Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A new approach to deformation of C*-algebras via coactions

Alcides Buss, Siegfried Echterhoff

TL;DR

The paper provides a unified deformation framework for $C^*$-algebras with coactions by locally compact groups, extending Kasprzak's cocycle approach to maximal, reduced, and exotic coactions and to circle-valued 2-cocycles. It exploits Landstad duality and generalized fixed-point algebras to realize deformations via twists, 2-cocycles, or Fell-bundle twists, without separability restrictions. The authors prove the expected structural properties, including nuclearity preservation, semicontinuity/continuity of deformation fields, and $K$-theory invariance under homotopies or $K$-amenability assumptions, linking the theory to Baum–Connes. They also develop a robust machinery for deformations by twists and continuous families, establish compatibility with product twists, and derive K-theoretic consequences for broad classes of groups and crossed-product functors. Overall, the work furnishes a comprehensive, flexible deformation paradigm that unifies and extends prior approaches while delivering practical invariance results and continuity behavior for a wide spectrum of group actions and coactions.

Abstract

We revisit the procedure of deformation of $C^*$-algebras via coactions of locally compact groups and extend the methods to cover deformations for maximal, reduced, and exotic coactions for a given group $G$ and circle-valued Borel $2$-cocycles on $G$. In the special case of reduced (or normal) coactions our deformation method substantially differs from -- but turns out to be equivalent to -- the ones used by previous authors, specially those given by Bhowmick, Neshveyev, and Sangha in [7]. Our approach yields all expected results, like a good behaviour of deformations under nuclearity, continuity of fields of $C^*$-algebras and $K$-theory invariance under mild conditions.

A new approach to deformation of C*-algebras via coactions

TL;DR

The paper provides a unified deformation framework for -algebras with coactions by locally compact groups, extending Kasprzak's cocycle approach to maximal, reduced, and exotic coactions and to circle-valued 2-cocycles. It exploits Landstad duality and generalized fixed-point algebras to realize deformations via twists, 2-cocycles, or Fell-bundle twists, without separability restrictions. The authors prove the expected structural properties, including nuclearity preservation, semicontinuity/continuity of deformation fields, and -theory invariance under homotopies or -amenability assumptions, linking the theory to Baum–Connes. They also develop a robust machinery for deformations by twists and continuous families, establish compatibility with product twists, and derive K-theoretic consequences for broad classes of groups and crossed-product functors. Overall, the work furnishes a comprehensive, flexible deformation paradigm that unifies and extends prior approaches while delivering practical invariance results and continuity behavior for a wide spectrum of group actions and coactions.

Abstract

We revisit the procedure of deformation of -algebras via coactions of locally compact groups and extend the methods to cover deformations for maximal, reduced, and exotic coactions for a given group and circle-valued Borel -cocycles on . In the special case of reduced (or normal) coactions our deformation method substantially differs from -- but turns out to be equivalent to -- the ones used by previous authors, specially those given by Bhowmick, Neshveyev, and Sangha in [7]. Our approach yields all expected results, like a good behaviour of deformations under nuclearity, continuity of fields of -algebras and -theory invariance under mild conditions.
Paper Structure (14 sections, 34 theorems, 136 equations)

This paper contains 14 sections, 34 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Exotic crossed products
  4. Coactions and Landstad duality
  5. Deformation by coactions
  6. Exotic deformation by coactions
  7. Abadie-Exel deformation
  8. Deformation by continuous cocycles
  9. Group twists
  10. Deformation by twists
  11. Compatibility with products
  12. Nuclearity
  13. Continuity
  14. K-theoretic results

Key Result

Theorem 1.1

Suppose $(A,\delta)$ is a coaction such that the dual action $\beta=\widehat{\delta}$ on $B=A\rtimes_\delta\widehat{G}$ is amenable (e.g., if $G$ is amenable). Then $(A,\delta)$ is normal and, for every $\sigma\in \mathrm{Twist}(G)$, the deformed *-algebra $A^\sigma$ is nuclear if and only if $A$ is

Theorems & Definitions (84)

  • Theorem 1.1: Nuclearity
  • Theorem 1.2: Continuity
  • Theorem 1.3: $K$-theory invariance
  • Definition 2.1
  • Theorem 2.4: exotic Landstad duality
  • Proposition 2.8: Buss-Echterhoff:Exotic_GFPA
  • Proposition 2.9
  • Remark 2.10
  • Remark 3.1
  • Lemma 3.3
  • ...and 74 more