Topologically free actions and ideals in twisted Banach algebra crossed products

TL;DR

This paper extends key $C^*$-algebraic results on topological freeness and ideals to twisted Banach algebra crossed products, providing new simplicity criteria and ideal structure insights.

Contribution

It generalizes the concept of topological freeness and intersection properties from $C^*$-algebras to Banach algebras with twisted actions, including $L^p$-operator algebras.

Findings

Topological freeness is equivalent to the intersection property in twisted Banach algebra crossed products.

Identifies the prime ideal space as the quasi-orbit space of the action.

Full and reduced twisted $L^p$-operator algebras coincide for amenable actions.

Abstract

We generalize the influential $C^*$-algebraic result of Kawamura-Tomiyama and Archbold-Spielberg for crossed products of discrete transformation groups to the realm of Banach algebras and twisted actions. We prove that topological freeness is equivalent to the intersection property for all reduced twisted Banach algebra crossed products coming from subgroups, and in the untwisted case to a generalised intersection property for a full $L^p$-operator algebra crossed product for any $p\in [1,\infty]$. This gives efficient simplicity criteria for various Banach algebra crossed products. We also use it to identify the prime ideal space of some crossed products as the quasi-orbit space of the action. For amenable actions we prove that the full and reduced twisted $L^p$-operator algebras coincide.