Equivariant Callias index theory via coarse geometry

Abstract

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 $C^{\ast }$-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 $C^{\ast }$-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 $C^{\ast }$-algebra.

47 pages

References in corpus (7)

• The topology of positive scalar curvature
• Positive curvature, partial vanishing theorems, and coarse indices
• Positive Scalar Curvature and Poincare Duality for Proper Actions
• Index of Equivariant Callias-Type Operators and Invariant Metrics of Positive Scalar Curvature
• Coarse geometry and Callias quantisation
• The coarse index class with support
• Callias-type operators in $C^{\ast }$-algebras and positive scalar curvature on noncompact manifolds

Cited by in corpus (3)

• Coarse geometry and Callias quantisation
• Positive scalar curvature and an equivariant Callias-type index theorem for proper actions
• Relative Equivariant Coarse Index Theorem and Relative $L^2$-Index Theorem