Quantitative K-theory for Banach algebras

Abstract

Abstract. Quantitative (or controlled) K-theory for C * -algebras was used by Guoliang Yu in his work on the Novikov conjecture, and later developed more formally by Yu together with Hervé Oyono-Oyono. In this paper, we extend their work by developing a framework of quantitative K-theory for the class of algebras of bounded linear operators on subquotients (i.e., subspaces of quotients) of Lp spaces. We also prove the existence of a controlled Mayer-Vietoris sequence in this framework.

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

“…We will follow the notation in [46]. One may refer to [6,29] for more detail about the controlled K-theory for C -algebras and L p -algebras.…”

“…Yeong-Chyuan Chung developed a quantitative K-theory for Banach algebras [6] and applied this theory to compute K-theory of L p crossed products [7]. Chung showed that the L p Baum-Connes conjecture for G with coefficient in C.X / is true if the dynamical system G Õ X has finite dynamical complexity, a property introduced by Guentner-Willett-Yu [17] and obtained a partial answer about p-independence of L p crossed products.…”

“…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].…”

Persistence Approximation Property for Maximal Roe Algebras

“…The ordinary K-theory developed in [1] of Banach algebras is focused on idempotents or invertibles, in comparison, quantitative K-theory studied in [2] for Banach algebras is focused on quasi-idempotents or quasi-invertibles. In this section, we recall some basic definitions and theorems of quantitative K-theory for filtered SQ p algebras from [2].…”

“…Based on their work, Yeong Chyuan Chung later extended the framework of quantitative K-theory to the class of algebras of bounded linear operators on subquotients of L p spaces for p ∈ [1, ∞) (i.e. SQ p algebras) in [2]. Since an L p operator algebra is obviously an SQ p algebra, we can derive a framework of quantitative K-theory for L p operator algebras by applying Chung's work to the L p operator algebras.…”