Equivariant versions of the Roe algebra and the coarse index have been developed for proper actions of discrete groups. Another refinement is a localized variant of the coarse index developed by Roe [37] that generalizes an index defined by Gromov and Lawson [16]. In future work, we will provide further applications of the index theory we develop in this paper.
This allows us to formulate a localized version of the surjectivity direction of the Baum-Connes conjecture. Proofs of the properties of admissible modules and Roe algebras from Section 2 are given in Section 5. Much of the material in this section is well known in the case of discrete groups, but we will see that the generalization to general locally compact groups requires some work.
The use of admissible modules ensures that the constructed algebras are independent of the choice of module. It also implies that the localized Roe algebra is independent of the choice of admissible modulus, as in (2.6). A consequence of this is that we can define a maximal version of the Roe equivariant algebra, in the following sense.
In particular, the relations (2.2) and (2.3) imply that the reduced and maximal algebras are independent of the choice of the admissible C0(X)-moduleHX.
Localised Roe algebras
In the case of G= Γ discrete and X/Γ compact, a proof can be found in the forthcoming book [40]. The algebraic, reduced, and maximally equivalent Roe algebras defined with respect to this module are all equal to K(l2(N)), which contains no group-theoretic information about K. Of the terms in Definition 2.17, the localized equivalent Roe algebra C∗(
Note that if Z/G is compact, the algebras Calg∗ (X;Z, HX)G and C∗(X;Z, HX)G are independent of Z as long as Z/G is compact. If HX is admissible, the algebraCker∗ (Pen(Z, r))G has a well-defined maximal norm with (2.1) for all r. The complement of this injective limit in the maximal norm will be called the maximal localized equivariant Roe algebra X and denoted by Cmax* (X)Gloc.
The localised equivariant index 1. Indices of abstract operators
- The equivariant coarse index of elliptic operators In the rest of this section, we work with the reduced version of the equi-
- The localised index of elliptic operators
- Admissible and non-admissible modules
- Special cases
In the reduced case, a natural choice for D∗(X)G is the algebraDred∗ (X)G, an equivariant version of the algebraD∗(X) used in [37]. The equivariant rough index of elliptic operators In the rest of this section we work with the reduced version of the equi- In the rest of this section we work with the reduced version of the equi- variant Roe algebra and specialize to the geometric environment we are most interested in. Letχ ∈C∞(M) be a cutoff function, in the sense that its support has compact intersections with allG orbits, and that for allm∈M,.
The following result of Roe is the basis of the index theory that we develop in this article. Let us explain the importance of using admissible modules in the definition of the equivariant coarse index. This will lead to an equivalent definition of the localized equivariant coarse index in the graded case (3.8) below.
The importance of using admissible modules in the definition of a (localized) equivariant rough index is clear in the simplest case, where G is compact, M is a point, E = V, an irreducible representation of G (with trivial scaling ), and D= 0V is the null operator on V. 3) There is a technical subtlety in the case where M is a finite set: the finite dimensionality of V means that it is not a standard module (see Definition 2.1). In this case, indexpt,VG (0V) is just the integer 1 when viewed as an element of Z = K0(CIV), but the theoretical information about the representation of this index is encoded in the map⊕0.
This can for example be guaranteed by constructing Φ from projections in the Higson corona algebra as in [17]. The coarse geometric viewpoint we develop in this paper could already be useful in the case of trivial groups. Possibly more useful in the non-equivariant setting is to consider the lift of a Callias-type operator DΦ on a manifoldM, with fundamental group Γ, to a Γ-equivariant Callias-type operator on the universal cover of M.
In future work we intend to show that the Fredholm index of DΦ can be recovered from the localized equivariant rough index of its lifting via an application of the von Neumann trace, or the summation trace in the context of maximal group C∗ -algebras. In the general equivariant case, index theory of Callias-type operators was developed in [17]; we will see in Theorem 4.2 that Definition 3.3 generalizes the index of [17]. As in the case of Callias operators, a special case is the lifting of an operator on a compact manifold with boundary to the universal cover, in which case one obtains a refinement of the Atiyah–Patodi–Singer index in the K∗(Cred∗ ( π)), where π is the fundamental group of the compact manifold.
Results
The equivariant Callias index
LetEe → Mfbe aG-equivariant, Z2-graded, Hermitian vector bundle and a module over the Clifford bundle of Mf. Assume that De is a Dirac-type operator onM and that onU is of the form. The index of Definition 3.6 is now an equivariant Atiyah-Patodi-Singer type index for correct actions and reduces to the original APS index ifGis trivially.
One can find a continuous, G-invariant, compactly supported function f on M such that DΦ2 +f is invertible in the sense of HilbertCred∗ (G) Ej modules as in Definition 1 in [17]. Here, as in [17], we assume that EisZ2-graded and DΦ is odd with respect to the gradation, and the above class of theory K is defined in terms of the even part of F, as in the second point in Definition 3.2. One of our main results in this paper is that this index is a special case of the localized equivariant index.
If M/Gis is compact, then the localized equivariant index of an elliptic operator is its image under the analytic composition map. Note that if M/Gis is compact, then the localized equivariant coarse index is equal to the usual equivariant coarse index. In the case of discrete groups, Corollary 4.3 is the well-known fact that the equivariant rough index for a proper action is equal to the analytic composition map for such groups [35].
Positive scalar curvature
In future work, we intend to prove index theorems that allow us to derive concrete topological obstructions to positive scalar curvature from Theorem 4.4. If M is a proper G-manifold, where G is an almost connected Lie group, then there exists a global cutNwhich is a K-manifold, in the G-manifoldM, where Kis is a maximally compact subgroup of G is. If N is a bounded geometry manifold with a K-invariant Riemannian metric of uniform positive scalar curvature, then M = G×K N is a bounded geometry manifold with a G-invariant Riemannian metric of uniform positive scalar curvature.
A localised Baum–Connes conjecture
The converse of Proposition 4.9 is directly related to the question of whether the equivariant localized index of Definitions 3.3 and 3.6 lands on the image of the Baum-Connes assembly map. This question was posed to the equivariant Callias index in [17], and is generally open.
Proofs of properties of equivariant Roe algebras In Subsections 1–3, we prove Theorem 2.7, which guarantees the ex-
Proof of Theorem 2.7
The remaining step in the proof of Theorem 2.7 is to show that Ψ relates the local compactness of operators on HX with respect to C0(X) to the local compactness of operators onL2(G)⊗H with respect to C0(G). A bounded operatorTonHX is locally compact with respect to the action fromC0(X) if and only if the bounded operator Ψ−1◦T◦ΨonL2(G)⊗H is locally compact with respect to the action from C0(G). SupposeT is a bounded operator on L2(E)⊗L2(G) that is locally compact with respect to multiplication by C0(X).
Now suppose Te= Ψ−1◦T◦Ψ is a bounded operator on L2(G)⊗H which is locally compact with respect to the multiplicative action of C0(G) on the first factor. The general case for cocompactX and more than one cut follows from the single cut case as in the proof of Proposition 5.5. Under the assumption in Theorem 2.7 that G/K or X/Gis is infinite, the space H in Proposition 5.3 is infinite-dimensional.
Reduced and maximal equivariant Roe algebras can alternatively be described in terms of continuous Schwartz kernels of operators. In this subsection, we will always identify HX with L2(G)⊗H for an infinite-dimensional separable Hilbert space H, using the isomorphism Ψ in Definition 2.4. LetCker∗ (X)G denote the algebra of bounded operators on HX defined by continuous G-invariant Schwartz kernels.
We will prove Proposition 5.12 by showing that T can be approximated by elements of Cker∗ (X)G in the operator norm.
The equivariant Callias index
Index maps in the cocompact case
Let M be the multiplier algebra for C∗(X)G and let L:=L(EC∗(G)) be the algebra of adjoint operators on EC∗(G). The mapping (η◦U∗◦index) is the usual G-index mapping for Fredholm operators in terms of Hilbert C∗(G)-modules on EC∗(G)-modules. Thus, Proposition 6.3 provides the identification of an index mapping in the picture of a Roe algebra with the usual notion of G-index for operators in aG-cocompact space.
The localised equivariant index
Here C∗(G)andC∗(X)Gloc can be taken as either the reduced or maximal version of the groupC∗ and Roe algebras. We now relate the reduced version of the equivariant Callias-type index defined in [17] to (the reduced version of) the localized coarse index. By replacing DΦ by the operator 1cDΦ, which has the same index as DΦ, we can alternatively make the slightly more convenient assumption that D2Φ >1 outside Z.
By Proposition 6.5, their indices in K0(Cred∗ (G)) can be seen equivalently through each of these figures, and they are equal to both sides of (4.3).
Proof of Theorem 4.2
Positive scalar curvature and the localised Baum–Connes conjecture
Positive scalar curvature
The localised Baum–Connes conjecture
Thus, gM is a G-invariant metric with a uniform positive scalar curvature onM. If Z ⊂X is G-cocompact, the naturalness of boundary maps with respect to∗-homomorphisms implies that the diagram. Higson, “Classification of space for right actions and K-theory of groupC∗-algebras”, inC∗-algebras San Antonio, 1993), Contemporary Mathematics, vol. Lawson, Jr., “Positive scalar curvature and the Dirac operator on full Riemannian manifolds,” Publ.
Przybyszewska, “Proper metrics on locally compact groups and proper affine isometric actions on Banach spaces”, https://arxiv.org/abs/. Song, “Equivariant index for regular actions III: Invariant and discrete serial indices”, Differ. Wang, “Equivariant Atiyah–Patodi–Singer index theorem for proper actions I: index formula,” https://arxiv.org/abs/1904 .
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