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

TL;DR

This paper demonstrates that for actions with finite dynamical complexity, an $L^p$ assembly map is an isomorphism, extending known results for $p=2$ to all $p$ in $[1, olinebreak\infty)$ using controlled $K$-theory.

Contribution

It extends the isomorphism result of the Baum-Connes assembly map from $p=2$ to all $p$ in $[1, olinebreak\\infty)$ for actions with finite dynamical complexity.

Findings

The $L^p$ assembly map is an isomorphism under finite dynamical complexity.

Extension of Baum-Connes isomorphism from $p=2$ to all $p$ in $[1, olinebreak\\infty)$.

Application of controlled $K$-theory to $L^p$ 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.