Controlled K-theory for groupoids & applications to Coarse Geometry

Abstract

We develop a generalization of quantitative K-theory, which we call controlled K-theory. It is powerful enough to study the K-theory of crossed product of C * -algebras by action ofétale groupoids and discrete quantum groups. In this article, we will use it to study groupoids crossed products. We define controlled assembly maps, which factorize the Baum-Connes assembly maps, and define the controlled Baum-Connes conjecture. We relate the controlled conjecture for groupoids to the classical conjecture, and to the coarse Baum-Connes conjecture. This allows to give applications to Coarse Geometry. In particular, we can prove that the maximal version of the controlled coarse Baum-Connes conjecture is satisfied for a coarse space which admits a fibred coarse embedding, which is a stronger version of a result of M. Finn-Sell.

$L^p$ coarse Baum–Connes conjecture and $K$-theory for $L^p$ Roe algebras

“…It encodes more geometric information, and it is a powerful tool to compute the K-theory of Roe algebras or other C -algebras coming from geometry. The quantitative K-theory has been generalized to general geometric C -algebras by Oyono-Oyono and Yu [29][30][31], to Banach algebras by Yeong-Chyuan Chung [6], and to groupoids by Clement Dell'Aiera [11]. It has many important applications in dynamical systems [7,17] and coarse geometry [8,25].…”

The equivariant coarse Novikov conjecture and coarse embedding

“…When the Γ-action is cocompact, i.e., X/Γ is compact, C * (X) Γ is Morita equivalent to C * r (G), the reduced C * -algebra of G, then the equivariant higher index map is the Baum-Connes map introduced by Baum, Connes and Higson (see [1,2]). When Γ is trivial, the equivariant higher index map is the coarse Baum-Connes map introduced by Roe, Higson and Yu (see [20,26,27,37,5,8,9,10]).…”