Twisted crossed products of Banach algebras

TL;DR

The work generalizes twisted crossed products to Banach algebras with a contractive approximate identity, establishing a universal framework via a uniformly bounded family of covariant representations. It develops full and reduced $L^p$-twisted crossed products for $L^p$-operator algebras, proving stability under exterior equivalence and providing a $p$-analogue of the Packer–Raeburn untwisting trick. The results extend core C*-algebra crossed-product theory to Banach and $L^p$-operator settings, yielding robust universal properties, covariant-representation correspondences, and stable untwisting formulas. The constructions enable a coherent analysis of twisted actions on Banach algebras and their $L^p$-operator algebra realizations, with potential applications to representation theory and non-C*-algebraic dynamics.

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.

Theorems & Definitions (65)

• Definition 2.5
• Definition 2.6
• Theorem 2.7
• Definition 2.8
• Definition 3.1
• Remark 3.2
• Remark 3.3
• Remark 3.4
• Definition 3.5
• Proposition 3.6