[Paper Review] Amenability and exactness for dynamical systems and their C*-algebras

Claire Anantharaman-Delaroche|ArXiv.org|May 2, 2000

Advanced Operator Algebra Research9 references19 citations

TL;DR

This paper establishes a precise correspondence between amenability of C*-dynamical systems and exactness/nuclearity of their crossed products, showing that a discrete group is exact if and only if it admits an amenable action on a compact space. The work provides direct, groupoid-free proofs of key results linking dynamical amenability to operator algebraic properties, with applications to uniform embeddability into Hilbert spaces for amenable-at-infinity groups.

ABSTRACT

In this survey, we study the relations between amenability (resp. amenability at infinity) of C*-dynamical systems and equality or nuclearity (resp. exactness) of the corresponding crossed products.

Motivation & Objective

• To clarify the relationship between amenability of C*-dynamical systems and the exactness or nuclearity of their crossed products.
• To provide accessible, direct proofs of these relationships without relying on groupoid theory.
• To establish that a discrete group is exact if and only if it admits an amenable action on a compact space.
• To extend the connection between dynamical amenability and uniform embeddability into Hilbert spaces.
• To generalize the notion of amenability at infinity for transformation groups and relate it to exactness.

Proposed method

• Introduces the concept of an approximate invariant continuous mean (a.i.c.m.) for transformation groups to define amenability.
• Uses positive type functions and Hilbert C*-module techniques to characterize positive type kernels via coefficient functions.
• Applies Fubini’s theorem and approximation arguments to relate nets of functions with compact support to invariant means.
• Employs Kirchberg’s characterization of exactness via unital completely positive maps into bounded operators on l²(G).
• Constructs a uniform embedding into a Hilbert space using sequences of kernels with controlled support and decay.
• Defines amenability at infinity via proper G-equivariant surjections from amenable G-spaces.

Experimental results

Research questions

• RQ1 Under what conditions does amenability of a C*-dynamical system imply nuclearity or exactness of its crossed product?
• RQ2 Can the equivalence between exactness of a discrete group and the existence of an amenable action on a compact space be proven without groupoid theory?
• RQ3 How does amenability at infinity of a group action relate to uniform embeddability into Hilbert spaces?
• RQ4 What is the role of positive type functions and coefficient maps in characterizing amenability in dynamical systems?
• RQ5 To what extent does amenability at infinity of a groupoid imply exactness of its reduced C*-algebra?

Key findings

• A discrete group is exact if and only if it admits an amenable action on a compact space, establishing a deep link between dynamical and algebraic properties.
• The full and reduced crossed products of an amenable C*-dynamical system are equal, and the crossed product is nuclear if and only if the system is amenable.
• Amenability at infinity of a finitely generated group implies uniform embeddability into a Hilbert space, as shown via a constructed embedding using kernel sequences.
• For discrete groups, the conditions of exactness, existence of a proper amenable extension, and the existence of a sequence of positive type kernels with controlled support are equivalent.
• The paper provides a self-contained proof of the equivalence between exactness and the existence of an amenable action on a compact space, independent of groupoid theory.
• The characterization of exactness via unital completely positive approximations is used to construct the uniform embedding into Hilbert space.