The main goal of this talk is to introduce twisted crossed products of Banach algebras by locally compact groups. 

Classical crossed products of Banach algebras have been extensively studied for different classes of representations, including contractive representations on L^p-spaces. In this talk, we will give a general formulation for Banach algebras associated with twisted dynamical systems. Recent developments in L^p-twisted crossed products have mostly focused on situations where either the algebra is the complex numbers or when the group is discrete (more generally for étale groupoids). We present a universal characterization of the twisted crossed product when the acting group is locally compact and the Banach algebra has a contractive approximate identity. 

As an application, we focus on the case when the representations are contractive ones acting on L^p spaces. We briefly discuss a reduced version for L^p-operator algebras and present conjectures regarding amenability and rigidity when p\neq 2. Time permitting, we will present a generalization of the so called Packer–Raeburn trick to the L^p-setting, by showing that the universal L^p twisted crossed product is ``stably'' isometrically isomorphic to an untwisted one. 

This is joint work with Carla Farsi and Judith Packer.