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Topologically free actions and ideals in twisted Banach algebra crossed products

K. Bardadyn, B. K. Kwaśniewski

TL;DR

This work extends the C*-algebraic simplicity program to twisted Banach algebra crossed products by generalising topological freeness and the intersection property to $F(\alpha,u)$ and its $L^p$- and $P$-parametrised analogues. The authors develop a Fourier-decomposition framework that replaces positivity-driven arguments and prove that topological freeness is equivalent to a generalised intersection property for reduced crossed products, with amenable actions ensuring full and reduced coincide. They provide precise descriptions of ideal structure via the quasi-orbit space and establish simplicity criteria for a broad class of Banach-algebra crossed products, including twisted and residually reduced variants. The results extend to fibre-bundle representations over $X$, yield quasi-orbit maps for prime ideal spaces, and connect dynamics to spectral data in the $L^p$-operator algebra setting, thereby unifying and expanding known C*-theoretic results to a wide Banach-algebra context.

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.

Topologically free actions and ideals in twisted Banach algebra crossed products

TL;DR

This work extends the C*-algebraic simplicity program to twisted Banach algebra crossed products by generalising topological freeness and the intersection property to and its - and -parametrised analogues. The authors develop a Fourier-decomposition framework that replaces positivity-driven arguments and prove that topological freeness is equivalent to a generalised intersection property for reduced crossed products, with amenable actions ensuring full and reduced coincide. They provide precise descriptions of ideal structure via the quasi-orbit space and establish simplicity criteria for a broad class of Banach-algebra crossed products, including twisted and residually reduced variants. The results extend to fibre-bundle representations over , yield quasi-orbit maps for prime ideal spaces, and connect dynamics to spectral data in the -operator algebra setting, thereby unifying and expanding known C*-theoretic results to a wide Banach-algebra context.

Abstract

We generalize the influential -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 -operator algebra crossed product for any . 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 -operator algebras coincide.
Paper Structure (6 sections, 33 theorems, 64 equations)

This paper contains 6 sections, 33 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reduced $L^p$-crossed products and amenable actions
  4. Fourier decomposition and reduced crossed products
  5. Topological freeness and intersection properties
  6. Simplicity and ideal structure

Key Result

Proposition 2.2

Cocycles $u,v\in Z^2(G,C_u(X))$ are equivalent if and only if $F(\alpha,u)\cong F(\alpha,v)$ by an isometric isomorphism that respects the $G$-gradings and it is the identity on $C_0(X)$. Any such isomorphism $\psi:F(\alpha,u) \to F(\alpha,v)$ is $*$-preserving and is of the form where $\xi:G\to C_u(X)$ is such that $u=vd^{2}(\xi)$. Thus such isomorphisms are parametrised by $1$-cocycles $Z^1(G,C

Theorems & Definitions (98)

  • Definition 2.1
  • Proposition 2.2: cf. Zeller-Meier
  • proof
  • Remark 2.3
  • Corollary 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8: cf. Zeller-Meier
  • proof
  • ...and 88 more