Twisted crossed products of Banach algebras

Abstract

Given a locally compact group $G$, a nondegenerate Banach algebra $A$ with a contractive approximate identity, a twisted action $(α, σ)$ of $G$ on $A$, and a family $\mathcal{R}$ of uniformly bounded representations of $A$ on Banach spaces, we define the twisted crossed product $F_\mathcal{R}(G,A,α, σ)$. When $\mathcal{R}$ consists of contractive representations, we show that $F_\mathcal{R}(G,A,α, σ)$ is a Banach algebra with a contractive approximate identity, which can also be characterized by an isometric universal property. As an application, we specialize to the $L^p$-operator algebra setting, defining both the $L^p$-twisted crossed product and the reduced version. Finally, we give a generalization of the so-called Packer-Raeburn trick to the $L^p$-setting, showing that any $L^p$-twisted crossed product is "stably" isometrically isomorphic to an untwisted one.