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- Title
A Lichnerowicz vanishing theorem for the maximal Roe algebra.
- Authors
Guo, Hao; Xie, Zhizhang; Yu, Guoliang
- Abstract
We show that if a countable discrete group acts properly and isometrically on a spin manifold of bounded Riemannian geometry and uniformly positive scalar curvature, then, under a suitable condition on the group action, the maximal higher index of the Dirac operator vanishes in K-theory of the maximal equivariant Roe algebra. The group action is not assumed to be cocompact. A key step in the proof is to establish a functional calculus for the Dirac operator in the maximal equivariant uniform Roe algebra. This allows us to prove vanishing of the index of the Dirac operator in K-theory of this algebra, which in turn yields the result for the maximal higher index.
- Subjects
VANISHING theorems; DIRAC operators; ALGEBRA; RIEMANNIAN manifolds; RIEMANNIAN geometry; K-theory; GROUP actions (Mathematics); DISCRETE groups
- Publication
Mathematische Annalen, 2023, Vol 385, Issue 1/2, p717
- ISSN
0025-5831
- Publication type
Article
- DOI
10.1007/s00208-021-02333-0