In this paper, we give a definition of the relative equivariant coarse index for proper actions and derive a relative equivariant coarse index theorem connecting this index with the localized equivariant coarse indices. This is an equivariant version of Roe's relative coarse index theorem in arXiv:arch-ive/1210.6100. Furthermore, we present a definition of the relative $L^2$-index and prove a relative $L^2$-index theorem which is a relative version of Atiyah's $L^2$-index theorem.
The equivariant coarse Novikov conjecture provides an algorithm for determining nonvanishing of equivariant higher index of elliptic differential operators on noncompact manifolds. In this article, we prove the equivariant coarse Novikov conjecture under certain coarse embeddability conditions. More precisely, if a discrete group $Γ$ acts on a bounded geometric space $X$ properly, isometrically, and with bounded distortion, $X/Γ$ and $Γ$ admit coarse embeddings into Hilbert space, then the $Γ$-equivariant coarse Novikov conjecture holds for $X$. Here bounded distortion means that for any $γ\in…
The equivariant coarse Baum-Connes conjecture was firstly introduced by Roe [29] as a unified way to approach both the Baum-Connes conjecture and its coarse counterpart. In this paper, we prove that if an a-T-menable group $Γ$ acts properly and isometrically on a bounded geometry metric space $X$ with controlled distortion such that the quotient space $X/Γ$ is coarsely embeddable, then the equivariant coarse Baum-Connes conjecture holds for this action. This answers a question posed in [8] affirmatively.
We derive the Pimsner-Voiculescu sequence calculating the K-theory of a $C^{*}$- algebra with $\mathbb{Z}$-action using constructions with equivariant coarse K-homology theory. We then investigate to which extend this idea extends to more general equivariant coarse homology theories.
Let $Γ$ be a discrete icc subgroup of PSL(2,R) of infinite covolume. and let M denote the quotient of the unit disc by $Γ$. We prove that a Toeplitz operator with $Γ$-invariant symbol f in C(M) is Brauer Fredholm if its symbol is invertible on the boundary of M and its Brauer index is equal to the winding number of f at the boundary. We construct the associated extension of the algebra of functions continuous on the boundary of M by the Brauer ideal in the C*-algebra generated by such operators.
The equivariant coarse index is well-understood and widely used for actions by discrete groups. We extend the definition of this index to general locally compact groups. We use a suitable notion of admissible modules over $C^*$-algebras of continuous functions to obtain a meaningful index. Inspired by work by Roe, we then develop a localised variant, with values in the $K$-theory of a group $C^*$-algebra. This generalises the Baum-Connes assembly map to non-cocompact actions. We show that an equivariant index for Callias-type operators is a special case of this localised index, obtain results …
The paper constructs the analytic index for an elliptic pseudodifferential family of $L^{m}_{ρ,\de}$-operators invariant under the proper action of a continuous family groupoid on a $G$-compact, $C^{\infty,0}$ $G$-space.
We prove a local index formula in conformal geometry by computing the Connes-Chern character for the conformal Dirac (twisted) spectral triple recently constructed by Connes-Moscovici. Following an observation of Moscovici, the computation reduces to the computation of the CM cocycle of an equivariant Dirac (ordinary) spectral triple. This computation is obtained as a straightforward consequence of a new proof of the local equivariant index theorem of Patodi, Donelly-Patodi and Gilkey. This proof is obtained by combining Getzler's rescaling with an equivariant version of Greiner's approach to …
Consider a proper, isometric action by a unimodular locally compact group $G$ on a Riemannian manifold $M$ with boundary, such that $M/G$ is compact. Then an equivariant Dirac-type operator $D$ on $M$ under a suitable boundary condition has an equivariant index $\operatorname{index}_G(D)$ in the $K$-theory of the reduced group $C^*$-algebra $C^*_rG$ of $G$. This is a common generalisation of the Baum-Connes analytic assembly map and the (equivariant) Atiyah-Patodi-Singer index. In part I of this series, a numerical index $\operatorname{index}_g(D)$ was defined for an element $g \in G$, in term…
We compute $K$-theory invariants of algebras of pseudodifferential operators on manifolds with corners and prove an equivariant index theorem for operators invariant with respect to an action of $\R^k.$ We discuss the relation between our results and the $η$-invariant.
Equivariant indices have previously been defined in cases where either the group or the orbit space in question is compact. In this paper, we develop an equivariant index without assuming the group or the orbit space to be compact. This allows us to generalise an index of deformed Dirac operators, defined for compact groups by Braverman. In parts II and III of this series, we will explore some properties and applications of this index.
For a proper, cocompact action by a locally compact group of the form $H \times G$, with $H$ compact, we define an $H \times G$-equivariant index of $H$-transversally elliptic operators, which takes values in $KK_*(C^*H, C^*G)$. This simultaneously generalises the Baum--Connes analytic assembly map, Atiyah's index of transversally elliptic operators, and Kawasaki's orbifold index. This index also generalises the assembly map to elliptic operators on orbifolds. In the special case where the manifold in question is a real semisimple Lie group, $G$ is a cocompact lattice and $H$ is a maximal comp…
Consider a proper, isometric action by a unimodular locally compact group $G$ on a Riemannian manifold $M$ with boundary, such that $M/G$ is compact. For an equivariant, elliptic operator $D$ on $M$, and an element $g \in G$, we define a numerical index $\operatorname{index}_g(D)$, in terms of a parametrix for $D$ and a trace associated to $g$. We prove an equivariant Atiyah-Patodi-Singer index theorem for this index. We first state general analytic conditions under which this theorem holds, and then show that these conditions are satisfied if $g=e$ is the identity element; if $G$ is a finitel…
Let $\GR \to B$ be a bundle of compact Lie groups acting on a fiber bundle $Y \to B$. In this paper we introduce and study gauge-equivariant $K$-theory groups $K_\GR^i(Y)$. These groups satisfy the usual properties of the equivariant $K$-theory groups, but also some new phenomena arise due to the topological non-triviality of the bundle $\GR \to B$. As an application, we define a gauge-equivariant index for a family of elliptic operators $(P_b)_{b \in B}$ invariant with respect to the action of $\GR \to B$, which, in this approach, is an element of $K_\GR^0(B)$. We then give another definition…
In the first part of this series, we defined an equivariant index without assuming the group acting or the orbit space of the action to be compact. This allowed us to generalise an index of deformed Dirac operators, defined for compact groups by Braverman. In this paper, we investigate properties and applications of this index. We prove that it has an induction property that can be used to deduce various other properties of the index. In the case of compact orbit spaces, we show how it is related to the analytic assembly map from the Baum-Connes conjecture, and an index used by Mathai and Zhan…
Let $T$ be a circle and $LT$ be its loop group. Let $\mathcal{M}$ be an infinite dimensional manifold equipped with a nice $LT$-action. We construct an analytic $LT$-equivariant index for $\mathcal{M}$, and justify it in terms of noncommutative geometry. More precisely, we construct a Hilbert space $\mathcal{H}$ consisting of "$L^2$-sections of a Clifford module bundle" and a "Dirac operator" $\mathcal{D}$ which acts on $\mathcal{H}$. Then, we define an analytic index of $\mathcal{D}$ valued in the representation group of $LT$, so called Verlinde ring. We also define a "twisted crossed product…
We construct a periodic cyclic cocycle on the symbol algebra of Boutet de Monvel operators and use it to interpret the index formula for elliptic pseudodifferential boundary value problems due to Fedosov as the Chern--Connes pairing of the classes in $K$-theory of elliptic symbols with this cyclic cocycle. We also consider the equivariant case. Namely, we construct a periodic cyclic cocycle on the crossed product of the algebra of symbols with a group acting on this algebra by automorphisms. Such crossed products arize in index theory of nonlocal boundary value problems with shift operators.
We develop a fundamental theory of compact quantum group equivariant finite extensions of C*-algebras. In particular we focus on the case of quantum homogeneous spaces and give a Tannaka-Krein type result for equivariant correspondences. As its application, we show that every Jones' value appears as the index of an equivariant conditional expectation. In the latter half of this paper, we give an imprimitivity theorem in some cases: for general compact quantum groups under a finiteness conditions, and for the Drinfeld-Jimbo deformation $G_q$ of a simply-connected compact Lie group $G$. As an ap…
Let $T$ be a circle group, and $LT$ be its loop group. We hope to establish an index theory for infinite-dimensional manifolds which $LT$ acts on, including Hamiltonian $LT$-spaces, from the viewpoint of $KK$-theory. We have already constructed several objects in the previous paper \cite{T}, including a Hilbert space $\mathcal{H}$ consisting of "$L^2$-sections of a Spinor bundle on the infinite-dimensional manifold", an "$LT$-equivariant Dirac operator $\mathcal{D}$" acting on $\mathcal{H}$, a "twisted crossed product of the function algebra by $LT$", and the "twisted group $C^*$-algebra of $L…
Let $T$ be the circle group and let $LT$ be its loop group. We formulate and investigate several topological aspects of the $LT$-equivariant index theory for proper $LT$-spaces, where proper $LT$-spaces are infinite-dimensional manifolds equipped with "proper cocompact" $LT$-actions. Concretely, we introduce "$\mathcal{R}KK$-theory for infinite-dimensional manifolds", and by using it, we formulate an infinite-dimensional version of the $KK$-theoretical Poincaré duality homomorphism, and an infinite-dimensional version of the $\mathcal{R}KK$-theory counterpart of the assembly map, for proper $L…
This paper studies higher index theory for a random sequence of bounded degree, finite graphs with diameter tending to infinity. We show that in a natural model for such random sequences the following hold almost surely: the coarse Baum-Connes assembly map is injective; the coarse Baum-Connes assembly map is not surjective; the maximal coarse Baum-Connes assembly map is an isomorphism. These results are closely tied to issues of expansion in graphs: in particular, we also show that such random sequences almost surely do not have geometric property (T), a strong form of expansion. The key geome…
We consider the algebra $A$ of bounded operators on $L^2(\mathbb{R}^n)$ generated by quantizations of isometric affine canonical transformations. The algebra $A$ includes as subalgebras all noncommutative tori and toric orbifolds. We define the spectral triple $(A, H, D)$ with $H=L^2(\mathbb R^n, Λ(\mathbb R^n))$ and the Euler operator $D$, a first order differential operator of index $1$. We show that this spectral triple has simple dimension spectrum: For every operator $B$ in the algebra $Ψ(A,H,D)$ generated by the Shubin type pseudodifferential operators and the elements of $A$, the zeta f…
In this paper we study the index theoretic interpretation of the analytical assembly map that appears in the Baum-Connes conjecture. In its general form it may be constructed using Kasparov's equivariant KK-theory. In the special case of a torsionfree group the domain simplifies to the usual K-homology of the classifying space BG of G and it is frequently used that in this case the analytical assembly map is given by assigning to an operator an equivariant index. We give a precise formulation of this statement and prove it.
Consider a proper, isometric action by a unimodular, locally compact group $G$ on a complete Riemannian manifold $M$. For equivariant elliptic operators that are invertible outside a cocompact subset of $M$, we show that a localised index in the $K$-theory of the maximal group $C^*$-algebra of $G$ is well-defined. 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 $C^*$-algebra instead of its reduced …
Recently, two of the authors of this paper constructed cyclic cocycles on Harish-Chandra's Schwartz algebra of linear reductive Lie groups that detect all information in the $K$-theory of the corresponding group $C^*$-algebra. The main result in this paper is an index formula for the pairings of these cocycles with equivariant indices of elliptic operators for proper, cocompact actions. This index formula completely determines such equivariant indices via topological expressions.
Let M be a complete Riemannian manifold, D a Dirac-type operator on M whose Weitzenbock curvature is uniformly positive on the complement of a subset Z of M. We show that the coarse index of D is localized to the K-theory of the coarse C*-algebra of Z. Applications are discussed, including a coarse form of the relative index theorem.
We define the gauge-equivariant index of a family of elliptic operators invariant with respect to the free action of a family $\GR \to B$ of Lie groups (these families are called ``gauge-invariant families'' in what follows). If the fibers of $\GR \to B$ are simply-connected and solvable, we compute the Chern character of the gauge-equivariant index, the result being given by an Atiyah-Singer type formula that incorporates also topological information about the bundle $\GR \to B$. The algebras of invariant pseudodifferential operators that we study, $\Psm {\infty}Y$ and $\PsS {\infty}Y$, are g…
