Coarse geometry and Callias quantisation
Abstract
Consider a proper, isometric action by a unimodular, locally compact group on a complete Riemannian manifold . For equivariant elliptic operators that are invertible outside a cocompact subset of , we show that a localised index in the theory of the maximal group algebra of is welldefined. The approach is based on the use of maximal versions of equivariant localised Roe algebras, and many of the technical arguments in this paper are used to handle the ways in which they differ from their reduced versions.
By using the maximal group algebra instead of its reduced counterpart, we can apply the trace given by integration over to recover an index defined earlier by the last two authors, and developed further by Braverman, in terms of sections invariant under the group action. As a very special case, this allows one to refine numerical obstructions to positive scalar curvature on a noncompact manifold defined via Callias index theory, to obstructions in the theory of the maximal algebra of the fundamental group .
As a motivating application in another direction, we prove a version of Guillemin and Sternberg’s quantisation commutes with reduction principle for equivariant indices of Calliastype operators.
Contents
1 Introduction
Background
Let be a complete Riemannian manifold, and let be an elliptic differential operator on a vector bundle . The coarse index [37] of lies in , the theory group of the Roe algebra of . This Roe algebra is the closure in the operator norm of the algebra of locally compact, bounded operators on that enlarge supports of sections by a finite amount. If is compact, then is the algebra of compact operators, and the coarse index of is its Fredholm index. A strength of the coarse index is that it applies very generally, without any assumptions on compactness of , or on the behaviour of at infinity. Coarse index theory has a range of applications, for example to Riemannian metrics of positive scalar curvature [39], and to the Novikov conjecture [43, 44]. A central role here is played by the coarse Baum–Connes conjecture [36].
The general applicability of the coarse index can come at the cost of computability. For that reason, it is worth looking for special cases, or variations, where a version of the coarse index is more explicit or computable. One useful approach is Roe’s localised coarse index [38]. If is positive outside a subset in a suitable sense, then Roe constructed a localised coarse index
The special case where is compact is already of interest: then is Fredholm, and its localised coarse index generalises the Gromov–Lawson index [11], the Atiyah–Patodi–Singer index on compact manifolds with boundary [2], and the index of Calliastype Dirac operators [1, 7, 25] , where is a Dirac operator, and is a vector bundle endomorphism making invertible at infinity.
The localised coarse index was generalised to an equivariant version in [14], for a proper, isometric action by a unimodular locally compact group on , preserving all structure including . Then, if is compact, one obtains a localised equivariant index
(1.1) 
where is the reduced group algebra of . The fact that this index lies in is useful, because that theory group is independent of , and it is a very wellstudied object that is central to many problems in geometry, topology and group theory. In particular, it is large enough to contain relevant grouptheoretic information. And importantly, there is a range of traces and higher cyclic cocycles on subalgebras of that allows one to obtain a number from the index (1.1), for which one can then find a topological expression. Examples of such expressions are the equivariant Atiyah–Patodi–Singer index theorems in [8, 24, 42], in the case of manifolds with boundary.
Results
This paper is about the construction and application of a maximal localised equivariant coarse index, taking values in the theory of the maximal group algebra
(1.2) 
The first result in this paper is that this index is welldefined: see Propositions 3.1 and 3.3.
The index (1.2) has several advantages over (1.1). From a general point of view, the natural map from to maps the index in to the one in , so the former is a more refined invariant. On a more practical level, the integration map extends to a trace on (not on ). That means it can be applied to the index (1.2), to yield the integer
(1.3) 
Morally, applying the integration trace should correspond to taking the invariant part of the equivariant index. The second result in this paper, Theorem 3.9, is that this is indeed the case in a precise sense:
(1.4) 
where the right hand side is the Fredholm index of restricted to invariant sections that are square integrable transversally to orbits in a certain sense. The latter index was defined in [18], and developed further by Braverman [6].
In the example where is an elliptic operator on a possibly noncompact manifold , invertible outside a compact set, and is its lift to the universal cover of , Theorem 3.9 implies that maps the equivariant, localised, maximal index of to the Fredholm index of . This means that the index (1.2) refines the Gromov–Lawson index, the index of Calliastype operators, as well as the Atiyah–Patodi–Singer index. One application of this fact is that it leads to refinements of obstructions to Riemannian metrics of positive scalar curvature defined through the Gromov–Lawson and Callias indices on manifolds. This is analogous to the way in which the image of under the analytic assembly map [3] for the maximal group algebra of refines the index of in the case where is compact. Explicit applications to positive scalar curvature will be explored in future work.
A completely different application that motivates the development of the index (1.2) and Theorem 3.9. is a version of Guillemin and Sternberg’s quantisation commutes with reduction principle [12] for Calliastype Dirac operators. That principle was initially stated and proved for compact Kähler and symplectic manifolds [31, 32, 34, 40]. This principle was extended in various directions, including results for proper actions by possibly noncompact groups, with possibly noncompact orbit spaces, see [18] for the symplectic case and [19] for manifolds. The index, or quantisation, used in those papers, was defined just in terms of sections invariant under the group action. Furthermore, the index was only welldefined after a suitable order zero term was added to the operator in question. The first of these issues was partially remedied in [23], where the quantisation commutes with reduction principle was proved for an index with values in the completed representation ring of a maximal compact subgroup of .
Since the work of Paradan and Vergne [35], the quantisation commutes with reduction principle is known to be a general property of equivariant indices of Dirac operators in general, and not just of geometric quantisation in the narrow sense. For a Calliastype operator , where is a Dirac operator, the third result in this paper, Theorem 3.11, states that the quantisation commutes with reduction principle holds, in the sense that
(1.5) 
where is a Dirac operator on a reduced space , a analogue of a reduced space in symplectic geometry, for high enough powers of the determinant line bundle of the structure. In this setting, the use of the maximal localised coarse index allows us to prove such a result in the setting of noncompact groups and orbit spaces, for a truly equivariant index in , which is defined without the need of an added term.
The equality (1.5) already appears to be new in the case where is compact. Then is Fredholm, and has an equivariant index in the usual sense. In this case, a version of the shifting trick in symplectic geometry applies to yield information about the multiplicities in that index of all irreducible representations of .
Techniques used
The key ingredient in the construction of the index (1.2) is the notion of a maximal localised equivariant Roe algebra for arbitrary unimodular, locally compact groups. This involves the notion of an admissible module, which was defined in [45] for discrete groups, and in [14] in general. In the nonequivariant, nonlocalised case, the natural maximal norm for such algebras was shown to be welldefined in [10]. In the equivariant, localised case, this is less clear, and getting around this is a step in the construction of the algebras we need.
The construction of the index (1.2) is very different form the construction of the reduced version (1.1) in [14]. Instead of viewing as an unbounded operator on , we view it as an unbounded operator on a maximal localised equivariant Roe algebra , viewed as a Hilbert module over itself. The reason for this is that the localisation results in [38] that make the definition of the localised coarse index possible do not directly carry over to the norm on the maximal Roe algebra. Indeed, it is not even clear if the operators involved lie in the unlocalised maximal Roe algebra, let alone if they localise in a suitable way.
We prove versions of Roe’s localisation results for as an operator on , thus allowing us to define (1.2). To do this we prove that the functional calculus for such operators on is welldefined. This was done in [15] for the uniform maximal Roe algebra; in our setting it works for usual maximal Roe algebras due to localisation at a cocompact set.
To prove the equality (1.4), we use various averaging maps, which map equivariant operators on to operators on . Comparing such maps for operators on and on to the integration trace then leads to a proof of (1.4).
Using (1.4), we see that the left hand side of (1.5) equals a more concrete index in terms of invariant sections. For the latter index, we obtain localisation estimates that allow us to show that this index equals the right hand side of (1.5). These localisation estimates build on those in [18, 19, 30, 40], but a fundamental difference is that we now need the key deformation term to go to zero at infinity, rather than grow towards infinity.
Outline of this paper
We start by defining equivariant localised maximal Roe algebras in Section 2. That allows us to state the three results in the paper mentioned above, in Section 3. Welldefinedness of the index (1.2) is proved in Section 4. To prepare for the proof of (1.4), we construct several averaging maps in section 5. In Section 6, we use these maps to prove (1.4). We conclude this paper by using (1.4) and some localisation estimates to prove (1.5) in Section 7.
Acknowledgements
The authors are grateful to Rufus Willett, Zhizhang Xie and Guoliang Yu for their helpful advice. Varghese Mathai was supported by funding from the Australian Research Council, through the Australian Laureate Fellowship FL170100020. Hao Guo was supported in part by funding from the National Science Foundation under grant no. 1564398.
2 Equivariant localised maximal Roe algebras
2.1 Equivariant modules
Let be a metric space in which all closed balls are compact. Let be a unimodular, locally compact group, acting properly and isometrically on . Let be a Haar measure on .
A equivariant module is a Hilbert space equipped with a unitary representation of , and a homomorphism , such that for all and ,
Here , for all . We will omit the representations and from the notation, and for example write , for and .
Fix a equivariant module . Let be the algebra of equivariant bounded operators on . An operator is said to be locally compact if for all , the operators and are compact. And has finite propagation if there is an such that for all whose supports are at least a distance apart,
In that case, the infimum of such numbers is the propagation of . The equivariant reduced Roe algebra of with respect to is the closure in the operator norm of the algebra of locally compact operators in with finite propagation. In this paper, we will use an algebra that differs from the equivariant reduced Roe algebra in two ways: we consider a localised version, and complete it in a maximal norm.
A relevant example of a equivariant module is the space of square integrable sections of a equivariant, Hermitian vector bundle , with respect to a invariant measure on . The algebra acts on by pointwise multiplication, and acts in the usual way. Consider the vector bundle . Let be the algebra of locally compact operators with finite propagation, for which there is a bounded, measurable^{4}^{4}4One can also work with continuous sections, the main reason why we use measurable sections is that the map in Proposition 4.5 does not preserve continuity. section of such that for all and ,
We will identify such operators with their kernels .
2.2 Admissible modules and the maximal Roe algebra
In Definition 2.2 in [45], the notion of an admissible equivariant module was introduced, for discrete groups . In Definition 2.4 in [14], this was extended to general unimodular, locally compact groups . The main difference between the discrete and general group case is the role played by local slices in the sense of Palais [33] in the nondiscrete case.
A equivariant module is defined to be admissible if there is a equivariant, unitary isomorphism
for a Hilbert space , such that locally compact operators on are mapped to locally compact operators on , and operators with finite propagation are mapped to operators with finite propagation, in both cases with respect to the pointwise action by .
The point of using admissible modules is that the resulting equivariant Roe algebras encode the relevant grouptheoretic information. It is clear that such information may be lost in the example where is a point, acted on trivially by a compact group, and one uses the nonadmissible module .
By Theorem 2.7 in [14], an example of an admissible module is . Here is as at the end of the previous subsection, acts pointwise on the factor , and acts diagonally, with respect to the left regular representation of in . By definition of admissibility, we have an isomorphism
(2.1) 
with the properties above. Let be the algebra of locally compact operators on with finite propagation, given by bounded, measurable kernels
via the isomorphism (2.1). Explicitly, for such a , the corresponding operator is defined by
for , and . If is compact, then Theorem 2.11 in [14] states that is isomorphic to a dense subalgebra of , where is either the reduced or maximal group algebra of . (To be precise, is isomorphic to the convolution algebra of compactly supported, bounded, measurable functions on with values in the algebra of compact operators on .) This implies that the maximal norm of an element ,
(2.2) 
where the supremum is over all representations
is finite. This norm is equal to the tensor product norm on . So the completion of in the maximal norm equals
(2.3) 
Since this algebra is independent of the admissible module used, we will denote it by
Remark 2.1.
In the case where is trivial, and is not assumed to be compact but is only assumed to have bounded geometry, finiteness of the maximal norm (2.2) was proved by Gong, Wang and Yu, see Lemma 3.4 in [10]. See also Lemma 1.10 in [41]. It is our understanding that this generalises directly to free actions by discrete groups, see Lemma 3.16 in [10].
2.3 The map and the maximal norm for nonadmissible modules
We will use a completion of in a version of the maximal norm. It is unclear a priori if an analogue of the supremum (2.2) is finite, however. We therefore define the norm we use via an embedding of into , which has a welldefined maximal norm if is compact, as we saw at the end of the previous subsection.
Let be the algebra of bounded, equivariant, locally compact operators on an equivariant module , with finite propagation. In Section 3.2 in [14], a map
(2.4) 
is defined as follows. Let be a function whose support has compact intersections with all orbits, and has the property that for all ,
(2.5) 
(The integrand is compactly supported by properness of the action.) Such a function will be called a cutoff function. The map , given by
for , and , is an isometric, equivariant embedding. Let be the orthogonal projection. The map
(2.6) 
that maps to is an injective homomorphism, and preserves equivariance, local compactness, and finite propagation. Hence it restricts to an injective homomorphism (2.4). (The notation reflects the fact that equals on the image of , and zero on its orthogonal complement.)
Lemma 2.2.
The map (2.4) maps into .
Proposition 4.5 is a refinement of this lemma. If is compact, then for , we define its maximal norm as
We denote the completion of in this norm by
Remark 2.3.
In the case of reduced Roe algebras, defined with respect to the operator norm for a module, the algebra is dense in . See Proposition 5.11 in [14]. In that case, kernels and operators can be used more or less interchangeably, but this is less clear for the maximal completions we use here.
2.4 Localised maximal Roe algebras
Let be a invariant subset. Let be a equivariant module. An operator is supported near if there is an such that for all whose support is at least a distance away from , the operators and are zero. Let be the algebra of elements of supported near .
For and any subset , we write
Then we have a natural isomorphism
(2.7) 
Now suppose that is compact. The algebra is then independent of , as long as is compact. For this reason, we write
For every , we have the norm on . Let be the resulting norm on via (2.7).
Definition 2.4.
The localised, maximal equivariant Roe algebra of for , denoted by , is the completion of in the norm .
The localised, maximal equivariant Roe algebra of , denoted by , is the completion of in the norm .
3 Results
Our first result is the fact that a maximal version of the localised equivariant index of [14] is welldefined, see Propositions 3.1 and 3.3 and Definition 3.4. We will show that that index is an equivariant refinement of the index defined in terms of invariant sections in [6, 18, 30], see Theorem 3.9. The quantisation commutes with reduction results for proper, noncocompact actions in [18, 20] only involved sections invariant under a group action. In Theorem 3.11, we generalise this to the equivariant index of Definition 3.4, in the case of Calliastype Dirac operators.
3.1 The localised maximal equivariant index
From now on, we suppose that , a complete Riemannian manifold, and and that is the Riemannian distance corresponding to a invariant Riemannian metric. We suppose that is a smooth, equivariant, Hermitian vector bundle and a symmetric, first order, elliptic, equivariant differential operator on sections of . Suppose that has finite propagation speed, i.e. if is its principal symbol, then
Then is essentially self adjoint as an unbounded operator on , see Proposition 10.2.11 in [17].
Let be a closed, cocompact invariant subset. Let be the algebra of smooth kernels in . Then acts on by
Here we used the fact that for every , is a smooth section of .
For a algebra and a Hilbert module, we write and for the algebras of bounded adjointable operators and compact operators on , respectively. We can view as a right Hilbert module over itself, with valued inner product.
(3.1) 
for . Then , with the isomorphism being given by identifying the operator
with left multiplication by . We also have that is the multiplier algebra of .
To simplify notation, we will from now on use to denote the maximal, localised equivariant Roe algebra . Then is a Hilbert module over itself. We will use functional calculus for selfadjoint, regular operators on the Hilbert module . (For a uniform version of the maximal Roe algebra, this was developed in [15].) This functional calculus applies to because of the following result.
Proposition 3.1.
The unbounded operator on the Hilbert module is essentially selfadjoint and regular.
This proposition is proved in Subsection 4.4. Because of Proposition 3.1, we can apply the following general result (see [26], [16] Theorem 3.1 and [9] Theorem 1.19) about functional calculus on Hilbert modules to the selfadjoint closure of .
Theorem 3.2.
Let be a algebra and a Hilbert module. Let be the algebra of complexvalued continuous functions on . For any regular, essentially selfadjoint operator on , there is a preserving linear map
with values in the set of regular operators on , such that:

restricts to a homomorphism ;

If for all , then ;

If is a sequence in for which there exists such that for all , and if converge to a limit function uniformly on compact subsets of , then for each ;

;

;
In the context of this theorem, we write .
Suppose that there are a equivariant, Hermitian vector bundle , a differential operator , a equivariant vector bundle endomorphism of , and a constant such that
(3.2) 
and , fibrewise outside . (The use of instead of is a convention here, which implies that outside in an appropriate sense.)
In this setting, and when is trivial but without assuming to be compact, Roe [38] developed localised index theory with values in the theory of a reduced completion of . We will use an equivariant version of this index theory for the maximal completion, in terms of admissible modules. The reason for using the maximal completion is that we then obtain an index in the theory of , to which we can apply an integration map to recover the invariant index from [18] as a special case, see Theorem 3.9. The construction of the localised index is based on the following analogue of Lemma 2.3 and Theorem 2.4 in [38].
Proposition 3.3.
If is supported in , then
This proposition is proved in Subsection 4.5
Let be a continuous, increasing, odd function, such that for all with . Then has the property of the function in Proposition 3.3. So, in particular, is invertible modulo , and hence has an index in
This index lies in even theory if is odd with respect to a invariant grading on , and in odd theory otherwise. See for example Definition 3.2 in [14] for details.
Explicitly, consider the case where is odd with respect to a invariant grading . Let be the algebra of kernels in supported near . Let be the restriction of to kernels in that are sections of . Then is invertible modulo , and its inverse is the restriction of to . Hence this operator defines a class , and the index of is defined as
(3.3) 
where is the boundary map in the sixterm exact sequence correspondig to the ideal . For ungraded operators, one uses the projection in and applies the boundary map to its class in even theory to obtain the index of in .
Definition 3.4.
The localised, maximal, equivariant index of is the image of the index of in described above under the map
It is denoted by .
Remark 3.5.
One could consider (3.3) (and its analogue in in the nongraded case) as a localised index of , defined in terms of the nonadmissible module . Two advantages of the index in Definition 3.4 over (3.3) are that it takes values in a theory group independent of or , and that the application of the map on theory means that the index of Definition 3.4 captures grouptheoretic information that is not encoded in (3.3). This is clear in the example where is compact, is a point, and is the zero operator on , as discussed in Example 3.8 in [14]. This illustrates why it is useful to use the admissible module .
Example 3.6.
If is a Diractype operator associated to a Clifford connection on , then
for a vector bundle endomorphism of . (If is a Dirac operator, then is scalar multiplication by a quarter of scalar curvature, by Lichnerowicz’ formula.) If outside , then the condition on holds, with and . This is the situation considered in [38], for trivial.
Example 3.7.
Let be a equivariant Dirac operator on , and let be a equivariant vector bundle endomorphism of . Suppose that is a vector bundle endomorphism of , and that
(3.4) 
fibrewise outside . Then satisfies the conditions on as above, with , and .
The main application of the maximal localised index in this paper, Theorem 3.11, is about the maximal localised index of Calliastype operators.
3.2 The invariant index
Integrating functions over extends to a trace on . We will see in Theorem 3.9 that applying this trace to the localised index of recovers an index defined in terms of invariant sections in [18]. This fact will be used in the proof of Theorem 3.11. It can also be used to obtain refined index theoretic information on noncompact manifolds; see Remark 3.10.
Let a function with the property (2.5). Consider the space of transversally compactly supported sections of , defined as the space of continuous, invariant sections of whose supports have compact images in under the quotient map. The Hilbert space of invariant, transversally sections of is the completion of in the inner product
The space is independent of the choice of ; see Lemma 4.4 in [18].
Suppose that is odd with respect to a invariant grading . In Proposition 4.7 in [18], it is shown that defines a Fredholm operator from a suitable Sobolev space inside into . In Proposition 4.8 in the same paper, it is deduced that the space
is finitedimensional, and that the index of equals
(3.5) 
Definition 3.8.
The invariant index of , denoted by , is the number (3.5).
In [6], Braverman further develops the theory of this index, when applied to Dirac operators with an added zeroorder term that is relevant to geometric quantisation, and in particular proves that it is invariant under a suitable notion of cobordism.
The map from to given by integrating functions over extends continuously to a homomorphism, or a trace
The integer
plays the role of the invariant part of the localised index of , and this will be made precise in Theorem 3.9 below.
If is compact, then all smooth sections of are transversally . Then the invariant index of equals
where is the restriction of to sections of . This index was developed and applied by Mathai and Zhang in [30], with an appendix by Bunke. In Theorem 2.7 and Proposition D.3 in that paper, it is shown that the index can be recovered from the equivariant index of in , defined via the analytic assembly map, if one applies the integration trace . We will show that this generalises to the index in Definition 3.4 in the noncocompact case.
Theorem 3.9.
We have
(3.6) 
This theorem is proved in Section 6.
Remark 3.10.
Theorem 3.9 allows us to construct a more refined invariant of operators that are invertible at infinity in the nonequivariant case than their Fredholm index. Suppose that is the universal cover of a manifold and that is the fundamental group of , acting on in the natural way. Let be an elliptic operator on that is invertible at infinity in the appropriate sense, so that it lifts to a equivariant operator on satisfying the conditions of Theorem 3.9. That theorem then implies that
In this sense, refines the Fredholm index of , much like the image of under the analytic assembly map for the maximal group algebra refines the Fredholm index of if is compact.
This can for example be used to obtain stronger obstructions to Riemannian metrics of positive scalar curvature that classical obstructions from Callias index theory [1] or the Gromov–Lawson index [11]. Applications to positive scalar curvature will be worked out in a future article.
More generally, for any proper, isometric action by a discrete group on , the quotient is an orbifold, and the refines an orbifold version of the index of .
Proposition 2.4 in [21] shows that the index defined in [22] is another refinement of the invariant index. That index applies to Dirac operators with certain deformation terms added that are relevant to geometric quantisation. It takes values in the completion of the representation ring of a maximal compact subgroup of the group acting.
3.3 Callias quantisation commutes with reduction
In [18, 20], the quantisation commutes with reduction principle of Guillemin and Sternberg [12, 31, 32, 34, 40] and its version [35] is generalised to proper actions by possibly noncompact groups, with possibly noncompact orbit spaces, for suitably high powers of the prequantum or determinant line bundle in question. These results in [18, 20] are stated in terms of the invariant index of Definition 3.8. The result in [18] in the symplectic setting generalises the result in [30] from compact to noncompact orbit spaces. This is generalised to the setting in [20].
These were the first results on a version of the quantisation commutes with reduction principle where both the group and orbit space were allowed to be noncompact, but two drawbacks were that the invariant index used

was only welldefined after a deformation term (Clifford multiplication by the Kirwan vector field) was added to the relevant Dirac operator;

only involved invariant sections, and therefore provided no information about the parts of the kernel of on which acts nontrivially.
The second point was partially addressed in Theorem 2.13 in [21], a quantisation commutes with reduction result for noncocompact actions, where quantisation takes values in the completion of the representation ring of a maximal compact subgroup.
We are now able to remedy both points, in the case of Calliastype Dirac operators.
Let be a Calliastype operator as in Example 3.7. Suppose that has a invariant grading, and that and , and hence , are odd for this grading. Suppose that is the spinor bundle of a equivariant structure on , and let be its determinant line bundle. (The assumption that is graded now means that is evendimensional.) Suppose that is a Dirac operator on . The Clifford connection on used to define can be constructed locally from a invariant, Hermitian connection on and the connection on the spinor bundle for a local structure; see e.g. Proposition D.11 in [28]. This also induces a Clifford connection on the spinor bundle , for any . Let be the corresponding Dirac operator on . Set
We have
(3.7) 
where denotes the anticommutator. In what follows, we will omit ‘’ from the notation. By (3.4) and (3.7), we have
(3.8) 
outside , for all . Hence has an index
(3.9) 
The quantisation commutes with reduction principle in general is an equality between the invariant part of the equivariant index of a Dirac operator and the index of a Dirac operator localised at the level set of a moment map. The invariant part of the index will now be represented by the image of (3.9) under the integration trace .
The moment map associated to is the map
such that for all ,
where denotes the Lie derivative with respect to , and is the vector field induced by . Our sign convention is that for all and ,
Suppose that is a regular value of , and that acts freely on . Suppose that the reduced space
is compact. In Lemma 3.3 in [20], a condition is given for to inherit a structure from , with determinant line bundle , with . This is true for example if is semisimple, see Proposition 3.5 and Example 3.6 in [20]. It is also true in the symplectic setting, where the structure on is associated to a invariant almost complex structure compatible with a invariant symplectic form, together with a equivariant, Hermitian line bundle on . From now on, we assume such a structure on exists. Let be a Dirac operator on for this structure.
Theorem 3.11 (Callias quantisation commutes with reduction).
There is a such that for all ,
This theorem is proved in Section 7.
Remark 3.12.
In the case of Calliastype operators , as in Example 3.7, an index in was constructed directly in [13]. Here can be either the reduced or maximal group algebra. Let us denote this index by
Theorem 4.2 in [14] states that, for the reduced group algebra and Roe algebra, this index of Calliastype operators is a special case of the localised index:
Via analogous arguments, one can show that this equality still holds for the maximal group algebra and Roe algebra. Then Theorem 3.11 implies that, under the conditions in that theorem,
Remark 3.13.
As far as we are aware, Theorem 3.11 was not known in the case where is compact, so that is Fredholm in the classical sense. In that case, by the standard shifting trick (see for example Corollary 1.2 in [31], and [35]), Theorem 3.11 implies expressions for the multiplicities of all irreducible representations of in the equivariant index of . One can handle cases where a reduced space is not smooth by

using orbifold line bundles and indices if is a regular value of in an appropriate sense;
4 Regularity and localisation
In this section, we prove Propositions 3.1 and 3.3, which imply that the localised maximal index of Definition 3.4 is welldefined.
4.1 Maximal operator modules
Our first goal is to make sense of as an unbounded, regular, essentially selfadjoint operator on certain maximal operator modules that we now introduce.
Let and be Riemannian manifolds equipped with proper, isometric actions. Let and be Hermitian vector bundles over and respectively. Consider the vector bundle .
Definition 4.1.
Denote by
the vector space of smooth invariant, cocompactly supported, finite propagation kernels. Here we say that a kernel has cocompact support if there exists cocompact subsets and of and respectively such that .
There is no natural product or operation on . However, it admits a natural action of from the right, given by composition of kernels. Further, it has a valued inner product given by
for , defined through the usual adjoint and multiplication of kernels. This makes