Dynamical Complexity and $K$-Theory of $L^p$ Operator Crossed Products

Abstract

We apply quantitative (or controlled) $K$-theory to prove that a certain $L^p$ assembly map is an isomorphism for $p\in [1,\infty )$ when an action of a countable discrete group $\Gamma$ on a compact Hausdorff space $X$ has finite dynamical complexity. When $p=2$, this is a model for the Baum-Connes assembly map for $\Gamma$ with coefficients in $C(X)$, and was shown to be an isomorphism by Guentner, Willett, and Yu.

32 pages. Added a new section at the end with a brief discussion of the domain of the assembly map and *-algebraic versions of the algebras considered. Also made corrections and improvements to exposition based on the referee's comments

References in corpus (4)

• Quasidiagonality of nuclear C*-algebras
• Representations of $p$-convolution algebras on $L^q$-spaces
• Rigidity of ${\ell }^p$ Roe-type algebras
• Quantitative $K$-theory for Banach algebras

Cited by in corpus (3)

• Topologically free actions and ideals in twisted Banach algebra crossed products
• Persistence approximation property for $L^p$ operator algebras
• $L^p$ coarse Baum-Connes conjecture via $C_0$ coarse geometry