Uniform Roe algebras and geometric RD property. (English) Zbl 1468.46075
Curto, Raul E. (ed.) et al., Operator theory, operator algebras and their interactions with geometry and topology. Ronald G. Douglas memorial volume. Proceedings of the international workshop on operator theory and applications (IWOTA 2018), Shanghai, China, July 23–27, 2018. Cham: Birkhäuser. Oper. Theory: Adv. Appl. 278, 359-371 (2020).
Summary: We survey an early work of the authors on the notion of geometric RD property for a uniformly locally finite metric space. We show that metric spaces of polynomial growth satisfy this property. Associated to a metric space \(X\) with the geometric RD property we define a Fréchet space \(BS_2(X)\) which in fact is a smooth and dense subalgebra of the uniform Roe algebra of the space \(X\). This resulted an alternate proof of a result of the first named author on the nonexistence of positive scalar curvature on certain manifolds.
For the entire collection see [Zbl 1455.47001].
For the entire collection see [Zbl 1455.47001].
MSC:
| 46L80 | \(K\)-theory and operator algebras (including cyclic theory) |
| 58J60 | Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) |
