C*-dynamical rapid decay

TL;DR

This paper extends the rapid decay property to reduced crossed products of C*-algebras by certain discrete groups, including hyperbolic groups and groups of polynomial growth, establishing key inequalities.

Contribution

It generalizes known rapid decay results to a broader class of C*-dynamical systems involving reduced crossed products.

Findings

Inequality holds for hyperbolic groups and polynomial growth groups

Extends rapid decay property to C*-dynamical setting

Provides bounds for finitely supported operators in reduced crossed products

Abstract

Some well known results by Haagerup, Jolissaint and de la Harpe may be extended to the setting of a reduced crossed product of a C*-algebra A by a discrete group $G.$ We show that for many discrete groups, which include Gromov's hyperbolic groups and finitely generated discrete groups of polynomial growth, an inequality of the form $$\|X\| \leq C \sqrt{\sum_{g \in G} (1+|g|)^4 \|X_g\|^2 } $$ holds for any finitely supported operator $X$ in the reduced crossed product.