Equivariant Callias index theory via coarse geometry

Abstract

The equivariant coarse index is well-understood and widely used for actions by discrete groups. We first 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 a work of 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 on existence and non-existence of Riemannian metrics of positive scalar curvature invariant under proper group actions, and show that a localised version of the Baum-Connes conjecture is weaker than the original conjecture, while still giving a conceptual description of the K-theory of a group C * -algebra.Résumé. -L'indice grossier équivariant est bien compris et utilisé pour les actions par les groupes discrets. On commence par étendre la définition de cet indice aux groupes localement compacts généraux. On utilise une notion de modules admissibles sur des C * -algèbres de functions continues, pour obtenir un indice utile. Inspirés par le travail de Roe, nous développons une variante localisée, à valeurs dans la K-théorie de la C * -algèbre d'un groupe, généralisant l'assembly map de Baum-Connes aux actions non-cocompactes. On montre qu'un indice pour des opérateurs de type Callias est un cas spécial de cet indice localisé; on obtient des résultats sur l'existence et la non-existence de métriques Riemanniennes à courbure scalaire positive, invariantes par des actions propres; et on montre qu'une version localisée de la conjecture de Baum-Connes est plus faible que la conjecture originale, et on donne une description conceptuelle de la K-théorie des C * -algèbres de groupes.

An equivariant Atiyah–Patodi–Singer index theorem for proper actions II: the K-theoretic index

“…(Here an action is called cocompact if its quotient is compact.) This is a generalisation of the usual equivariant index in the compact case, and takes values in K * (C * r G), the K -theory of the reduced group C * -algebra of G. In [18], a generalisation of the assembly map was constructed and studied, which applies to possibly non-cocompact actions, as long as the operator it is applied to is invertible outside a cocompact set in the appropriate sense. This index also generalises the Gromov-Lawson index [16], an equivariant index of Callias-type operators [17], the (equivariant) APS index on manifolds with boundary [1,13], and the index used by Ramachandran for manifolds with boundary [37].…”

“…If X /G is compact, then L 2 (E) ⊗ L 2 (G) is an admissible equivariant C 0 (X )-module, under the non-essential assumption that either X /G or G/K , for a maximal compact subgroup K < G, is infinite. See Theorem 2.7 in [18]. This type of C 0 (X )-module is central to the constructions in [18].…”

“…The localised equivariant Roe algebra of X is the closure in [18]). And, assuming either Z /G or G/K is an infinite set,…”

“…from [18], where C * r G is the reduced group C * -algebra of G. This index was defined in [18] in a more general setting, and applied to, for example, Callias-type operators and positive scalar curvature [19] and the quantisation commutes with reduction problem [20].…”

“…• If N = ∅, then (1.2) is the image of D under the Baum-Connes analytic assembly map [4], see Corollary 4.3 in [18]. That is the most natural and widely-used generalisation of the classical equivariant index to proper, cocompact actions.…”