Topologically free actions and ideals in twisted Banach algebra crossed products

Abstract

We generalize the influential $C^{\ast }$-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.

Accepted for publication (and already published online) by Proceedings of the Royal Society of Edinburgh. Small changes, e.g. notation for L^P-algebras changed

References in corpus (1)

• The ideal structure of reduced crossed products

Cited by in corpus (4)

• Banach algebras associated to twisted étale groupoids: inverse semigroup disintegration and representations on $L^p$-spaces
• On the topological ranks of Banach ${}^{\ast }$-algebras associated with groups of subexponential growth
• Twisted crossed products of Banach algebras
• C*-like modules and matrix $p$-operator norms