Equivariant Callias index theory via coarse geometry

Guo, Hao; Hochs, Peter; Mathai, Varghese

Annales de l'Institut Fourier, Tome 71 (2021) no. 6, pp. 2387-2430.

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.

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.

Classification : 19K56, 58J20, 46L80, 22D15
Keywords: Roe algebra, equivariant index, proper group action, locally compact group, Callias-type operator
Mot clés : Algèbre de Roe, indice équivariant, action propre, groupe localement compact, opérateur de type Callias

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