Equivariant Callias index theory via coarse geometry
aa r X i v : . [ m a t h . K T ] F e b Equivariant Callias index theory via coarsegeometry
Hao Guo, ∗ Peter Hochs † and Varghese Mathai ‡ February 21, 2019
Dedicated to the memory of John Roe
Abstract
The equivariant coarse index is well-understood and widely used foractions by discrete groups. We extend the definition of this index togeneral locally compact groups. We use a suitable notion of admissiblemodules over C ∗ -algebras of continuous functions to obtain a meaning-ful index. Inspired by work by Roe, we then develop a localised variant,with values in the K -theory of a group C ∗ -algebra. This generalisesthe Baum–Connes assembly map to non-cocompact actions. We showthat an equivariant index for Callias-type operators is a special case ofthis localised index, obtain results on existence and non-existence ofRiemannian metrics of positive scalar curvature invariant under propergroup actions, and show that a localised version of the Baum–Connesconjecture is weaker than the original conjecture, while still giving aconceptual description of the K -theory of a group C ∗ -algebra. Contents ∗ Texas A&M University, [email protected] † University of Adelaide, [email protected] ‡ University of Adelaide, [email protected] The localised equivariant index 12 G -representations . . . . . . . . . . . . . . 255.2 Propagation in X and in G . . . . . . . . . . . . . . . . . . . 275.3 Proof of Theorem 2.7 . . . . . . . . . . . . . . . . . . . . . . . 305.4 Kernels and group C ∗ -algebras . . . . . . . . . . . . . . . . . 31 G -Callias-type operators and Roe’s localised index . . . . . . 396.4 Proof of Theorem 4.2 . . . . . . . . . . . . . . . . . . . . . . . 41 Background
Coarse geometry is the study of large-scale structures of metric spaces. Im-portant invariants in this area are various versions of the
Roe algebra andtheir K -theory groups. The coarse index , with values in such K -theorygroups, is a powerful tool that has been studied and applied by many au-thors. A standard introduction is [32]. A central problem is the coarseBaum–Connes conjecture [31], which has a range of important consequences.Important areas of applications of coarse index theory are obstructions to2iemannian metrics of positive scalar curvature (see [36] for a survey) andthe Novikov conjecture (see for example [40, 41]).Equivariant versions of the Roe algebra and the coarse index have beendeveloped for proper actions by discrete groups. Another refinement is alocalised variant of the coarse index developed by Roe [35], generalisingan index defined by Gromov and Lawson [16]. This localised coarse indexapplies, in a certain precise sense, to operators that are invertible outsidea given subset of the space. For actions by discrete groups, an equivariantversion of this localised index theory in terms of coarse K -homology wasrecently developed by Bunke and Engel [10].Equivariant coarse index theory for general locally compact groups wouldbe useful in the study of such groups and their actions. In particular, alocalised approach to this index, which takes values in K -theory of the group C ∗ -algebra, offers greater flexibility compared to the standard equivariantindex for actions with compact quotients [4]. However, the topology of anon-discrete group poses some technical challenges in the development ofequivariant coarse index theory for such groups. Results
Our main goal in this paper is to develop equivariant coarse index theoryfor proper actions by general locally compact groups G , and in particular alocalised version with values in K ∗ ( C ∗ red ( G )). Here C ∗ red ( G ) is the reducedgroup C ∗ -algebra of G . Our secondary goal is to demonstrate the usefulnessof this theory by showing how it simultaneously generalises other versionsof index theory, and by obtaining applications to Riemannian metrics ofpositive scalar curvature invariant under proper actions. Our main resultsare:1. constructions of equivariant Roe algebras and an equivariant coarseindex, and in particular localised versions of these objects (Definitions2.10, 2.17, 3.4 and 3.6);2. a proof that the analytic assembly map [4] and the equivariant in-dex of Callias-type operators in [17] are special cases of the localisedequivariant coarse index (Theorem 4.2 and Corollary 4.3);3. obstructions to and existence of Riemannian metrics with positivescalar curvature, invariant under proper group actions (Proposition4.4 and Theorem 4.6); 3. the formulation of a localised version of the surjectivity part of theBaum–Connes conjecture [4], and relations between the localised andoriginal conjectures (Conjecture 4.8 and Proposition 4.9).In forthcoming work [18], we will give further applications of the indextheory we develop in this paper, showing that it refines the indices in [6, 22],and obtaining an application to the quantisation commutes with reductionproblem. The localised equivariant coarse index
Let M be a complete Riemannian manifold, on which a locally compactgroup G acts properly and isometrically. Let D be an elliptic differentialoperator on M . For the definition of the localised index, we assume thatthere is a G -invariant subset Z ⊂ M such that D has a uniform positivelower bound outside Z . Then we obtain the localised equivariant coarseindex index ZG ( D ) ∈ K ∗ ( C ∗ ( Z ) G ) . (See Definition 3.6.) Here C ∗ ( Z ) G is the equivariant Roe algebra of Z . Weare particularly interested in the case where Z/G is compact, so that C ∗ ( Z ) G is stably isomorphic to C ∗ red ( G ). While there is no technical reason a priorito restrict to the case where Z/G is compact, this special case is interestingfor several reasons, namely, in this case:1. the localised equivariant coarse index and the K -theory group it liesin are independent of Z ;2. the receptacle of the index, namely K ∗ ( C ∗ red ( G )), is a rich and relevantobject (in particular, nonzero); there exist many tools to extract in-formation from it, such as traces and higher cyclic cohomology classeson (smooth subalgebras of) C ∗ red ( G );3. various existing indices, including the analytic assembly map, are spe-cial cases, as we discuss below.Operators D to which the localised equivariant index applies include thefollowing three important special cases:1. Callias-type operators of the form D = ˜ D + Φ, where ˜ D is a Diracoperator and Φ is a vector bundle endomorphism making D uniformlypositive outside Z . The study of these operators, their indices andtheir applications was initiated by Callias [11], and extended in various4irections by many authors, see e.g. [2, 3, 5, 7, 8, 9, 12, 13, 25, 26, 28,39]. The equivariant case for proper actions was treated in [17];2. Dirac operators D whose curvature term R in the Weitzenb¨ock formula D = ∆ + R is uniformly positive outside Z [16, 35];3. Dirac operators D on manifolds with boundary that are invertible onthe boundary, extended to cylinders attached to these boundaries.For Callias-type operators, the coarse-geometric approach in this paper mayalready be useful in the case of trivial groups. In addition, we can nowconsider the lift of a (non-equivariant) Callias-type operator on a manifold M to the universal cover of M , and obtain an equivariant index in the K -theory of the fundamental group of M . This is a more refined invariant thanthe Fredholm index of the initial operator.In the case where Z/G is compact, the localised equivariant index gener-alises the Baum–Connes assembly map from the case of actions with compactquotients to the cases above. This allows us to formulate a localised ver-sion of the surjectivity direction of the Baum–Connes conjecture. We willshow that this localised surjectivity is implied by standard Baum–Connessurjectivity.One of the technical challenges in constructing a meaningful index inthis context is to develop the appropriate notion of an admissible module .For actions by discrete groups, this was done in Definition 2.2 in [42]. TheRoe algebra of a metric space X acted on by a locally compact group G isdefined in terms of operators a on Hilbert space H X with compatible actionsby C ( X ) and G . The resulting algebra should ideally be independent of thechoice of H X , and its K -theory should contain relevant information about G , and possibly also X . (A natural initial choice would be H X = L ( X )for a Borel measure on X , but this does not contain enough information if,for example, X is a point or if G is compact and acts trivially on X .) Weachieve these two things by taking H X to be an admissible module, in thesense that we define in Sections 2 and 5. We indicate how Roe algebras andthe associated K -theory groups and indices defined in terms of admissiblemodules on the one hand, and more geometric, but non-admissible moduleson the other, are related in Subsection 3.4. Outline of this paper
We introduce admissible modules and the associated Roe algebras in Section2. We use these notions in Section 3 to define the equivariant coarse index5nd its localised version. In Section 4 we apply these notions to show thatthe equivariant Callias-type index in [17] is a special case, and we establishresults on positive scalar curvature, as well as state a localised Baum–Connesconjecture. Proofs of the properties of admissible modules and Roe algebrasfrom Section 2 are given in Section 5. Proofs of the results in Section 4 aregiven in Sections 6 and 7.
Acknowledgements
The authors are grateful to Rufus Willett, Zhizhang Xie, and Guoliang Yufor their helpful advice. Varghese Mathai was supported by funding from theAustralian Research Council, through the Australian Laureate FellowshipFL170100020. Hao Guo was supported in part by funding from the NationalScience Foundation under Grant No. 1564398.
A key idea in this paper is to use coarse geometry and Roe algebras toconstruct a localised equivariant index for proper actions with values inthe K -theory of a group C ∗ -algebra. We start by discussing the necessarybackground in coarse geometry. Much of the material in this section is well-known in the case of discrete groups, but we will see that the generalisationto general locally compact groups requires some work.Throughout this section, ( X, d ) will denote a proper metric space, i.e.a metric space in which closed balls are compact, and G a locally compactgroup acting properly and isometrically on X . We assume G to be unimodu-lar and fix a Haar measure dg on G . Throughout this paper, we will use leftand right invariance of dg without mentioning this explicitly, in argumentsinvolving substitutions in integrals over G . (The contents of this paper canlikely be generalised to non-unimodular group if the modular function is in-serted where appropriate.) We will sometimes assume X/G to be compact,but not always. We always view L ( G ) as a unitary representation of G viathe left-regular representation.The two properties of the modules and algebras we define here that aremost important to the construction of the equivariant localised coarse indexin Section 3 are Theorems 2.7 and 2.11. These are proved in Subsections5.3 and 5.4, respectively. 6 .1 Admissible modules We will construct the reduced and maximal equivariant Roe algebras of X in terms of admissible C ( X ) -modules , and a particularly useful typeof such modules we call geometric admissible modules . Using admissiblemodules ensures that the algebras constructed are independent of the choiceof module. Their purpose is also to ensure that the Roe algebras we usecontain sufficient information about the group G , as illustrated in Examples2.16 and 3.8. Admissible modules were first defined in the case of discretegroups by Guoliang Yu in Definition 2.2 in [42]. For non-discrete G , thedefinition needs to take into account the topology of G .Admissible modules are special cases of ample, equivariant C ( X )-modules. Definition 2.1. An equivariant C ( X ) -module is a Hilbert space H X witha unitary representation of G , together with a ∗ -homomorphism π : C ( X ) → B ( H X )such that for all g ∈ G and f ∈ C ( X ), π ( g · f ) = g ◦ π ( f ) ◦ g − . Here g · f is the function mapping x ∈ X to f ( g − x ).An equivariant C ( X )-module is nondegenerate if π ( C ( X )) H X is densein H X . It is standard if π ( f ) is a compact operator only if f = 0. Themodule is ample if it is nondegenerate and standard.We will usually omit the homomorphism π from the notation, and write f · ξ := π ( f ) ξ for f ∈ C ( X ) and ξ ∈ H X . Example 2.2.
The space H G = L ( G ) ⊗ H , for a separable infinite-dimensionalHilbert space H equipped with the trivial G -representation, is an ampleequivariant C ( G )-module with respect to the multiplicative action of C ( G )on L ( G ).The action by C ( X ) on any ample, equivariant C ( X )-module H X hasa unique extension to an action by the algebra L ∞ ( X ) of bounded Borelfunctions, characterised by the property that for a uniformly bounded se-quence in L ∞ ( X ) converging pointwise, the corresponding operators on H X converge strongly. All functions we will apply this extension to are bounded,continuous functions on closed sets in X , such as the indicator function Y of a closed subset Y ⊂ X . 7 efinition 2.3. Let H X be an ample C ( X )-module. An operator T ∈B ( H X ) has finite propagation if there is an r > f , f ∈ C ( X ) whose supports are further than r apart, we have f T f = 0.An operator T ∈ B ( H X ) is locally compact if for all f ∈ C ( X ), theoperators f T and T f are compact.We now formulate the general notion of admissible module over a space.
Definition 2.4.
Let G is a locally compact group. Let H X be an ample,equivariant C ( X )-module. If X/G is compact, H X is said to be admissible if there is a G -equivariant unitary isomorphismΨ : L ( G ) ⊗ H ∼ = H X , for a separable infinite-dimensional Hilbert space H equipped with the trivial G -representation, such that for any bounded, G -equivariant operator T on H X ,1. T has finite propagation with respect to the action by C ( X ) if andonly if Ψ − ◦ T ◦ Ψ has finite propagation with respect to the actionby C ( G );2. T is locally compact with respect to the action by C ( X ) if and onlyif Ψ − ◦ T ◦ Ψ is locally compact (with respect to the action by C ( G ).If X/G is not necessarily compact, then an ample, equivariant C ( X )-module is admissible if for every closed, G -invariant subset Y ⊂ X such that Y /G is compact, Y H X is an admissible module over C ( Y ). Remark 2.5. If G = Γ is a discrete group, Definition 2.4 is an alternativeto the definition of admissible modules given in the conditions in Definition2.4 are implied by the definition of admissible modules given in [42]. Inparticular, if H X is an admissible module in the sense of Definition 2.2 in[42], then it is also admissible in our sense. Remark 2.6.
The idea behind admissible modules is to provide a class ofgeneral spaces on which to carry out analysis of operators on X , but whichalso allow us to define an equivariant index of these operators. For example,we will use Conditions 1 and 2 of Definition 2.4 in an essential way to definethe equivariant coarse index and the localised equivariant coarse index in K ∗ ( C ∗ ( G )) (see Subsections 6.1 and 6.2). See the proof of Proposition 5.11.
8n this paper, we will mostly work with a particular geometric type ofadmissible module.Consider a covering of X by sets of the form G × K j Y j , for compactsubgroups K j < G and compact, K j -invariant slices Y j ⊂ X . Suppose thatthe intersections between these sets have measure zero. For each j , fix a K -invariant measure dy j on Y j . Together with the Haar measure dg , theyinduce a G -invariant measure dx on X . We call such a measure inducedfrom slices . Such measures are natural choices; see for example Lemma 4.1in [23]. Theorem 2.7.
Suppose
X/G is compact. Suppose that at least one of thesets
G/K and
X/G is infinite. Let H X = L ( E ) ⊗ L ( G ) , for a Hermitian G -vector bundle E → X , defined with respect to the a measure on X inducedfrom slices. Then H X , equipped with the diagonal representation of G andthe C ( X ) -action on the factor L ( E ) by pointwise multiplication, is anadmissible C ( X ) module. Definition 2.8. If X/G is compact, then L ( E ) ⊗ L ( G ), for a Hermitian G -vector bundle E → X , defined with respect to the a measure on X inducedfrom slices is a geometric admissible C ( X ) -module .Note that the difference between a general admissible module in Defini-tion 2.4 and a geometric module is that on a geometric admissible module,the group G acts diagonally, whereas on a general admissible module in theform L ( G ) ⊗ H , G acts only on the first factor. The advantage of workingwith a geometric module is that the action by C ( X ) is explicit. We will infact usually work with geometric admissible modules.The notion of a geometric admissible module, and hence Theorem 2.7, isa key component of our construction of (localised) coarse indices of ellipticoperators in Subsections 3.2 and 3.3. Remark 2.9.
The condition in Definition 2.4 that H is infinite-dimensional,and the corresponding condition in Theorem 2.7 that G/K or X/G is infinite,are assumed to ensure that the equivariant coarse index theory of Section3 is rich enough to capture information about the group G (see Examples2.16 and 3.8). More specifically, infinite-dimensionality of H guaranteesthat the algebras D ∗ ( X ) G , which are used to define the coarse index (seeSubsection 3.1) exist and have the properties needed to define a useful index.It also implies that the localised Roe algebra is independent of the choice ofadmissible module, as in (5). (Although for finite-dimensional H , the factor K in (5) would become a finite-dimensional matrix algebra, which makes nodifference at the level of K -theory.) 9n Theorem 2.7, if both G/K and
X/G are finite, then one can still formthe admissible module L ( E ) ⊗ L ( G ) ⊗ l ( N ). Fix an equivariant C ( X )-module H X . We denote the algebra of G -equivariantbounded operators H X by B ( H X ) G . Definition 2.10.
The algebraic equivariant Roe algebra for H X of X isthe algebra C ∗ alg ( X ; H X ) G consisting of the locally compact operators in B ( H X ) G with finite propagation. The equivariant Roe algebra for H X of X is the closure C ∗ ( X ; H X ) G of C ∗ alg ( X ) G in B ( H X ).If H X is an admissible module and X/G is compact, then C ∗ alg ( X ) G := C ∗ alg ( X ; H X ) G is the algebraic equivariant Roe algebra of X , and C ∗ ( X ) G := C ∗ ( X ; H X ) G is the equivariant Roe algebra of X . Theorem 2.11. If X/G is compact and H X is , then C ∗ alg ( X ) G is ∗ -isomorphic to a dense subalgebra of C ∗ ( G ) ⊗ K , where C ∗ ( G ) denotes eitherthe reduced or maximal group C ∗ -algebra, and K is the algebra of compactoperators on a separable Hilbert space. This theorem will be proved in Subsection 5.4. This involves realisingRoe algebras in terms of Schwartz kernels of operators.There is also a maximal version of the equivariant Roe algebra. For any ∗ -algebra A , and any a ∈ A , we write k a k max := sup π k π ( a ) k B ( H π ) , where the supremum runs over all irreducible ∗ -representations π of A inHilbert spaces H π . This supremum may be infinite. Since this maximalnorm is always finite for group C ∗ -algebras, Theorem 2.11 has the followingconsequence. Corollary 2.12. If X/G is compact, then k a k max < ∞ for all a ∈ C ∗ alg ( X ) G . Definition 2.13. If X/G is compact, then the maximal equivariant Roealgebra of X is the completion of C ∗ alg ( X ) G in the maximal norm k · k max . Remark 2.14.
Gong, Wang and Yu [15] proved that if G = Γ is discrete,then the maximal norm on C ∗ alg ( X ) Γ is always finite, even if X/ Γ is notcompact, as long as X has bounded geometry. We expect this to be truealso when G is locally compact. (In [15], the Roe algebras are defined interms of kernels; see Subsection 5.4.)10f X/G is compact then Theorem 2.11 implies that there are ∗ -isomorphisms C ∗ ( X ) G ∼ = C ∗ red ( G ) ⊗ K ; (1) C ∗ max ( X ) G ∼ = C ∗ max ( G ) ⊗ K . (2) Remark 2.15.
The relations (1) and (2) in particular imply that the re-duced and maximal algebras are independent of the choice of the admissible C ( X )-module H X . We expect this to be true even if X/G is not compact.In the case of G = Γ discrete and X/ Γ compact, a proof can be found in theforthcoming book [38].
Example 2.16.
Suppose that G = K is compact, and X = pt is a point.Then l ( N ) is an ample module over C (pt) = C . It is equivariant if weequip it with the trivial action by K . The algebraic, reduced and maxi-mal equivariant Roe algebras defined with respect to this module all equal K ( l ( N )), which contains no group-theoretic information about K . This isbecause the module l ( N ) is not admissible. The localised index that we will define in Definition 3.3 involves a localisedversion of Roe algebras.Let H X be an equivariant C ( X )-module. Let Z ⊂ X be a G -invariant,closed subset. Definition 2.17.
An operator T ∈ B ( H X ) is supported near Z if there isan r > f ∈ C ( X ) whose support is at least a distance r away from Z , we have f T = T f = 0.The algebraic equivariant Roe algebra for H X of X , localised at Z , de-noted by C ∗ alg ( X ; Z, H X ) G , consists of the operators in C ∗ alg ( X, H X ) G sup-ported near Z .The equivariant Roe algebra for H X of X , localised at Z , denoted by C ∗ ( X ; Z, H X ) G , is the closure of C ∗ alg ( X ; Z, H X ) G in B ( H X ).If Z/G is compact, then we call C ∗ alg ( X ; H X ) G loc := C ∗ alg ( X ; Z, H X ) G the localised algebraic equivariant Roe algebra for H X of X , and C ∗ ( X ; H X ) G loc := C ∗ ( X ; Z, H X ) G the localised equivariant Roe algebra for H X of X . If H X isan admissible module, then we omit it from the notation and terminology,and obtain the localised algebraic equivariant Roe algebra C ∗ alg ( X ) G loc and the localised equivariant Roe algebra C ∗ ( X ) G loc of X .Of the terms in Definition 2.17, the localised equivariant Roe algebra C ∗ ( X ) G loc is the one we are most interested in. Note that if Z/G is compact,11hen the algebras C ∗ alg ( X ; Z, H X ) G and C ∗ ( X ; Z, H X ) G are independent of Z , as long as Z/G is compact.For r >
0, we writePen(
Z, r ) := { x ∈ X ; d ( x, Z ) ≤ r } . (3)In terms of these sets, we have C ∗ alg ( X ; Z, H X ) G = lim −→ r C ∗ alg (Pen( Z, r ); H X ) G ; C ∗ ( X ; Z, H X ) G = lim −→ r C ∗ (Pen( Z, r ); H X ) G . (4)If H X is admissible, then the algebra C ∗ alg (Pen( Z, r )) G has a well-definedmaximal norm by Theorem 2.11, for all r . Hence, by (4), so does C ∗ alg ( X ) G loc . Definition 2.18. If H X is admissible, the maximal localised equivariant Roealgebra of X , denoted by C ∗ max ( X ) G loc , is the completion of C ∗ alg ( X ) G loc in themaximal norm k · k max .Theorem 2.11 and (4) imply that C ∗ ( X ) G loc ∼ = C ∗ red ( G ) ⊗ K ; C ∗ max ( X ) G loc ∼ = C ∗ max ( G ) ⊗ K . (5) The (non-localised) equivariant coarse index is defined completely analo-gously to the case for discrete groups, but with Roe algebras defined interms of the admissible modules from Subsection 2.1.Let H X be an equivariant C ( X )-module. Let C ∗ ( X ; H X ) G denote thereduced or maximal version (if it exists ) of the equivariant Roe algebrafor H X . Let D ∗ ( X ) G be any C ∗ -algebra containing C ∗ ( X ; H X ) G as a two-sided ideal. For example, we can take D ∗ ( X ) G to be the multiplier algebraof C ∗ ( X ; H X ) G , or the C ∗ -algebra generated by C ∗ ( X ; H X ) G and a singleoperator in B ( H X ) G . For a general space, one first needs to show finiteness of the maximal norm to knowthat C ∗ max ( X ; H X ) G is well-defined. emark 3.1. In the reduced case, a natural choice for D ∗ ( X ) G is thealgebra D ∗ red ( X ) G , an equivariant version of the algebra D ∗ ( X ) used in[35]. This is defined as the closure in B ( H X ) of the algebra of operators T ∈ B ( H X ) G with finite propagation such that [ T, f ] is compact for all f ∈ C ( X ).Let ∂ : K ∗ +1 ( D ∗ ( X ) G /C ∗ ( X ) G ) → K ∗ ( C ∗ ( X ) G ) (6)be the boundary map associated to the short exact sequence0 → C ∗ ( X ) G → D ∗ ( X ) G → D ∗ ( X ) G /C ∗ ( X ) G → . Definition 3.2.
Let F ∈ D ∗ ( X ) G , and suppose that F − F ∗ and F − C ∗ ( X ; H X ) G . • If no grading on H X given, consider the projection P = ( F + 1) in D ∗ ( X ) G /C ∗ ( X ; H X ) G and the class [ P ] ∈ K ( D ∗ ( X ) G /C ∗ ( X ; H X ) G ).Then the equivariant coarse index of F isindex G ( F ) = ∂ [ P ] ∈ K ( C ∗ ( X ; H X ) G ) . • If a grading on H X is given that is preserved by C ( X ) and G , andinterchanged by F , let F + be the restriction of F to the even-degreepart of H X . Then F + is invertible modulo C ∗ ( X ) G , and we have[ F + ] ∈ K ( D ∗ ( X ) G /C ∗ ( X ; H X ) G ). The equivariant coarse index of F is index G ( F ) = ∂ [ F + ] ∈ K ( C ∗ ( X ; H X ) G ) . More generally, we will also write index G for the boundary map (6).Let Z ⊂ X be a closed G -invariant subset. Let D ∗ ( X ) G ⊂ B ( H X ) G bea C ∗ -algebra containing C ∗ ( X ; Z, H X ) G as a two-sided ideal. The algebra D ∗ red ( X ) G defined above has this property. Definition 3.3.
Let F ∈ D ∗ ( X ) G , and suppose that F − F ∗ and F − C ∗ ( X ; Z, H X ) G . The equivariant coarse index of F , localised at Z ,index ZG ( F ) ∈ K ∗ ( C ∗ ( X ; Z, H X ) G )is defined analogously to index G ( F ) in Definition 3.2, with C ∗ ( X, H X ) G replaced by C ∗ ( X ; Z, H X ) G everywhere.If Z/G is compact and H X is an admissible module, we writeindex loc G ( F ) := index ZG ( F ) ∈ K ∗ ( C ∗ ( X ) G loc ) = K ∗ ( C ∗ ( G )) , and call this the localised equivariant coarse index of F . Here C ∗ ( G ) denoteseither the reduced or maximal group C ∗ -algebra.13e will also denote the boundary map (6), with C ∗ ( X, H X ) G replacedby C ∗ ( X ; Z, H X ) G , by index ZG , or by index loc G if Z/G is compact and H X isan admissible module. In the rest of this section, we work with the reduced version of the equiv-ariant Roe algebra and specialise to the geometric setting that we are mostinterested in. Let X = M be a Riemannian manifold, and d the Riemanniandistance. Suppose, as before, that G acts properly and isometrically on M .Let E → M be a G -equivariant Hermitian vector bundle. Let D be a G -equivariant, first order elliptic differential operator on E that is essentiallyself-adjoint on L ( E ).To apply Definitions 3.2 and 3.3 to D , we embed L ( E ) into the (geo-metric) admissible module H M := L ( E ) ⊗ L ( G ) of Theorem 2.7. We willillustrate why using admissible modules is necessary in Example 3.8 (seealso Example 2.16).Let χ ∈ C ∞ ( M ) be a cutoff function, in the sense that its support hascompact intersections with all G -orbits, and that for all m ∈ M , Z G χ ( gm ) dg = 1 . (7)The map j : L ( E ) → H M , (8)given 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.It intertwines the actions by C ( M ) on L ( E ) and H M . Define the maps ⊕ , ⊕ B ( L ( E )) → B ( H M ) (9)by identifying operators on L ( E ) with operators on j ( L ( E )) via conjuga-tion by j , and extending them by zero or the identity operator, respectively,on the orthogonal complement of j ( L ( E )) in H M .Let b ∈ C b ( R ) be a normalising function, i.e. an odd function with val-ues in [ − ,
1] such that lim x →∞ b ( x ) = 1. Let D ∗ ( M ; L ( E )) G be a uni-tal ∗ -subalgebra of B ( L ( E )) containing b ( D ), and C ∗ ( M ; L ( E )) G as atwo-sided ideal. Similarly, let D ∗ ( M ) G be a unital ∗ -subalgebra of B ( H M )containing C ∗ ( M ; H M ) G as a two-sided ideal. Suppose that the image of14 ∗ ( M ; L ( E )) G under the map ⊕ D ∗ ( M ) G . This is the case ifwe use multiplier algebras, or the algebra D ∗ red ( M ) G as in Remark 3.1.If F ∈ D ∗ ( M ; L ( E )) G and F ∗ − F and F − C ∗ ( M ; L ( E )) G ,then F ⊕ ∈ D ∗ ( X ) G by assumption, and ( F ⊕ ∗ − F ⊕ F ⊕ − C ∗ ( M ; H M ) G . Hence F ⊕ K ∗ ( C ∗ ( M ; H M ) G ) as inDefinition 3.2. Definition 3.4.
The equivariant coarse index of D isindex G ( D ) := index G ( b ( D ) ⊕ ∈ K ∗ ( C ∗ ( M ; H M ) G ) , where the index on the right hand side is as in Definition 3.2.As in Definition 3.2, index G ( D ) lies in even or odd K -theory dependingon the presence of a grading on E with respect to which D is odd. Again, let Z ⊂ M be a closed, G -invariant subset. Suppose that there is aconstant c > s ∈ Γ ∞ c ( E ) supported outside Z , k Ds k L ≥ c k s k L . Let b ∈ C ∞ ( R ) be an odd, increasing function taking values in {± } on R \ [ − c, c ]. Form the operator b ( D ) by functional calculus. The followingresult by Roe is the basis of the index theory we develop in this paper. Proposition 3.5.
The operator b ( D ) ∈ B ( L ( E )) satisfies b ( D ) − ∈ C ∗ ( M ; Z, L ( E )) G . See Lemma 2.3 in [35]. As above Definition 3.4 This proposition impliesthat b ( D ) ⊕ Definition 3.6.
The equivariant coarse index of D , localised at Z isindex ZG ( D ) := index ZG ( b ( D ) ⊕ ∈ K ∗ ( C ∗ ( X ; Z, H M ) G ) , (10)where the index on the right hand side is as in Definition 3.3.If Z/G is compact, then the index (10) is by definition the localisedequivariant coarse index of D , and denoted byindex loc G ( D ) := index loc G ( b ( D ) ⊕ ∈ K ∗ ( C ∗ red ( G )) .
15s before, index ZG ( D ) and index loc G ( D ) lie in even or odd K -theory de-pending on the presence of a grading.The localised equivariant coarse index of an elliptic operator is the ob-ject we are most interested in here. It is a natural generalisation of theBaum–Connes analytic assembly map [4] from cocompact to non-cocompactactions; see Corollary 4.3. Let us clarify the relevance of using admissible modules in the definition ofthe equivariant coarse index. This will lead to an equivalent definition ofthe localised equivariant coarse index in the graded case, (13) below.The map ⊕ Z , and hence restricts to an injective ∗ -homomorphism ⊕ C ∗ ( M ; Z, L ( E )) G → C ∗ ( M ; Z, H M ) G . (11)We denote the map induced on K -theory by ⊕ C ∗ ( M ; Z, L ( E )) G as a subalgebra of C ∗ ( M ; Z, H M ) G via themap (11), we find that the map ⊕ ⊕ D ∗ ( M ; L ( E )) G /C ∗ ( M ; L ( E )) G → D ∗ ( M ) G /C ∗ ( M ; H M ) G . (12)This induces a homomorphism on odd K -theory, which we still denote by ⊕ Lemma 3.7.
The following diagram commutes: K (cid:0) D ∗ ( M ; L ( E )) G /C ∗ ( M ; Z, L ( E )) G (cid:1) ∂ / / ⊕ (cid:15) (cid:15) K ( C ∗ ( M ; Z, L ( E )) G ) ⊕ (cid:15) (cid:15) K ( D ∗ ( M ) G /C ∗ ( M ; Z, H M ) G ) ∂ / / K ( C ∗ ( M ; Z, H M ) G ) , where the maps ∂ are boundary maps in the respective six-term exact se-quences.Proof. The boundary map can be described explicitly as follows. Supposethat u is an invertible element in D ∗ ( M ; L ( E )) G /C ∗ ( X ; Z, L ( E )) G , andlet v be its inverse. Let U and V respectively be representatives of u and v in D ∗ ( M ; L ( E )) G . Define W = (cid:18) U (cid:19) (cid:18) − V (cid:19) (cid:18) U (cid:19) (cid:18) −
11 0 (cid:19) . ∂ [ u ] = (cid:20) W (cid:18) (cid:19) W − (cid:21) − (cid:20)(cid:18) (cid:19)(cid:21) ∈ K ( C ∗ ( M ; Z, L ( E )) G ) . Using the analogous description of index G ([ u ] ⊕ D be as in Subsection 3.3, and suppose that E has a Z grading withrespect to which D is odd. Letindex Z,L ( E ) G ( D ) := ∂ [ b ( D )] ∈ K ( C ∗ ( M ; Z, L ( E )) G )be the image of [ b ( D )] ∈ K (cid:0) D ∗ ( M ; L ( E )) G /C ∗ ( M ; Z, L ( E )) G (cid:1) under theboundary map. Lemma 3.7 implies that the localised equivariant index of D equals index ZG ( D ) = index Z,L ( E ) G ( D ) ⊕ . (13) Example 3.8.
The importance of using admissible modules in the definitionof the (localised) equivariant coarse index is clear in the simplest case, where G is compact, M is a point, E = V , an irreducible representation of G (withthe trivial grading), and D = 0 V is the zero operator on V . We take Z = M ,so the localised index equals the non-localised index. We have L ( E ) = V ,and [ b ( D )] = [0 V ] ∈ K (cid:0) D ∗ ( M ; V ) G /C ∗ ( M ; V ) G (cid:1) . (14)In that case, Schur’s lemma implies that C ∗ ( M ; V ) G = End( V ) G = C I V , where I V is the identity operator on V . The map j in (8) is now given by j ( v ) = 1 ⊗ v, for v ∈ V , where 1 is the constant function 1 on G . The map ⊕ C ∗ ( M ; V ) G = C I V → M W ∈ ˆ G End( W ) = C ∗ ( G ) There is a technical subtlety in the case where M is a finite set: finite-dimensionalityof V implies that it is not a standard module (see Definition 2.1). The conditions inTheorem 2.7 are not satisfied, since G/K and
M/G are both finite sets. Furthermore,we now have D ∗ ( M ; V ) G = C ∗ ( M ; V ) G if D ∗ ( M ; V ) G is a subalgebra of B ( V ) G . Then D ∗ ( M ; V ) G /C ∗ ( M ; V ) G is the zero algebra. All of these issues can be solved by tensoring V by l ( N ), see also Remark 2.9.
17s given by the inclusion map C I V ֒ → End( V ). At the level of K -theory, themap ⊕ K ( C ∗ ( M ; V ) G ) = Z → R ( G ) = K ( C ∗ ( G ))mapping k ∈ Z to k [ V ] ∈ R ( G ), the representation ring of G . The imageunder index VG of the class (14) is the Fredholm index of 0 V , which is [ V ] ∈ K ( C I V ). Under the identification of this K -theory group with Z , that classis mapped to 1. Henceindex pt ,VG (0 V ) ⊕ V ] ∈ R ( G ) . On the other hand, index G (0 V ⊕
1) is the equivariant Fredholm index of theoperator 0 V ⊕
1, which also equals [ V ] ∈ R ( G ).This example shows that:1. we need to use an admissible module to obtain a single K -theory group K ( C ∗ ( X ) G ) = R ( G ) containing all localised, G -equivariant indices on X ;2. commutativity of the diagram in Lemma 3.7 means that index L ( E ) G ,defined via a natural C ( X )-module, but landing in a non-canonical K -theory group, determines the localised equivariant index with valuesin K ( C ∗ ( G )). In this example, index pt ,VG (0 V ) is just the integer 1when viewed as an element of Z = K ( C I V ), but the representationtheoretic information about this index is encoded in the map ⊕ Operators D as in Subsection 3.3 occur naturally in at least three settings. Callias-type operators
First of all, let ˜ D be a Dirac-type operator on E . Let Φ be a G -equivariantvector bundle endomorphism of E such that ˜ D Φ + Φ ˜ D is a vector bundleendomorphism, and ˜ D Φ + Φ ˜ D + Φ ≥ c (15)outside a cocompact subset Z ⊂ M , for a constant c >
0. This can, forexample, be guaranteed by constructing Φ from projections in the Higsoncorona algebra as in [17]. (In that case, the pointwise norm of ˜ D Φ+Φ ˜ D goes18o zero at infinity, while the norm of Φ goes to one.) Then the Callias-typeoperator D = D Φ := ˜ D + Φ (16)has the properties in Subsection 3.3.If G is the trivial group, i.e. in the non-equivariant case, index theory ofthese operators was studied and applied in various places [9, 11, 28]. Thecoarse geometric viewpoint we develop in this paper could already be use-ful in the case of trivial groups. Possibly more useful in the non-equivariantsetting is to consider the lift of a Callias-type operator D Φ on a manifold M ,with fundamental group Γ, to a Γ-equivariant Callias-type operator on theuniversal cover of M . The localised equivariant coarse index in K ∗ ( C ∗ red (Γ))of this lift is a more refined invariant than the Fredholm index of D Φ it-self. This could for example yield more refined obstructions to Riemannianmetrics of positive scalar curvature. In future work, we intend to show thatthe Fredholm index of D Φ can be recovered from the localised equivariantcoarse index of its lift via an application of the von Neumann trace, or thesummation trace in the context of maximal group C ∗ -algebras.In [13], Callias-type index theory is extended to operators on bundles ofHilbert modules over C ∗ -algebras A , with indices in K ∗ ( A ). If A = C ∗ red ( G )this seems related to the index we study here; in fact we suspect that thetwo coincide if G is the fundamental group of M/G , and M is its universalcover, and the Hilbert module bundle in question is constructed from thenatural bundle M × G C ∗ red ( G ) → M/G . In the general equivariant case,index theory of Callias-type operators was developed in [17]; we will see inTheorem 4.2 that Definition 3.3 generalises the index of [17].
Positive curvature at infinity
Secondly, suppose that D is a Dirac-type operator satisfying a Weitzenb¨ock-type formula D = ∇ ∗ ∇ + R, for a vector bundle endomorphism R satisfying R ≥ c outside Z . Then D satisfies the conditions in Subsection 3.3, and therefore has a well-definedindex in K ( C ∗ red ( G )). The case where G is trivial (so that Z is compact)was studied by Gromov and Lawson [16] and applied to questions aboutRiemannian metrics of positive scalar curvature. The case where G is trivialand Z may be noncompact was treated by Roe [35] using coarse geometry.19 anifolds with boundary Finally, let ˜ M be a Riemannian manifold with boundary, on which G actsproperly, isometrically and cocompactly. Suppose that a neighbourhood U of ∂ ˜ M is G -equivariantly and isometrically diffeomorphic to a collar ∂ ˜ M × [0 , ε ). Let ˜ E → ˜ M be a G -equivariant, Z -graded, Hermitian vector bundle,and a module over the Clifford bundle of ˜ M . Suppose that ˜ E | U ∼ = ˜ E | ∂ ˜ M × [0 , ε ) as equivariant, Hermitian vector bundles. Suppose that ˜ D is a Dirac-type operator on M , and that on U it is of the form D | U = σ ◦ (cid:0) ∂∂t + D ∂ ˜ M (cid:1) , (17)where σ : ˜ E + | ∂ ˜ M → ˜ E − | ∂ ˜ M is an equivariant vector bundle isomorphism, t is the coordinate in [0 , ε ), and D ∂ ˜ M is a Dirac operator on ˜ E + | ∂ ˜ M .Form M by attaching a cylinder ∂ ˜ M × [0 , ∞ ) to ˜ M . Extend the Rie-mannian metric and the action by G to M in the natural way. Let E → M be the natural extension of ˜ E , and let D be the extension of ˜ D to E equalto (17) on ∂ ˜ M × [0 , ∞ ).Suppose that D ∂ ˜ M is invertible. Then D ∂ ˜ M ≥ c for some c >
0, and D ≥ c outside Z = ˜ M . Hence D satisfies the conditions in Subsection3.3. The index of Definition 3.6 is now an equivariant Atiyah–Patodi–Singertype index for proper actions, and reduces to the original APS index if G is trivial. An index theorem for this index is proved in [24]. As in the caseof Callias operators, a special case is the lift of an operator on a compactmanifold with boundary to the universal cover, in which case one obtains arefinement of the Atiyah–Patodi–Singer index in the K ∗ ( C ∗ red ( π )), where π is the fundamental group of the compact manifold. We will show that the index of Definition 3.6 generalises the equivariantindex of Callias-type operators introduced in [17] (Theorem 4.2). As ap-plications, we obtain results on existence and non-existence of Riemannianmetrics of positive scalar curvature in Subsection 4.2, and discuss a localisedversion of the Baum–Connes conjecture (Conjecture 4.8).
Suppose that D = D Φ is a Callias-type operator as in (16). Let E denotethe Hilbert C ∗ red ( G )-module defined by completing the space Γ ∞ c ( E ) with20espect to the C c ( G )-valued inner product h s, t i ( g ) := h s, gt i L ( E ) and the right action of C c ( G ) defined by s · b := Z G g − ( b ( g ) s ) dg, for s , s ∈ Γ ∞ c ( E ) and g ∈ G . One can find a continuous, G -invariant,cocompactly supported function f on M such that D + f is invertible inthe sense of the Hilbert C ∗ red ( G )-modules E j as in Definition 1 in [17]. Wecan then form the normalised G -Callias-type operator F := D Φ ( D + f ) − / . (18)Then F lies in the C ∗ -algebra L ( E ) of bounded adjointable operators on E .It was shown in Theorem 25 in [17] that F is invertible modulo the algebra K ( E ) of compact operators on E , and thus defines a class[ F ] ∈ K ( L ( E ) / K ( E )) . Here, as in [17], we assume that E is Z -graded and D Φ is odd with respectto the grading, and the above K -theory class is defined in terms of the evenpart of F , as in the second point in Definition 3.2.Let ∂ : K ( L ( E ) / K ( E )) → K ( K ( E )) = K ( C ∗ red ( G )) (19)be the boundary map associated to the short exact sequence0 → K ( E ) → L ( E ) → L ( E ) / K ( E ) → . In (19), we have used the Morita equivalence K ( E ) ∼ C ∗ red ( G ).In [17], the following index was constructed and applied. Definition 4.1.
The equivariant Callias-index of D Φ isindex CG ( D Φ ) := ∂ [ F ] ∈ K ( C ∗ red ( G )) . One of our main results in this paper is that this index is a special case ofthe localised equivariant index. This gives a new approach to Callias indextheory.
Theorem 4.2.
We have index loc G ( D Φ ) = index CG ( D Φ ) ∈ K ( C ∗ red ( G )) . (20)21his result is proved in Section 6.If M/G is compact, then we may take Φ = 0, and index CG equals the ana-lytic assembly map [4]. Therefore, Theorem 4.2 has the following immediateconsequence. Corollary 4.3. If M/G is compact, then the localised equivariant index ofan elliptic operator is its image under the analytic assembly map.
Note that if
M/G is compact, then the localised equivariant coarse indexequals the usual equivariant coarse index. In the case of discrete groups,Corollary 4.3 is the well-known fact that the equivariant coarse index for aproper action equals the analytic assembly map for such groups [33].
In the second special case in Subsection 3.5, if R is uniformly positive, i.e. Z = ∅ , then its localised coarse index vanishes by standard arguments.Thus index loc G ( D ) ∈ K ∗ ( C ∗ red ( G )) is an obstruction to G -invariant Rieman-nian metrics of positive scalar curvature. There are many techniques forextracting more concrete, numerical obstructions from this K -theory class,such as pairing with traces and higher cyclic cocycles on (smooth subalge-bras of) C ∗ red ( G ).In the case where Z/G is non-compact, the localised equivariant coarseindex allows us to use the following method to find obstructions to G -invariant Riemannian metrics of positive scalar curvature. This generalisesthe comments at the start of Section 3 in [35]. Proposition 4.4.
Let M be a complete Riemannian manifold, with a proper,isometric action by a locally compact group G . Let D be a Dirac-type opera-tor whose curvature term R in the Weitzenb¨ock-type formula D = ∆ + R isuniformly positive outside a G -invariant subset Z ⊂ M , for which the inclu-sion map C ∗ ( M ; Z ) G → C ∗ ( M ) G induces the zero map on K -theory, with re-spect to an admissible C ( M ) -module H M and its restriction H Z := Z H M .Then index G ( D ) = 0 . When Z is cocompact, it is clear that the inclusion K ∗ ( C ∗ ( M ; Z, H M ) G ) = K ∗ ( C ∗ ( Z, Z H M ) G ) . induces the identity map on K theory. More generally, we expect that, as inthe discrete group case, the equivariant Roe algebras of coarsely equivalent See the forthcoming book [38]. K -theory, and hence that this identityholds for general Z (see e.g. Lemma 1 in Section 5 of [21], or Proposition6.4.7 in [20] for the non-equivariant case.) Then the condition on the set Z in the above proposition is satisfied for example if Z is contained in asubset Y ⊂ M such that K ∗ ( C ∗ ( Y ) G ) = 0. In future work, we aim to proveindex theorems that allow us to deduce concrete topological obstructions topositive scalar curvature from Proposition 4.4.We now turn to an existence result. Recall the following theorem from[1], which we need only for Lie groups: Theorem 4.5 (Abels) . If M is a proper G -manifold, where G is an almostconnected Lie group, then there exists a global slice N which is a K -manifold,in the G -manifold M , where K is a maximal compact subgroup of G . By this theorem, M is G -equivariantly diffeomorphic to G × K N . Theorem 4.6.
Let G be an almost connected Lie group, and let K be amaximal compact subgroup of G . If N is a bounded geometry manifold witha K -invariant Riemannian metric of uniform positive scalar curvature, then M = G × K N is a bounded geometry manifold with a G -invariant Rieman-nian metric of uniform positive scalar curvature. The Baum–Connes conjecture [4] describes K ∗ ( C ∗ red ( G )) in terms of equivari-ant indices of elliptic operators for cocompact actions by G . The surjectivitypart of this conjecture is a particularly hard problem. Using the localisedequivariant index of Definition 3.3, we will formulate a localised version ofBaum–Connes surjectivity. We show that this is implied by Baum–Connessurjectivity in the usual sense (Proposition 4.9 below). It is therefore aweaker statement - and potentially easier to prove because one is allowed touse equivariant indices for non-cocompact actions - but which neverthelessdescribes the group K ∗ ( C ∗ red ( G )).Let D ∗ red ( X ) G be the algebra defined in Subsection 3.1. Definition 4.7.
The localised equivariant K -homology of X is K G ∗ ( X ) loc := K ∗ +1 ( D ∗ red ( X ) G /C ∗ ( X ) G loc ) . This terminology is motivated by Paschke duality (see e.g. page 85 of [34]and Theorem 8.4.3 in [20]), which implies that K G ∗ ( X ) loc equals the usual23quivariant K -homology of X in the opposite degree, if X/G is compact.The index of Definition 3.3 defines a localised equivariant indexindex loc G : K G ∗ ( X ) loc → K ∗ ( C ∗ red ( G )) . Now let EG be a universal example for proper G -actions [4]. Conjecture 4.8 (Localised Baum–Connes surjectivity) . The map index loc G : K G ∗ ( EG ) loc → K ∗ ( C ∗ red ( G )) is surjective. Recall that the representable equivariant K -homology of X is RK G ∗ ( X ) := lim −→ Z K ∗ G ( Z ) , where Z runs over the G -invariant closed subsets of X such that Z/G iscompact. The Baum–Connes conjecture is the statement that the analyticassembly map µ G : RK G ∗ ( EG ) → K ∗ ( C ∗ red ( G ))is bijective. Proposition 4.9.
Surjectivity of µ G implies Conjecture 4.8. The converse of Proposition 4.9 is directly related to the question ofwhether the equivariant localised index of Definitions 3.3 and 3.6 lands inthe image of the Baum–Connes assembly map. This question was posedfor the equivariant Callias index in [17], and is open in general. See alsoRemark 7.3 below.
In Subsections 5.1–5.3, we prove Theorem 2.7, which guarantees the exis-tence of geometric admissible modules. We then use this in Subsection 5.4to prove Theorem 2.11. 24 .1 An isomorphism of G -representations Since G is a non-compact group, we decompose the space X using Palais’local slice theorem [30]. Let K x denote the stabiliser of a point x ∈ X .This result states that for every point x ∈ X , there is a K x -invariant subset Y x ⊂ X such that the action map G × Y x → X descends to a G -equivarianthomeomorphism from G × K x Y x onto a G -invariant open neighbourhood W x of x . Note that the stabiliser K x is compact, since the action is proper.Consider a covering of X by sets of the form G × K j Y j , for compactsubgroups K j < G and compact, K j -invariant slices Y j ⊂ X . Suppose thatthe intersections between these sets have measure zero. For each j , fix a K -invariant measure dy j on Y j . Together with the Haar measure dg , theyinduce a G -invariant measure dx on X . We will call such a measure inducedfrom slices . Such measures are natural choices; see for example Lemma 4.1in [23].We will use the following fact, whose proof is straightforward. Lemma 5.1 (Fell absorption) . If π : G → U( H ) is a unitary representation,and λ : G → U( L ( G )) is the left-regular representation, then the map Φ : L ( G ) ⊗ H → L ( G ) ⊗ H , defined by Φ( f )( g ) = π ( g ) f ( g ) , for f ∈ L ( G, H ) and g ∈ G , is a unitary isomorphism intertwining therepresentations λ ⊗ π and λ ⊗ . Suppose first that X = G × K Y , for a K -space Y . Let dx be a measureon X be induced from the measure dg on G and a K -invariant measure dy on Y . Consider the measure d ( Kg ) on K \ G . Choose a measurable section φ : K \ G → G . Lemma 5.2.
The map ψ : X × G = ( G × K Y ) × G ∼ = G × K \ G × Y, given by ψ ([ g, y ] , h ) = ( h ( φ ( Kg − h ) − ) , Kg − h, ( φ ( Kg − h ) h − g ) y ) is a G -equivariant, measurable bijection. It relates the measures dx × dg and dg × d ( Kg ) × dy to each other. roof. There is a K -equivariant isomorphism of measure spaces (by whichwe mean a measurable bijection relating the given measures on the twospaces) G → K × ( K \ G ) , g ( g ( φ ( Kg ) − ) , Kg ) . Thus we have a G -equivariant isomorphism of measure spaces G × K ( G × Y ) ∼ = G × K ( K × K \ G × Y ) , [( g, ( h, y ))] [( g, h ( φ ( Kh ) − ) , Kh, y )] . Combining this with the K -equivariant isomorphism K × Y → K × Y, ( k, y ) ( k, k − y )(where K acts diagonally on the left and only on the first factor on theright), this gives a G -equivariant isomorphism of measure spaces G × K ( G × Y ) ∼ = G × K ( K × K \ G × Y ) , [( g, ( h, y ))] [( g, h ( φ ( Kh ) − ) , Kh, ( φ ( Kh ) h − ) y )] , where K now acts trivially on Y on the right. Using the identification G × K K ∼ = G , [( g, k )] gk , we get the G -equivariant isomorphism G × K ( G × Y ) ∼ = G × K \ G × Y, [( g, ( h, y ))] ( gh ( φ ( Kh ) − ) , Kh, ( φ ( Kh ) h − ) y ) . Here G acts only on the first factor of both sides. Note that( G × K Y ) × G ∼ = G × K ( G × Y ) , ([( g, y )] , h ) [( g, ( g − h, y ))](where G acts diagonally on the left and only on the first factor on theright). The first claim then follows.Set H := L ( K \ G ) ⊗ L ( Y ). Pulling back functions along the map ψ inLemma 5.2 induces a unitary, G -equivariant isomorphism ψ ∗ : L ( G ) ⊗ H → L ( X ) ⊗ L ( G ) . Suppose that
X/G is compact. Then, in general, X is a finite unionof sets of the form G × K j Y j for compact subgroups K j < G and compact26ubsets Y j . These can be chosen so that the overlaps between these setshave measure zero. Lemma 5.2 yields isomorphisms ψ ∗ j : L ( G ) ⊗ H j → L ( G × K j Y j ) ⊗ L ( G ) , where H j = L ( K j \ G ) ⊗ L ( Y j ). They combine into a global isomorphismΨ : L ( G ) ⊗ H ∼ = −→ M j L ( G × K j Y j ) ⊗ L ( G ) ∼ = L ( X ) ⊗ L ( G ) , (21)where H = L j H j . We have proved the following: Proposition 5.3.
There exists a G -equivariant unitary isomorphism Ψ : L ( G ) ⊗ H → L ( X ) ⊗ L ( G ) for a separable Hilbert space H . The space H is infinite-dimensional if G/K or X/G is infinite. X and in G Next, we show that Ψ coarsely relates propagation on L ( G ) ⊗ H with respectto C ( G ) to propagation on L ( X ) ⊗ L ( G ) with respect to C ( X ). Proposition 5.4.
Let Ψ be the isomorphism in Proposition 5.3. An opera-tor T on L ( X ) ⊗ L ( G ) has finite propagation in X if and only if Ψ − ◦ T ◦ Ψ has finite propagation in G . To show this, we suppose first that X consists of just one slice. That is, X = G × K Y , for a compact K -space Y . We will reduce the general case tothis case. Let diam( K ) be the diameter of K . Lemma 5.5.
Let ψ : G × K Y × G → G × K \ G × Y be the bijective map from Lemma 5.2. Let ψ be its first component, mappinginto G . Then for all g, g ′ , h, h ′ ∈ G and y, y ′ ∈ Y , d G ( g, g ′ ) − K ) ≤ d G ( ψ ([ g, y ] , h ) , ψ ([ g ′ , y ′ ] , h ′ )) ≤ d G ( g, g ′ )+2 diam( K ) . Proof. If g, h ∈ G and y ∈ Y , then ψ ([ g, y ] , h ) = hφ ( Kg − h ) − , Here we have taken the bundle E → X to be the trivial line bundle. The generalcase can be proved completely analogously. φ : K \ G → G is a section. This means that there is a k ∈ K suchthat φ ( Kg − h ) = kg − h . Hence ψ ([ g, y ] , h ) = gk − . Let g, g ′ , h, h ′ ∈ G and y, y ′ ∈ Y . Then we have just seen that there are k, k ′ ∈ K such that d G ( ψ ([ g, y ] , h ) , ψ ([ g ′ , y ′ ] , h ′ )) = d G ( gk − , g ′ k ′− ) . This lies in the range specified by the triangle inequality and left invarianceof d . Lemma 5.6.
For all s > there are r, r ′ > such that for all g, g ′ ∈ G and y, y ′ ∈ Y , d G ( g, g ′ ) ≤ s ⇒ d ( gy, g ′ y ′ ) ≤ r.d ( gy, g ′ y ′ ) ≤ s ⇒ d G ( g, g ′ ) ≤ r ′ . Proof.
Let s > r := max { d ( gy, y ′ ); y, y ′ ∈ Y, g ∈ G, d G ( g, e ) ≤ s } . Here we use compactness of Y . Then for all g, g ′ ∈ G with d G ( g, g ′ ) ≤ s , wehave d G ( g ′− g, e ) ≤ s , so for all y, y ′ ∈ Y , d ( gy, g ′ y ′ ) = d ( g ′− gy, y ′ ) ≤ r. To prove the second claim, note that properness of the action by G on X and compactness of Y imply that the set A s := { g ∈ G ; gY ∩ Pen(
Y, s ) = ∅} is compact (with notation as in (3)). Set r ′ := max { d G ( g, e ); g ∈ A s } . Then for all g, g ′ ∈ G and y, y ′ ∈ Y with d ( gy, gy ′ ) ≤ s , we have g ′− g ∈ A s ,so d G ( g, g ′ ) ≤ r ′ . Lemma 5.7.
If an operator T on L ( X ) ⊗ L ( G ) has finite propagation in X , then Ψ − ◦ T ◦ Ψ has finite propagation in G . roof. Suppose that T is an operator on L ( X ) ⊗ L ( G ) with finite prop-agation s in X . By the second part of Lemma 5.6, there is an r > g, g ′ ∈ G and y, y ′ ∈ Y , d G ( g, g ′ ) ≥ r ⇒ d ( gy, g ′ y ′ ) ≥ s + 1 . Let χ , χ ∈ C c ( G ) be given, with d G (supp( χ ) , supp( χ )) ≥ r + 2 diam( K ).For j = 1 ,
2, let g j ∈ supp( χ j ), h j ∈ G and y j ∈ Y be given. Write(˜ g j ˜ y j , ˜ h j ) = ψ − ( g j , Kh j , y j ) , for ˜ g j , ˜ h j ∈ G and ˜ y j ∈ Y . Then by Lemma 5.5, d G (˜ g , ˜ g ) ≥ d G ( g , g ) − K ) ≥ r. So d (˜ g ˜ y , ˜ g ˜ y ) ≥ s + 1. Let π X : X × G → X be the projection onto thefirst factor. We have just seen that d (cid:0) π X ( ψ − (supp χ × K \ G × G )) , π X ( ψ − (supp χ × K \ G × G )) (cid:1) ≥ s + 1 . Hence we can choose ϕ j ∈ C c ( X ), for j = 1 ,
2, such that ϕ j ≡ π X ( ψ − (supp χ j × K \ G × G )), and d (supp( ϕ ) , supp( ϕ )) ≥ s. We conclude that χ (Ψ − ◦ T ◦ Ψ) χ = Ψ − ◦ ( ψ ∗ ( χ ⊗ K \ G × Y ) ϕ T ϕ ψ ∗ ( χ ⊗ K \ G × Y )) ◦ Ψ = 0 , since ϕ T ϕ = 0. Lemma 5.8.
If an operator ˜ T on L ( G ) ⊗ H has finite propagation in G ,then Ψ ◦ ˜ T ◦ Ψ − has finite propagation in X .Proof. Let ˜ T be an operator on L ( G ) ⊗ H with finite propagation s in G .The first part of Lemma 5.6 implies that there is an r > g, g ′ ∈ G and y, y ′ ∈ Y , d ( gy, g ′ y ′ ) ≥ r ⇒ d G ( g, g ′ ) ≥ s + 1 + 2 diam( K ) . (22)For j = 1 ,
2, let ϕ j ∈ C c ( X ) be such that d (supp( ϕ ) , ϕ ) ≥ r . Let π G : G × K \ G × Y → G be the projection onto the first factor. By Lemma 5.5, wehave, for all g j , h j ∈ G and y j ∈ Y such that g j y j ∈ supp( ϕ j ), d G (cid:0) π G ( ψ ( g y , h )) , π G ( ψ ( g y , h )) (cid:1) ≥ d G ( g , g ) − K ) ≥ s + 1 , χ j ∈ C c ( G ) such that χ j ≡ π G ( ψ (supp( ϕ j ) × G )) and d G (supp( χ ) , supp( χ )) ≥ s . Then ϕ (Ψ ◦ ˜ T ◦ Ψ − ) ϕ = Ψ ◦ (( ψ − ) ∗ ( ϕ ⊗ G ) χ ˜ T χ ( ψ − ) ∗ ( ϕ ⊗ G )) ◦ Ψ − = 0 , since χ ˜ T χ = 0. Proof of Proposition 5.4. If X = G × K Y for a single, compact slice Y ⊂ X ,then the claim is precisely Lemmas 5.7 and 5.8.In the general case when the cocompact space X consists of finitely manyslices, L ( X ) ⊗ L ( G ) is a finite direct sum L i L ( G ) ⊗ H i as in (21). Anoperator T on this space can be written as a finite matrix ( T i,j ), where eachentry T i,j is an operator T i,j : L ( G ) ⊗ H i → L ( G ) ⊗ H j . The result then follows from the case of a single slice.
The remaining step in the proof of Theorem 2.7 is to show that Ψ relates localcompactness of operators on H X with respect to C ( X ) to local compactnessof operators on L ( G ) ⊗ H with respect to C ( G ). Proposition 5.9.
A bounded operator T on H X is locally compact withrespect to the action by C ( X ) if and only if the bounded operator Ψ − ◦ T ◦ Ψ on L ( G ) ⊗ H is locally compact with respect to the action by C ( G ) .Proof. First, suppose that X = G × K Y for a single slice Y .Suppose T is a bounded operator on L ( X ) ⊗ L ( G ) that is locally com-pact with respect to multiplication by C ( X ). Let χ ∈ C c ( G ) be given.As in the proof of Lemma 5.7, Lemmas 5.5 and 5.6 imply that the sub-set π X ( ψ − (supp χ × K \ G × G )) of X is bounded. Hence we can choose φ ∈ C c ( X ) such that φ ≡ π X ( ψ − (supp χ × K \ G × G )). Thus(Ψ − ◦ T ◦ Ψ) χ = Ψ − ◦ ( T φψ ∗ ( χ ⊗ K \ G × Y )) ◦ Ψ ∈ K ( L ( G ) ⊗ H ) , since T φ ∈ K ( L ( X ) ⊗ L ( G )).Now suppose ˜ T = Ψ − ◦ T ◦ Ψ is a bounded operator on L ( G ) ⊗ H that islocally compact with respect to the multiplicative action of C ( G ) on the firstfactor. Let φ ∈ C c ( X ) be given. As in the proof of Lemma 5.8, Lemmas 5.530nd 5.6 imply that the subset π G ( ψ (supp( φ ) × G )) of G is bounded. Hencewe can choose χ ∈ C c ( G ) such that χ ≡ π G ( ψ (supp( φ ) × G )). Then T φ = (Ψ ◦ ˜ T ◦ Ψ − ) φ = Ψ ◦ ( ˜ T χ ( ψ − ) ∗ ( φ ⊗ G )) ◦ Ψ − ∈ K ( L ( X ) ⊗ L ( G )) , since ˜ T χ ∈ K ( L ( G ) ⊗ H ) . The general case for cocompact X and more than one slice follows fromthe single slice case as in the proof of Proposition 5.4.Theorem 2.7 is the combination of Propositions 5.4 and 5.9. Under theassumption in Theorem 2.7 that G/K or X/G is infinite, the space H inProposition 5.3 is infinite-dimensional. C ∗ -algebras The reduced and maximal equivariant Roe algebras can alternatively bedescribed in terms of continuous Schwartz kernels of operators. We workthis out in detail this subsection and use it to prove Theorem 2.11.Let X and G be as before. Suppose that X/G is compact. Let H X be any admissible equivariant C ( X )-module over X . In this subsection, wewill always identify H X with L ( G ) ⊗ H for an infinite-dimensional separableHilbert space H , using the isomorphism Ψ in Definition 2.4. Definition 5.10.
Let C ∗ ker ( X ) G denote the algebra of bounded operatorson H X defined by continuous G -invariant Schwartz kernels κ : G × G → K ( H )that have finite propagation in G .By the ‘if’ part of Proposition 5.4, we have C ∗ ker ( X ) G ⊂ C ∗ alg ( X ) G . Ourgoal is to prove the following proposition. Proposition 5.11. C ∗ ker ( X ) G is dense in C ∗ alg ( X ) G with respect to the op-erator norm on H X . Because of this proposition, the Roe algebra C ∗ ( X ) G can alternativelybe defined as the closure of C ∗ ker ( X ) G in B ( H X ). This immediately impliesTheorem 2.11, since C ∗ ker ( X ) G ∼ = C c ( G ) ⊗ K ( H )via the isomorphism sending κ ∈ C ∗ ker ( X ) G to the map g κ ( g − , e ).31et T ∈ C ∗ G, alg ( X ). We will prove Proposition 5.11 by showing that T can be approximated by elements of C ∗ ker ( X ) G in the operator norm.Fix χ ∈ C ∞ c ( G ) such that for all h ∈ G , Z G χ ( g − h ) dg = 1 . Let H be as in Definition 2.4. By the ‘only if’ part of Proposition 5.4, T hasfinite propagation in G . So there exist functions χ , χ ∈ C c ( G ) such that χ T χ = T χ,χχ = χ when T is viewed as an operator on L ( G ) ⊗ H .Let { e j } ∞ j =1 be a Hilbert basis for H X ∼ = L ( G, H ) such that e j ∈ C c ( G, H ) for every j . Let { e k } ∞ k =1 be the dual basis. We view e k as theelement of C c ( G, H ∗ ) such that for all g ∈ G and v ∈ H , e k ( g )( v ) = ( e k ( g ) , v ) H . By the definition of C ∗ G, alg ( X ) and Proposition 5.9, we have T χ ∈ K ( H X ).Thus we can write T χ = X j,k a jk e j ⊗ e k (23)for some constants a jk , with the sum converging in operator norm in B ( L ( G ) ⊗ H ). Define T jk ∈ B ( L ( G ) ⊗ H ) to be the operator given by the Schwartzkernel κ jk : G × G → K ( H ) , ( h, h ′ ) a jk Z G χ ( g − h ) χ ( g − h ′ ) e j ( g − h ) ⊗ e k ( g − h ′ ) dg, (24)where h, h ′ ∈ G . Since e j ( g ) ⊗ e k ( g ′ ) (for g, g ′ ∈ G ) is a finite-rank operatoron H and the integrand in (24) is compactly supported, we find that indeed κ jk ( h, h ′ ) ∈ K ( H ) for all h, h ′ ∈ G . Furthermore, κ jk is continuous, G -invariant, and has finite propagation in G . Lemma 5.12.
For every f ∈ L ( G, H ) and h ∈ G , ( T f )( h ) = ∞ X j,k =1 T jk f ( h ) . roof. Let f ∈ L ( G, H ). Then for every g ∈ G we have T ◦ ( g · χ ) = g ( T χ ) g − = gχ T χχ g − . Thus for all h ∈ G ,( T f )( h ) = Z G ( T ( g · χ ) f ) ( h ) dg = Z G (cid:0) ( gχ T χχ g − ) f (cid:1) ( h ) dg = X j,k a jk Z G χ ( g − h ) e j ( g − h ) (cid:18)Z G (cid:0) e k ( l ) , χ ( l ) f ( gl ) (cid:1) H dl (cid:19) dg = X j,k Z G κ jk ( h, m ) f ( m ) dm, where we substitute m = gl . Note that all integrands are continuous andcompactly supported, so we may indeed interchange integrals and sums. Lemma 5.13.
The sum ∞ X j,k =1 T jk converges in B ( L ( G ) ⊗ H ) with respect to the operator norm.Proof. We have for all j, k ∈ N and f ∈ C c ( G, H ), T jk f = a jk Z G ( g · ( χ e j ))( g · ( χ e k ) , f ) L ( G,H ) dg = a jk Z G ( g · χ )( g ◦ ( e j ⊗ e k ) ◦ g − )( g · χ ) f dg. (25)Hence for all M, N, M ′ , N ′ ∈ N with M ≤ M ′ and N ≤ N ′ , (cid:13)(cid:13)(cid:13) M ′ X j = M N ′ X k = N T jk f (cid:13)(cid:13)(cid:13) L ( G,H ) = (cid:13)(cid:13)(cid:13) M ′ X j = M N ′ X k = N a jk Z G ( g · χ )( g ◦ ( e j ⊗ e k ) ◦ g − )( g · χ ) f dg (cid:13)(cid:13)(cid:13) L ( G,H ) = (cid:13)(cid:13)(cid:13)Z G ( g · χ ) (cid:16) g ◦ (cid:16) M ′ X j = M N ′ X k = N a jk e j ⊗ e k (cid:17) ◦ g − (cid:17) ( g · χ ) f dg (cid:13)(cid:13)(cid:13) L ( G,H ) . T M ′ ,N ′ M,N := M ′ X j = M N ′ X k = N a jk e j ⊗ e k . Define F : G → L ( G, H ) by F ( g ) = ( g · χ )( g ◦ T M ′ ,N ′ M,N ◦ g − )( g · χ ) f, for g ∈ G . If g, g ′ ∈ G , and ( F ( g ) , F ( g ′ )) L ( G,H ) = 0, then g supp( χ ) ∩ g ′ supp( χ ) = ∅ . By properness of the action, this means that g − g ′ lies in a compact set S ⊂ G , only depending on χ . By Lemma 1.5 in [14], this implies that (cid:13)(cid:13)(cid:13)Z G F ( g ) dg (cid:13)(cid:13)(cid:13) L ( G,H ) ≤ vol( S ) Z G k F ( g ) k L ( G,H ) dg. Hence, since G acts unitarily on L ( G, H ), (cid:13)(cid:13)(cid:13) M ′ X j = M N ′ X k = N T jk f (cid:13)(cid:13)(cid:13) L ( G,H ) ≤ vol( S ) k χ k ∞ k T M ′ ,N ′ M,N k B ( L ( G,H )) Z G k ( g · χ ) f k L ( G,H ) dg ≤ vol( S ) k χ k ∞ k T M ′ ,N ′ M,N k B ( L ( G,H )) k χ k G k f k L ( G,H ) , where k χ k ∞ := max g ∈ G || χ ( g ) || B ( H ) , k χ k G := s max h ∈ G Z G || χ ( g − h ) || B ( H ) dg. We conclude that the operator M ′ X j = M N ′ X k = N T jk on L ( G, H ) is bounded, with norm at mostvol( S ) / k χ k ∞ k χ k G k T M ′ ,N ′ M,N k B ( L ( G,H )) . Since the sum (23) converges in the operator norm and B ( L ( G, H )) is com-plete, the claim follows. 34 roof of Proposition 5.11.
By Lemmas 5.12 and 5.13, we have T = ∞ X j,k =1 T jk , where the sum converges in the operator norm. Hence C ∗ G, ker ( X ) is dense in C ∗ G, alg ( X ). In this section, we prove Theorem 4.2, showing that the equivariant indexof a G -Callias-type operator, as defined in [17], identifies naturally with itslocalised equivariant index given by Definition 3.6. We begin in Subsec-tion 6.1 by relating the equivariant coarse index defined in Subsection 3.1for cocompact actions to the usual G -equivariant index obtained throughthe assembly map, before relating the localised equivariant index to thenon-cocompact G -equivariant index in Subsection 6.2. With respect to thenotation in Subsection 3.1, we are working with D ∗ ( X ) G = M ( C ∗ ( X ) G ) or M ( C ∗ ( X ) G loc ), depending on context.The results in the first two subsections of this section are of a generalnature and apply to both the maximal and reduced versions of the index,and we will use C ∗ ( G ) will denote either C ∗ red ( G ) or C ∗ max ( G ), and C ∗ ( X ) G (resp. C ∗ ( X ) G loc ) for either the reduced or maximal version of the Roe algebra(resp. localised Roe algebra). Suppose that X is G -cocompact. Equip the dense subspace C c ( G, L ( E ))of L ( E ) ⊗ L ( G ) with the C c ( G )-valued inner product h s, t i ( g ) := h s, gt i L ( E ) ⊗ L ( G ) and the right action of C c ( G ) defined by s · b := Z G g − ( b ( g ) s ) dg. Taking the completion gives rise to a Hilbert C ∗ ( G )-module E C ∗ ( G ) . Lemma 6.1. E C ∗ ( G ) is isomorphic to the standard Hilbert C ∗ ( G ) -module C ∗ ( G ) ⊗ H , for a separable Hilbert space H . roof. Let H be the Hilbert space in the isomorphism L ( E ) ⊗ L ( G ) ∼ = L ( G ) ⊗ H (26)from Theorem 5.3. Let E ′ C ∗ ( G ) denote the Hilbert C ∗ ( G )-module completionof C c ( G ) ⊗ H with respect to the C c ( G )-valued inner product and right C c ( G )-action h s, t i ( g ) := h s, gt i L ( G ) ⊗ H , s · b := Z G g − ( b ( g ) s ) dg, where s, t ∈ L ( G, H ) . Then the isomorphism (26), restricted to the densesubspace C c ( G, L ( E )) ⊆ L ( E ) ⊗ L ( G ), extends to an isomorphism E C ∗ ( G ) ∼ = E ′ C ∗ ( G ) . Further, one can check that the map E ′ C ∗ ( G ) → C ∗ ( G ) ⊗ H, s ˜ s, where ˜ s takes g s ( g − ) , is an isometric isomorphism of E ′ C ∗ ( G ) onto thestandard Hilbert C ∗ ( G )-module equipped with its usual inner product andright C ∗ ( G )-action.Using Lemma 6.1, we can write down an identification U : K ( E C ∗ ( G ) ) ∼ = K ( C ∗ ( G ) ⊗ H ) . Now let C ∗ ( X ) G denote the G -equivariant Roe algebra on L ( E ) ⊗ L ( G ) ∼ = L ( G ) ⊗ H , and let C ∗ ker ( X ) G be its dense subalgebra of G -invariantkernels from Definition 5.10. Let W denote the identification W : C ∗ ker ( X ) G ∼ = C c ( G ) ⊗ K ( H )below Proposition 5.11. This map identifies C ∗ ker ( X ) G with a subalgebra ofthe compact operators on the standard Hilbert C ∗ ( G )-module C ∗ ( G ) ⊗ H : W : C ∗ ker ( X ) G ∼ = −→ C c ( G ) ⊗ K ( H ) | {z } ⊆K (cid:16) E ′ C ∗ ( G ) (cid:17) ∼ = −→ C c ( G ) ⊗ K ( H ) | {z } ⊆K ( C ∗ ( G ) ⊗ H ) . This extends to an identification W : C ∗ ( X ) G ∼ = −→ K ( C ∗ ( G ) ⊗ H ).Let M be the multiplier algebra of C ∗ ( X ) G , and let L := L ( E C ∗ ( G ) ) bethe algebra of adjointable operators on E C ∗ ( G ) .36 orollary 6.2. We have an isomorphism U − ◦ W : C ∗ ( X ) G ∼ = −→ K ( E C ∗ ( G ) ) . This induces an isomorphism on the multiplier algebras and an isomorphismon K -theory of the quotient algebras: ( U − ◦ W ) ∗ : K ( M /C ∗ ( X ) G ) ∼ = −→ K ( L / K ( E C ∗ ( G ) )) . Now let η : K ( K ( C ∗ ( G ) ⊗ H )) → K ( C ∗ ( G ))be the stabilisation isomorphism on K -theory, and write φ := η ◦ W ∗ , where W ∗ is the map on K -theory induced by W . After making theseidentifications, the following proposition follows directly from naturality ofboundary maps with respect to ∗ -homomorphisms. Proposition 6.3.
The following diagram commutes: K ( M /C ∗ ( X ) G )) ( U − ◦ W ) ∗ (cid:15) (cid:15) index / / K ( C ∗ ( X ) G ) φ * * ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ K ( C ∗ ( G )) ,K ( L / K ( E C ∗ ( G ) )) index / / K ( K ( E C ∗ ( G ) )) η ◦ U ∗ ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ (27) where U ∗ is the map induced by U on K -theory. The map ( η ◦ U ∗ ◦ index) is the usual G -index map for Fredholm operatorsin the sense of Hilbert C ∗ ( G )-modules on the module E C ∗ ( G ) . Thus Propo-sition 6.3 provides an identification of the index map in the Roe algebrapicture with the usual notion of G -index for operators on a G -cocompactspace. Now suppose that
X/G is possibly noncompact. As before, let E be a G -vector bundle over X . Similar to the previous subsection, equip the dense37ubspace C c ( G, L ( E )) of L ( E ) ⊗ L ( G ) with the C c ( G )-valued inner prod-uct and right C c ( G )-action given by h s, t i ( g ) := h s, gt i L ( E ) ⊗ L ( G ) , s · b := Z G g − ( b ( g ) s ) dg. Taking the completion gives rise to a Hilbert C ∗ ( G )-module E C ∗ ( G ) . Let Z ⊂ X be closed and G -invariant. Let H X be an admissible equiv-ariant C ( X )-module. The restriction map C ( X ) → C ( Z ) allows us toview H X a as a C ( Z )-module, which will be degenerate (see Definition 2.1)in general. Let H Z = Z H X , an admissible C ( Z )-module. Write H X \ Z forthe orthogonal complement of H Z in H X . The map ϕ XZ : B ( H Z ) → B ( H X ) , (28)defined by extending operators by zero on H X \ Z , restricts to a ∗ -homomorphism ϕ XZ : C ∗ ( Z ) G → C ∗ ( X ; H X ) G , (29)whose image lies in C ∗ ( X ; Z, H X ) G .Let C ∗ ( X ) G loc be the localised equivariant Roe algebra of Definition 2.17. Proposition 6.4.
We have C ∗ ( X ) G loc ∼ = K ( E C ∗ ( G ) ) . Proof.
Fix Z ⊆ X a closed, G -invariant cocompact subset. For i >
0, letPen(
Z, i ) be as in (3). Then Pen(
Z, i ) is cocompact and G -stable, so by (1),Let ϕ i be the map ϕ i := ϕ Pen(
Z,i +1)Pen(
Z,i ) : C ∗ (Pen( Z, i )) G → C ∗ (Pen( Z, i + 1)) G as in (29). Then { C ∗ (Pen( Z, i )) G , ϕ i } i ∈ N is a directed system of C ∗ -algebras whose direct limit is C ∗ ( X ) G loc . Hence C ∗ ( X ) G loc ∼ = C ∗ ( G ) ⊗ K ( H ) , where H is the Hilbert space from the isomorphism L ( E ) ⊗ L ( G ) ∼ = L ( G ) ⊗ H in Definition 2.4. 38ow let E| Pen(
Z,i ) be the restriction of the Hilbert module E C ∗ ( G ) toPen( Z, i ). By Corollary 6.2, for each i , we have an isomorphism K ( E| Pen(
Z,i ) ) ∼ = C ∗ (Pen( Z, i )) G . These maps fit into a commutative diagram C ∗ (Pen( Z, i )) G ∼ = / / (cid:127) _ ϕ i (cid:15) (cid:15) K ( E| Pen(
Z,i ) ) (cid:127) _ (cid:15) (cid:15) C ∗ (Pen( Z, i + 1)) G ∼ = / / K ( E| Pen(
Z,i +1) ) . Finally, note that each element of K ( E C ∗ ( G ) ) is a limit of finite-rank opera-tors, hence K ( E C ∗ ( G ) ) = lim i K (cid:0) E| Pen(
Z,i ) (cid:1) .It follows that we have an isomorphism L ( E C ∗ ( G ) ) / K ( E C ∗ ( G ) ) ∼ = M ( C ∗ ( X ) G loc ) /C ∗ ( X ) G loc . Applying Proposition 6.3 to each of the G -cocompact spaces Pen( Z, i ) andtaking the direct limit, we have shown:
Proposition 6.5.
The following two index maps are equal: index G : K ( L / K ( E C ∗ ( G ) )) lim( η ◦ U ∗ ◦ index) −−−−−−−−−−−→ K ( C ∗ ( G )) and index G : K ( M / ( C ∗ ( X ) G loc )) lim( φ ◦ index) −−−−−−−−→ K ( C ∗ ( G )) . Here C ∗ ( G ) and C ∗ ( X ) G loc can be taken to be either the reduced or maximalversion of the group C ∗ and Roe algebras. G -Callias-type operators and Roe’s localised index We now relate the reduced version of the equivariant Callias-type indexdefined in [17] to (the reduced version of) the localised coarse index.Recall the setting of Subsection 4.1, where D = D Φ is a Callias-type op-erator. The operator F in (18) defines a class [ F ] ∈ K ( L ( E ) / K ( E )), whoseimage under the boundary map for the six-term exact sequence correspond-ing to the ideal L ( E ) ⊂ K ( E ) is by definition index CG ( D Φ ) ∈ K ( C ∗ red ( G )).Consider the embedding E ֒ → E C ∗ red ( G ) , defined on the dense subspace C c ( E ) by the map j in (8). The image of E is a complemented submodule of E C ∗ red ( G ) . Extend the operator F to all39f E C ∗ red ( G ) by defining the extension to be the identity on the orthogonalcomplement. We denote this extended operator by F .The assumption (15) on Φ and ˜ D implies that there is a G -cocompactsubset Z ⊂ X such that D ≥ c outside Z , for some c >
0. By replacing D Φ by the operator c D Φ , which has the same index as D Φ , we may alternativelymake the slightly more convenient assumption that D ≥ Z . Asin Definition 3.6, the localised equivariant coarse index of D Φ isindex G ( D Φ ) = index G ([ b ( D Φ )] ⊕ ∈ K ( C ∗ red ( G )) , (30)for an odd, continuous function b on R withsupp( b − ⊆ [ − , , We now make a specific choice for the function b : b ( x ) = − x ∈ ( −∞ , − x if x ∈ ( − , x ∈ [1 , ∞ ) . We will write F := b ( D Φ ) ⊕
1, for this function b . Then we have a class[ F ] ∈ K (( M /C ∗ ( X ) G loc )) . The index (30) equals the image of [ F ] under the relevant boundary map, ∂ [ F ] ∈ K ( C ∗ ( X ) G loc ) . Here the localised equivariant Roe algebra C ∗ ( X ) G loc is realised on the ad-missible module L ( E ) ⊗ L ( G ).The operators F and F define elements of K ( M /C ∗ ( X ) G loc ) and K ( L / K ( E C ∗ red ( G ) ))respectively. By Proposition 6.5, their indices in K ( C ∗ red ( G )) can be viewedequivalently through either of these pictures, and they equal the two sidesof (20). To prove Theorem 4.2, it therefore suffices to prove the followingequality. Proposition 6.6.
We have index G [ F ] = index G [ F ] ∈ K ( C ∗ red ( G ))40 .4 Proof of Theorem 4.2 We now prove Proposition 6.6, and hence Theorem 4.2.For s >
0, define the functions b s ∈ C b ( R ) and ψ s ∈ C ( R ) by b s ( x ) = x ( | x | /s + 1) s ; ψ s ( x ) = 1( | x | /s + 1) s , for x ∈ R . Then lim s ↓ k b s − b k ∞ = 0 . (31)Let ζ : (0 , → (0 ,
1] be a continuous function such that ζ (1) = 1 andlim s ↓ ζ ( s ) k ψ s/ ( D Φ ) k = 0 . (32)For s ∈ (0 , F s := b s/ ( D Φ ) + ζ ( s ) ψ s/ ( D Φ )Φon E . For s ∈ (0 , D Φ + ζ ( s )Φ is of G -Callias type. Hencethere is a continuous, G -invariant, cocompactly supported function f s on M such that D Φ + ζ ( s )Φ p ( D Φ + ζ ( s )Φ) + f s is invertible modulo K ( E ). Since the operator p ( D Φ + ζ ( s )Φ) + f s ψ s/ ( D Φ )is invertible, the operator˜ F s = D Φ + ζ ( s )Φ p ( D Φ + ζ ( s )Φ) + f s ! p ( D Φ + ζ ( s )Φ) + f s ψ s/ ( D Φ )is invertible modulo K ( E ) as well.We have ˜ F = D Φ + Φ p ( D Φ + Φ) + f p ( D Φ + Φ) + f p ( D Φ + Φ) + 1 . Hence ˜ F has the same index as D Φ + Φ p ( D Φ + Φ) + f , D Φ .Finally, (31) and (32) imply thatlim s ↓ k ˜ F s ⊕ − F k = 0 . So s ˜ F s is a continous path of operators that are invertible modulo K ( E )connecting F to the operator ˜ F , which has the same index in K ( C ∗ red ( G ))as F . This implies Proposition 6.6. Proof of Proposition 4.4.
In the setting of the proposition, the operator D has a well-defined localised indexindex ZG ( D ) ∈ K ∗ ( C ∗ ( M ; Z ) G ) . Then index G ( D ) ∈ K ∗ ( C ∗ ( M ) G ) is the image of index ZG ( D ) under the map K ∗ ( C ∗ ( Z ) G ) = K ∗ ( C ∗ ( M ; Z ) G ) → K ∗ ( C ∗ ( M ) G ) , and hence equal to zero.To prove Theorem 4.6, we use the following equivariant version of atheorem of Vilms [37] that was proved in [19]. Theorem 7.1.
Let π : M → B be a fibre bundle with fibre N and structuregroup K . Suppose that M and B both have bounded geometry and proper,isometric G -actions making π G -equivariant. Let g N be a K -invariant Rie-mannian metric on N . Then there is a G -invariant Riemannian metric g M on M such that π is a G -equivariant Riemannian submersion with totallygeodesic fibres.Proof of Theorem 4.6. Let κ G/K denote the scalar curvature of the G -invariantRiemannian metric g G/K on the base of the fibre bundle M → G/K . Notethat since
G/K is a homogeneous space, κ G/K is a finite constant. Let H ⊆ T M be an Ehresmann connection. Then as in the proof of Theorem7.1 above, we may lift g G/K to a G -invariant metric g H on H , as well aslift the K -invariant Riemannian metric g N on N to a metric on the verticalsubbundle V ⊆ T M . Define a G -invariant metric on M by g M := g H ⊕ g V .42ince N has uniformly positive scalar curvature κ N , it satisfies inf { κ N } =: κ >
0. Now let T and A denote the O’Neill tensors of the submersion π (their definitions can be found in [29]). By Theorem 7.1 above, the fibresof M are totally geodesic, so T = 0. Pick an orthonormal basis of horizon-tal vector fields { X i } . A result of Kramer ([27], p. 596), relates the scalarcurvatures by κ M ( p ) = κ G/K + κ N ( p ) − X i,j || A X i ( X j ) || p . Since both M and N have bounded geometry, it follows that their scalarcurvatures κ M and κ N are uniformly bounded. Thereforesup p ∈ M X i,j || A X i ( X j ) || p ≤ A < ∞ for some positive constant A . Upon scaling the fibre metric on N by apositive factor t , we obtain κ M ( p ) ≥ κ G/K + t − κ − A > < t < r κ − κ G/K + A , where we choose A > − κ G/K + A >
0. Thus g M is a G -invariant metric of uniform positive scalar curvature on M . Let Z ⊂ X be a closed, G -invariant, cocompact subset. Let H X be anadmissible equivariant C ( X )-module. The map ϕ XZ in (28) restricts to amap ϕ XZ : D ∗ red ( Z ) G → D ∗ red ( X ) G that maps C ∗ ( Z ) G into C ∗ ( X ; Z ) G . Hence we obtain ϕ XZ : D ∗ red ( Z ) G /C ∗ ( Z ) G → D ∗ red ( X ) G /C ∗ ( X ; Z ) G . (33)By Paschke duality, the analytic K -homology of Z equals K G ∗ ( Z ) = K ∗ +1 ( D ∗ red ( Z ) G /C ∗ ( Z ) G ) . So (33) induces ( ϕ XZ ) ∗ : K G ∗ ( Z ) → K G ∗ ( X ) loc . (34)Using the maps (34), we obtain ϕ X ∗ : RK G ∗ ( X ) → K G ∗ ( X ) loc . (35)43 emma 7.2. The following diagram commutes, where µ XG denotes the an-alytic assembly map for X : RK G ∗ ( X ) µ XG / / ϕ X ∗ (cid:15) (cid:15) K ∗ ( C ∗ red G ) .K G ∗ ( X ) loc index loc G ♣♣♣♣♣♣♣♣♣♣♣ Proof. If Z ⊂ X is G -cocompact, then naturality of boundary maps withrespect to ∗ -homomorphisms implies that the diagram K G ∗ ( Z ) index G / / ( ϕ XZ ) ∗ (cid:15) (cid:15) K ∗ ( C ∗ red G ) .K G ∗ ( X ) loc index loc G ♣♣♣♣♣♣♣♣♣♣♣ commutes. By Corollary 4.3, the top horizontal arrow equals µ ZG , so theclaim follows after we take direct limits.This lemma directly implies Proposition 4.9. Remark 7.3.
The arguments in this subsection have two more conse-quences.1. If the map (35) is injective, then injectivity of the Baum–Connes as-sembly map implies injectivity of the map in Conjecture 4.8.2. If the map (35) is surjective, then Conjecture 4.8 implies surjectivityof the Baum–Connes assembly map.
References [1] Herbert Abels. Parallelizability of proper actions, global K -slices andmaximal compact subgroups. Math. Ann. , 212:1–19, 1974/75.[2] Nicolae Anghel. Remark on Callias’ index theorem.
Rep. Math. Phys. ,28(1):1–6, 1989.[3] Nicolae Anghel. On the index of Callias-type operators.
Geom. Funct.Anal. , 3(5):431–438, 1993. 444] Paul Baum, Alain Connes, and Nigel Higson. Classifying space forproper actions and K -theory of group C ∗ -algebras. In C ∗ -algebras:1943–1993 (San Antonio, TX, 1993) , volume 167 of Contemp. Math. ,pages 240–291. American Mathematical Society, Providence, RI, 1994.[5] R. Bott and R. Seeley. Some remarks on the paper of Callias: “Axialanomalies and index theorems on open spaces” [Comm. Math. Phys.62 (1978), no. 3, 213–234; MR 80h:58045a].
Comm. Math. Phys. ,62(3):235–245, 1978.[6] Maxim Braverman. The index theory on non-compact manifolds withproper group action.
J. Geom. Phys. , 98:275–284, 2015.[7] Maxim Braverman and Simone Cecchini. Callias-type operators in vonNeumann algebras.
J. Geom. Anal. , 28(1):546–586, 2018.[8] Jochen Br¨uning and Henri Moscovici. L -index for certain Dirac-Schr¨odinger operators. Duke Math. J. , 66(2):311–336, 1992.[9] Ulrich Bunke. A K -theoretic relative index theorem and Callias-typeDirac operators. Math. Ann. , 303(2):241–279, 1995.[10] Ulrich Bunke and Alexander Engel. The coarse index class with sup-port. ArXiv:1706.06959, 2018.[11] Constantine Callias. Axial anomalies and index theorems on openspaces.
Comm. Math. Phys. , 62(3):213–234, 1978.[12] Catarina Carvalho and Victor Nistor. An index formula for perturbedDirac operators on Lie manifolds.
J. Geom. Anal. , 24(4):1808–1843,2014.[13] Simone Cecchini. Callias-type operators in C ∗ -algebras and positivescalar curvature on noncompact manifolds. ArXiv:1611.01800.[14] Alain Connes and Henri Moscovici. The L -index theorem for homoge-neous spaces of Lie groups. Ann. of Math. (2) , 115(2):291–330, 1982.[15] Guihua Gong, Qin Wang, and Guoliang Yu. Geometrization of thestrong Novikov conjecture for residually finite groups.
J. Reine Angew.Math. , 621:159–189, 2008.[16] Mikhael Gromov and H. Blaine Lawson, Jr. Positive scalar curvatureand the Dirac operator on complete Riemannian manifolds.
Inst. Hautes´Etudes Sci. Publ. Math. , (58):83–196 (1984), 1983.4517] Hao Guo. Index of equivariant Callias-type operators and invariantmetrics of positive scalar curvature. ArXiv:1803.05558.[18] Hao Guo, Peter Hochs, and Varghese Mathai. Coarse equivariant Cal-lias index theory and quantisation. In preparation.[19] Hao Guo, Mathai Varghese, and Hang Wang. Positive scalar curvatureand Poincar´e duality for proper actions.
J. Noncommut. Geometry ,2018. To appear, ArXiv:1609.01404.[20] Nigel Higson and John Roe.
Analytic K -homology . Oxford Mathe-matical Monographs. Oxford University Press, Oxford, 2000. OxfordScience Publications.[21] Nigel Higson, John Roe, and Guoliang Yu. A coarse Mayer-Vietorisprinciple. Math. Proc. Cambridge Philos. Soc. , 114(1):85–97, 1993.[22] Peter Hochs and Varghese Mathai. Geometric quantization and familiesof inner products.
Adv. Math. , 282:362–426, 2015.[23] Peter Hochs and Yanli Song. An equivariant index for proper actionsIII: The invariant and discrete series indices.
Differential Geom. Appl. ,49:1–22, 2016.[24] Peter Hochs, Bai-Ling Wang, and Hang Wang. An equivariant Atiyah–Patodi–Singer index theorem for proper actions. In preparation.[25] Chris Kottke. An index theorem of Callias type for pseudodifferentialoperators.
J. K-Theory , 8(3):387–417, 2011.[26] Chris Kottke. A Callias-type index theorem with degenerate potentials.
Comm. Partial Differential Equations , 40(2):219–264, 2015.[27] W. Kramer. The scalar curvature on totally geodesic fiberings.
Ann.Global Anal. Geom. , 18(6):589–600, 2000.[28] Dan Kucerovsky. A short proof of an index theorem.
Proc. Amer. Math.Soc. , 129(12):3729–3736, 2001.[29] Barrett O’Neill. The fundamental equations of a submersion.
MichiganMath. J. , 13:459–469, 1966.[30] Richard S. Palais. On the existence of slices for actions of non-compactLie groups.
Ann. of Math. (2) , 73:295–323, 1961.4631] John Roe. Coarse cohomology and index theory on complete Rieman-nian manifolds.
Mem. Amer. Math. Soc. , 104(497):x+90, 1993.[32] John Roe.
Index theory, coarse geometry, and topology of manifolds ,volume 90 of
CBMS Regional Conference Series in Mathematics . Pub-lished for the Conference Board of the Mathematical Sciences, Wash-ington, DC; by the American Mathematical Society, Providence, RI,1996.[33] John Roe. Comparing analytic assembly maps.
Q. J. Math. , 53(2):241–248, 2002.[34] John Roe.
Lectures on coarse geometry , volume 31 of
University LectureSeries . American Mathematical Society, Providence, RI, 2003.[35] John Roe. Positive curvature, partial vanishing theorems and coarseindices.
Proc. Edinb. Math. Soc. (2) , 59(1):223–233, 2016.[36] Thomas Schick. The topology of positive scalar curvature. In
Pro-ceedings of the International Congress of Mathematicians—Seoul 2014.Vol. II , pages 1285–1307. Kyung Moon Sa, Seoul, 2014.[37] Jaak Vilms. Totally geodesic maps.
J. Differential Geometry , 4:73–79,1970.[38] Rufus Willett and Guoliang Yu. Coarse index theory. In preparation,2019.[39] Robert Wimmer. An index for confined monopoles.
Comm. Math.Phys. , 327(1):117–149, 2014.[40] Guoliang Yu. The Novikov conjecture for groups with finite asymptoticdimension.
Ann. of Math. (2) , 147(2):325–355, 1998.[41] Guoliang Yu. The coarse Baum-Connes conjecture for spaces which ad-mit a uniform embedding into Hilbert space.
Invent. Math. , 139(1):201–240, 2000.[42] Guoliang Yu. A characterization of the image of the Baum-Connesmap. In
Quanta of maths , volume 11 of