆ M ; L ( ˆ E )) G loc . (The 2 × E = ˆ E + ⊕ ˆ E − .) See also page 353 of [10]. Wewrite p := (cid:18) (cid:19) . Let ι : C ∞ ( ˆ M ; L ( ˆ E )) G loc → C ∗ ( ˆ M ; L ( ˆ E )) G loc be the inclusion map. Letindex L ( E ) G ( ˆ D ) ∈ K ( C ∗ ( ˆ M ; L ( ˆ E )) G loc )be defined as in (2.3). Lemma 5.2.
We have index L ( ˆ E ) G ( ˆ D ) = ι ∗ ([ e ] − [ p ]) . (5.3) Proof.
The right hand side of (5.3) equals ∂ [ ˆ D ] , where ∂ : K ( M ( C ∞ ( ˆ M ; L ( ˆ E ) G loc ) /C ∞ ( ˆ M ; L ( ˆ E ) G loc ) → K ( C ∞ ( ˆ M ; L ( ˆ E ) G loc )is the boundary map in the six-term exact sequence. The image of ∂ [ ˆ D ] in K ( C ∗ ( ˆ M ; L ( ˆ E ) G ) loc ) equals [¯ e ] − [ p ] , where ¯ e is the idempotent definedas the right hand side of (5.2), with R replaced by ¯ R , and S j by ¯ S j , for any30ultiplier ¯ R of C ∞ ( ˆ M ; L ( ˆ E ) G loc such that ¯ S := 1 − ¯ R ˆ D + and ¯ S := 1 − ˆ D + ¯ R are in C ∞ ( ˆ M ; L ( ˆ E )) G loc . In other words, for any such ¯ R ,[ e ] − [ p ] = [¯ e ] − [ p ] . (5.4)Let b de the function used in Subsection 2.2. We now choose b such that b ( x ) = O ( x ) as x →
0, so that the function x b ( x ) /x has a continuousextension to R . The function b is odd, and the function x b ( x ) /x is even.So the operator b ( ˆ D )ˆ D is even with respect to the grading on E , whereas b ( ˆ D )is odd. We denote restrictions of operators to sections of E ± be subscripts ± , respectively. We choose ¯ R := b ( ˆ D ) − (cid:16) b ( ˆ D )ˆ D (cid:17) − . Then we obtain operators ¯ S and ¯ S which equal the restrictions of 1 − b ( ˆ D ) to even and odd graded sections of E , respectively. We claim that¯ S and ¯ S lie in C ∞ ( ˆ M ; L ( ˆ E ) G loc . Indeed, by Lemma 2.3 in [34], theseoperators lie in C ∗ ( ˆ M ; L ( ˆ E )) G loc . And 1 − b is compactly supported, soˆ D j (1 − b ( ˆ D ) ) is a bounded operator on L ( ˆ E ) for all j ∈ N . Hence, byelliptic regularity, 1 − b ( ˆ D ) maps L ( ˆ E ), and any Sobolev space defined interms of ˆ D , continuously into Γ ∞ ( E ). So this operator has a smooth kernel.For this choice of ¯ R , we have¯ e = ¯ S ¯ S (1 + ¯ S ) b ( ˆ D ) − (cid:16) b ( ˆ D )ˆ D (cid:17) − ¯ S ˆ D + − ¯ S ! For s ∈ [0 , A s := (cid:16) b ( ˆ D )ˆ D (cid:17) − s/ (cid:16) b ( ˆ D )ˆ D (cid:17) s/ − , and consider the idempotent e s := A s ¯ eA − s = ¯ S ¯ S (1 + ¯ S ) b ( ˆ D ) − (cid:16) b ( ˆ D )ˆ D (cid:17) − s − ¯ S ˆ D + (cid:16) b ( ˆ D )ˆ D (cid:17) s + − ¯ S in M ( C ∗ ( ˆ M ; L ( ˆ E ) G loc . Via this continuous path of idempotents, we con-clude from (5.4) that[ e ] − [ p ] = [¯ e ] − [ p ] = [ e ] − [ p ] .
31y the definition (2.3) of index L ( ˆ E ) G ( ˆ D ), this index equals [ e ] − [ p ]. Themap ι ∗ may be inserted here because the entries of e have smooth kernels. f TR In this subsection, we temporarily return to the general setting of Subsec-tion 2.1. Because
Z/G is compact, the equivariant Roe algebra C ∗ ( Z ; L ( E | Z )) G equals the closure in B ( L ( E | Z )) of the algebra of bounded operators on L ( E | Z ) with finite propagation, and G -invariant, continuous kernels κ ∈ Γ( Z × Z, End( E | Z )) . (5.5)This can be proved analogously to the arguments in Section 5.4 in [16]. Wewill not need this fact, however, since the operators in C ∗ ( Z ; L ( E | Z )) G loc wework with always have continuous kernels.Let χ ∈ C ( X ) be a cutoff function for the action by G , as in (2.4). Definethe map g TR : C ∗ ( Z ; L ( E | Z )) G → C ∗ r G ⊗ K ( L ( E | Z ))by g TR( κ )( h ) = T ( χ ⊗ χ ) h · κ , for h ∈ G and κ as in (5.5). Here T ( χ ⊗ χ ) h · κ is the operator whose Schwartzkernel is given by(( χ ⊗ χ ) h · κ )( z, z ′ ) = χ ( z ) χ ( z ′ ) hκ ( h − z, z ′ ) , for all h ∈ G and z, z ′ ∈ Z . (The map g TR is not a trace, the notation ismotivated by Lemma 5.4 below.)
Lemma 5.3.
The map g TR is an injective ∗ -homomorphism.Proof. The fact that g TR is a ∗ -homomorphism follows from direct compu-tations involving G -invariance of κ . It follows from G -invariance of κ that κ = 0 if ( χ ⊗ χ ) h · κ = 0 for all h ∈ G .Let C ∗ Tr ( Z ; L ( E | Z )) G ⊂ C ∗ ( Z ; L ( E | Z )) G be the subalgebra of opera-tors with kernels κ such that g TR( κ ) ∈ C ∗ r G ⊗ L ( L ( E | Z )), where L standsfor the space of trace-class operators.Analogously to Subsection 3.4 of [21], we defineTR( κ )( x ) := Z Z χ ( xm ) tr( xκ ( x − m, m )) dm, for κ ∈ Γ ∞ (End( E | Z )) G and x ∈ G for which the integral converges.32 emma 5.4. For all κ ∈ C ∗ Tr ( Z ; L ( E | Z )) G and x ∈ G , TR( κ )( x ) = Tr( g TR( κ )( x )) . Proof.
For any G -equivariant operator T on L ( E | Z ) with smooth kernel κ ∈ C ∗ Tr ( Z ; L ( E | Z )) G , and any x ∈ G , the trace property of the operatortrace Tr and G -equivariance of T imply thatTR( T )( x ) = Tr( xχ T ) = Tr( χxT χ ) = Tr( g TR( κ )( x )) . Lemma 5.5.
For all κ ∈ C ∗ Tr ( Z ; L ( E | Z )) G such that Tr ◦ g TR( κ ) ∈ A , τ g ◦ Tr ◦ g TR( κ ) = Tr g ( κ ) . Proof.
It is immediate from the definitions that Tr g = τ g ◦ TR. So the claimfollows from Lemma 5.4. C ∗ r G ⊗ K To apply τ g to the localised coarse index of an operator, one needs a specificisomorphism (2.1). The key step in the proof of Proposition 5.1 is the factthat two maps from localised Roe algebras to group C ∗ -algebras tensoredwith the algebra of compact operators lead to the same result when oneapplies τ g . See Proposition 5.6. One of these maps is the one applied in [16]to map the localised equivariant coarse index into the K -theory of a group C ∗ -algebra. The other is defined in terms of the map g TR from Subsection5.2, and is suitable for computing g -traces.Let X be a proper, isometric, Riemannian G -manifold, and let Z ⊂ X be a cocompact subset. Suppose that Z = G × K Y for a slice Y ⊂ Z and acompact subgroup K < G . (We comment on how to remove this assumptionin Remark 5.7.) Fix a Borel section φ : K \ G → G . The map ψ : Z × G → G × K \ G × Y (5.6)given by ψ ( gy, h ) = (cid:0) hφ ( Kg − h ) − , Kg − h, φ ( Kg − h ) h − gy (cid:1) for g, h ∈ G and y ∈ Y , is G -equivariant and bijective, with respect to thediagonal action by G on Z × G and the action by G on the factor G on theright hand side of (5.6). We always use the action by G on itself by left33ultiplication. The map ψ relates the measures dz dg and dg d ( Kg ) dy toeach other, as shown in Lemma 5.2 in [16].Let E → X be a G -equivariant, Hermitian vector bundle. Write H := L ( K \ G ) ⊗ L ( E | Y ) . Then pulling back along ψ defines a G -equivariant, unitary isomorphism ψ ∗ : L ( G ) ⊗ H → L ( E | Z ) ⊗ L ( G ) . (5.7)Let ψ and ψ be the projections of ψ onto G and K \ G × Y , respectively.Define the map η : Z → K \ G × Y by η ( z ) = ψ ( z, e ) . This induces a unitary isomorphism η ∗ : H → L ( E | Z ) . Let C ∗ ker ( Z ) G be the algebra as in Definition 5.10 in [16], of continuouskernels κ G : G × G → K ( H )with finite propagation, and the invariance property that for all g, g ′ , h ∈ G , κ G ( hg, hg ′ ) = κ G ( g, g ′ ) . (5.8)Such a kernel defines an operator on L ( G ) ⊗ H , which corresponds to anoperator on L ( E | Z ) ⊗ L ( G ) via (5.7). This gives a map a : C ∗ ker ( Z ) G → C ∗ ( Z ) G with dense image; see Proposition 5.11 in [16]. We also have an injective ∗ -homomorphism W : C ∗ ker ( Z ) G → C ∗ r G ⊗ K ( H )with dense image, given by W ( κ G )( g ) = κ G ( g − , e ) , for κ G ∈ C ∗ ker ( Z ) G and g ∈ G .There are natural maps ϕ : C ∗ ( Z ) G → C ∗ ( X ) G loc ; ϕ E : C ∗ ( Z ; L ( E | Z )) G → C ∗ ( X ; L ( E )) G loc , (5.9)defined by extending operators by zero outside Z , that induce isomorphismson K -theory; see Section 7.2 in [16]. Consider the map ⊕ roposition 5.6. The diagram C ∗ ( X ; L ( E )) G loc ⊕ / / C ∗ ( X ) G loc C ∗ ( Z ; L ( E | Z )) Gϕ E O O g TR (cid:15) (cid:15) C ∗ ( Z ) Gϕ O O C ∗ ker ( Z ) Ga O O W (cid:15) (cid:15) C c ( G ) ⊗ K ( L ( E | Z )) τ g ⊗ (cid:15) (cid:15) C c ( G ) ⊗ K ( H ) τ g ⊗ (cid:15) (cid:15) K ( L ( E | Z )) η ∗ / / K ( H ) . (5.10) commutes in the following sense: the maps a , ϕ E and ϕ are injective, withdense images, and the diagram commutes on the relevant dense subalgebrasfor the inverses of these maps. More explicitly, if κ ∈ C ∗ ( Z ; L ( E | Z )) G , κ G ∈ C ∗ ker ( Z ) G and ϕ E ( κ ) ⊕ ϕ ◦ a ( κ G ) , then η ∗ ◦ ( τ g ⊗ ◦ g TR( κ ) = ( τ g ⊗ ◦ W ( κ G ) . Remark . In general, Z is a finite disjoint union of subsets of the form Z j = G × K j Y j ; see [31]. We can generalise Proposition 5.6 to that setting,by viewing operators on L ( E | Z ) as finite matrices of operators between thespaces L ( E | Z j ), and comparing them with analogous matrices of operatorsbetween the spaces H j := L ( K j \ G ) ⊗ L ( E | Y j ). For simplicity, we will prove Proposition 5.6 in the case where E is the trivialline bundle. The general case can be proved analogously.By definition of the maps (5.9), as in [16], the diagram C ∗ ( X ; L ( E )) G loc ⊕ / / C ∗ ( X ) G loc C ∗ ( Z ; L ( E | Z )) Gϕ E O O ⊕ / / C ∗ ( Z ) Gϕ O O Z .Let an element of C ∗ ( Z ; L ( Z )) G be given by a continuous kernel κ : Z × Z → C with finite propagation. Lemma 5.8.
For all ζ ∈ L ( Z ) ⊗ L ( G ) , g ∈ G and z ∈ Z , (( ϕ E ( κ ) ⊕ ζ )( z, g ) = (cid:16)Z G g TR( κ )( h )( h − g − · ζ ( − , gh )) dh (cid:17) ( g − z ) . In this lemma, ζ ( − , gh ) ∈ L ( Z ), on which G acts via its action on Z . Proof.
Consider the map (2.5) in this setting, j : L ( Z ) → L ( Z ) ⊗ L ( G ) . Then ⊕ L ( Z ) to the correspondingoperators on j ( L ( Z )) by conjugation with j , and extending them by zero onthe orthogonal complement of j ( L ( Z )). Let p : L ( Z ) ⊗ L ( G ) → j ( L ( Z ))be the orthogonal projection. Then ϕ E ( κ ) ⊕ j ◦ ϕ E ( κ ) ◦ j − ◦ p. (5.11)One checks directly that for all ζ ∈ L ( Z ) ⊗ L ( G ) and z ∈ Z ,( j − ◦ p )( ζ )( z ) = Z G χ ( g − z ) ζ ( z, g ) dg. (5.12)The lemma can now be proved via a straightforward computation involving(5.11), (5.12), G -invariance of κ , and left invariance of the Haar measure on G . Next, fix κ G ∈ C ∗ ker ( Z ) G . Lemma 5.9.
For all ζ ∈ L ( Z ) ⊗ L ( G ) , g ∈ G and z ∈ Z , (( ϕ ◦ a )( κ G ) ζ )( z, g ) = (cid:16)Z G W ( κ G ) (cid:0) ψ ( z, g ) − hψ ( z, g ) (cid:1) ζ ( ψ − ( hψ ( z, g ) , − )) dh (cid:17) ( ψ ( z, g )) . Proof.
This is a straightforward computation involving G -invariance of κ G and right invariance of the Haar measure on G .36 emma 5.10. Let η : X → X be a measurable bijection between measurespaces ( X , µ ) and ( X , µ ) , such that η ∗ µ = µ . Let σ : X → G be anymap. Define Ψ : C c ( G ) ⊗ K ( L ( X )) → C c ( G ) ⊗ K ( L ( X )) by (cid:0) (Ψ( f )( g )) u (cid:1) ( x ) = (cid:0)(cid:0) η ∗ ◦ f ( σ ( x ) − gσ ( x )) ◦ ( η − ) ∗ (cid:1) u (cid:1) ( x ) for all f ∈ C c ( G ) ⊗ K ( L ( X )) , g ∈ G , u ∈ L ( X ) and x ∈ X . Then thefollowing diagram commutes: C c ( G ) ⊗ K ( L ( X )) τ g ⊗ (cid:15) (cid:15) C c ( G ) ⊗ K ( L ( X )) Ψ o o τ g ⊗ (cid:15) (cid:15) K ( L ( X )) η ∗ / / K ( L ( X )) . Proof.
This is a straightforward computation, involving G -invariance of themeasure d ( hZ g ) on G/Z g . Remark . The map Ψ in Lemma 5.10 is not a homomorphism in general,unless σ is constant.Applying Lemma 5.10 with X = Z , X = K \ G × Y and σ ( z ) = ψ ( z, e ),we obtain a commutative diagram C c ( G ) ⊗ K ( L ( Z )) τ g ⊗ (cid:15) (cid:15) C c ( G ) ⊗ K ( H ) Ψ o o τ g ⊗ (cid:15) (cid:15) K ( L ( Z )) η ∗ / / K ( H ) . (5.13) Proof of Proposition 5.6.
As before, fix an element of C ∗ ( Z ; L ( Z )) G givenby a continuous kernel κ : Z × Z → C with finite propagation, and κ G ∈ C ∗ ker ( Z ) G . Suppose that( ϕ ◦ a )( κ G ) = ( ϕ E ( κ ) ⊕ ∈ C ∗ ( X ; Z ) G . (5.14)Then Lemmas 5.8 and 5.9, applied with g = e , imply that for all ζ ∈ L ( Z ) ⊗ L ( G ) and z ∈ Z , (cid:16)Z G g TR( κ )( h )( h − · ζ ( − , h )) dh (cid:17) ( z )= (cid:16)Z G η ∗ ◦ W ( κ G ) (cid:0) ψ ( z, e ) − hψ ( z, e ) (cid:1) ζ ( ψ − ( hψ ( z, e ) , − )) dh (cid:17) ( z ) . (5.15)37ne has for all z ∈ Z and h ∈ G , ψ ( hz, h ) = ( hψ ( z, e ) , η ( z )) . (Recall that ψ is the projection of ψ onto G .) Hence the right hand side of(5.15) equals (cid:16)Z G η ∗ ◦ W ( κ G ) (cid:0) ψ ( z, e ) − hψ ( z, e ) (cid:1) ◦ ( η − ) ∗ ( h − · ζ ( − , h )) dh (cid:17) ( z ) . Therefore, if ζ = u ⊗ v , for u ∈ L ( Z ) and v ∈ L ( G ), then (5.15) impliesthat for all z ∈ Z , Z G v ( h ) g TR( κ )( h )( h − · u ) dh (cid:17) ( z )= (cid:16)Z G v ( h ) (cid:0) η ∗ ◦ W ( κ G ) (cid:0) ψ ( z, e ) − hψ ( z, e ) (cid:1) ◦ ( η − ) ∗ (cid:1) ( h − · u ) dh (cid:17) ( z ) . Hence for all u ∈ L ( Z ), h ∈ G and z ∈ Z , (cid:0)g TR( κ )( h ) u (cid:1) ( z ) = (cid:16) η ∗ ◦ W ( κ G ) (cid:0) ψ ( z, e ) − hψ ( z, e ) (cid:1) ◦ ( η − ) ∗ ( u ) (cid:17) ( z )= (cid:0) Ψ( W ( κ G ))( h ) u (cid:1) ( z ) . So Ψ( W ( κ G )) = g TR( κ ), and commutativity of diagram (5.13) implies theclaim. The isomorphism C ∗ ( X ) G loc ∼ = C ∗ r G ⊗ K used in [16] to identify localisedcoarse indices with classes in K ∗ ( C ∗ r G ) is the map W ◦ a − ◦ ϕ − , defined on a dense subalgebra and extended continuously. Hence we explic-itly haveindex G ( ˆ D ) = ( W ∗ ◦ a − ∗ ◦ ϕ − ∗ )(index L ( E ) G ( ˆ D ) ⊕ ∈ K ( C ∗ r G ) . (5.16)Therefore, Lemma 5.2 and Proposition 5.6 (see Remark 5.7) imply that τ g (index G ( ˆ D )) = τ g (cid:0)g TR ◦ ( ϕ E ) − ∗ ([ e ] − [ p ]) (cid:1) . The trace map on the sub-algebra of trace-class operators in K ( L ( E | Z ))induces the isomorphism K ∗ ( C ∗ r G ⊗ K ( L ( E | Z ))) ∼ = K ∗ ( C ∗ r G ) . Hence Proposition 5.1 follows by Lemma 5.5.38 .6 Proof of Theorem 2.7
Proposition 5.12.
If the operators e − t ˜ D and e − t ˜ D ˜ D and S and S are g -trace class, then Tr g ( S ) − Tr g ( S ) = Tr g ( S ) − Tr g ( S ) . (5.17) Proof.
We have S R = RS , and hence S − S = S (1 − S ) = RS ˆ D + ; S − S = S (1 − S ) = S ˆ D + R. Because e − t ˜ D and e − t ˜ D ˜ D are g -trace class, Lemma 5.3 and Proposition 5.7in [21] imply that S and S are g -trace class. So the operators RS ˆ D + and S ˆ D + R are g -trace class.By Lemma 3.1, S ˆ D + = ϕ ˜ S ˜ D + ψ − ϕ ˜ S σψ ′ − ϕ ′ σ ˜ Q ˜ D + ψ + ϕ ′ σ ˜ Qσψ ′ − ϕ ′ σQ C D C, + ψ + ϕ ′ Q C σψ ′ . (5.18)Since ˜ S and σ − ˜ S ˜ D + are g -trace class by assumption, ϕ ′ j has disjoint sup-port from ψ j , and all operators occurring are pseudo-differential operators,and therefore have smooth kernels off the diagonal, we find that σ − S ˆ D + is g -trace class. (And the last four terms on the right hand side of (5.18) have g -trace zero.) And Rσ is has a distributional kernel, so Lemma 3.4 impliesthatTr g ( RS ˆ D + ) = Tr g ( Rσσ − S ˆ D + ) = Tr g ( σ − S ˆ D + Rσ ) = Tr g ( S ˆ D + R ) . Hence (5.17) follows.Theorem 2.7 follows from Propositions 4.1, 5.1 and 5.12. D N We have so far assumed that the Dirac operator D N on the boundary N is invertible. We now discuss how that assumption can be weakened to theassumption that 0 is isolated in the spectrum of D N . The arguments arerelated to those in Section 6 of [21]. 39 .1 A shifted Dirac operator Let ε > − ε, ε ] ∩ spec( D N )) \ { } = ∅ . Let ψ ∈ C ∞ ( ˆ M ) G be a nonnegative function such that ψ ( n, u ) = (cid:26) u if n ∈ N and u ∈ (1 / , ∞ );0 if n ∈ N and u ∈ (0 , / ψ ( m ) = 0 if m ∈ M \ U . (Recall that U ∼ = N × (0 ,
1] is a neighbourhood of N in M .)As in Section 6 of [21], we consider the G -equivariant, odd, elliptic op-erator ˆ D ε := e εψ ˆ De − εψ . The operator ˆ D ε is G -equivariant, essentially self-adjoint, odd-graded andelliptic. Its restriction to ˆ M \ M equals σ (cid:16) − ∂∂u + D N + ε (cid:17) . (6.1)It therefore satisfies the condition (2.2), and has a well-defined indexindex G ( ˆ D ε ) ∈ K ( C ∗ r G ) . Let a be as in (2.15). Theorem 2.7 generalises as follows. Theorem 6.1.
Suppose that ˆ D ε is g -Fredholm, and that the heat kerneldecay (2.13) holds for the operators mentioned. If either(a) G/Z g is compact; or(b) G = Γ is discrete and finitely generated, and (2.14) holds for a k < a ε ,then τ g (index G ( ˆ D ε )) = index g ( ˆ D ε ) . Conditions for ˆ D ε to be g -Fredholm were given in Theorem 6.1 andCorollary 6.2 in [21].Corollary 2.10 also generalises to this setting. This involves Corollary6.2 in [21]. 40 .2 A shifted parametrix Let ˜ ψ be any smooth, G -invariant extension of ψ | M to the double ˜ M of M .As in Subsection 6.3 of [21], we use the operators˜ D ε = e ε ˜ ψ ˜ De − ε ˜ ψ ;˜ Q ε := 1 − e − t ˜ D ε, − ˜ D ε, + ˜ D ε, − ˜ D ε, + ˜ D ε, − ;˜ S ε, := 1 − ˜ Q ε ˜ D ε, + = e − t ˜ D ε, − ˜ D ε, + ;˜ S ε, := 1 − ˜ D ε, + ˜ Q ε = e − t ˜ D ε, + ˜ D ε, − . Let D C,ε be the restriction of ˆ D ε to N × (1 / , ∞ ), and let Q C,ε be the inverseof its self-adjoint closure, restricted to sections of E − .Let the functions ϕ j and ψ j be as in Subsection 3.1, with the differencethat they change values between 0 and 1 on the interval (1 / ,
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