Amenable action on a C*-algebra - Imported Knowlpedia Content

Definition

Let $(A,G,\alpha)$ be a $C^*$ -dynamical system. In the Anantharaman-Delaroche convention, $\alpha$ is an amenable action if its normal extension to $A^{**}$ admits a $G$ -equivariant conditional mean

E:L^\infty(G)\,\bar\otimes\,A^{**}\longrightarrow A^{**}

with $E(1\otimes x)=x$ . Here $G$ acts diagonally: by left translation on $L^\infty(G)$ and by $\alpha^{**}$ on $A^{**}$ . The map $E$ is required to be unital, completely positive, and contractive. This is amenability of the action, not merely amenability of the acting group; the coefficient algebra participates essentially in the averaging process.

Crossed-product consequence

Amenability makes the regular representation faithful: the canonical surjection

A\rtimes_\alpha G\longrightarrow A\rtimes_{\alpha,r}G

from the full crossed product to the reduced crossed product is an isomorphism. The converse requires care: equality of these two completions for a particular action is generally called weak containment and need not, without extra hypotheses, imply amenability.

If $G$ itself is amenable, every action of $G$ is amenable. Conversely, the trivial action on $\mathbb C$ is amenable exactly when $G$ is amenable.

Commutative and discrete pictures

For a countable discrete $G$ acting on a locally compact Hausdorff space $X$ , the definition for $A=C_0(X)$ agrees with topological amenability of the action, equivalently amenability of the transformation groupoid $G\ltimes X$ . In the discrete case it can also be expressed by approximately equivariant, positive-type functions with coefficients in the center of $A^{**}$ . These formulations explain why an action of a nonamenable group may still be amenable Anantharaman-Delaroche–Renault, Chapter 3.

Conventions and scope

Several nonequivalent notions are called amenability for actions outside the standard hypotheses: measurewise amenability, topological amenability, strong amenability, and variants for exact or nonunital systems. A theorem should specify its convention. The conditional-mean formulation above is the one used for $C^*$ -dynamical systems here.

References

Claire Anantharaman-Delaroche and Jean Renault, Amenable Groupoids, Monographies de L’Enseignement Mathématique 36, Geneva, 2000. Publisher record. Relevant: Chapter 3 on amenable actions and their operator-algebraic consequences.
Nathanial P. Brown and Narutaka Ozawa, $C^*$ -Algebras and Finite-Dimensional Approximations, Graduate Studies in Mathematics 88, American Mathematical Society, 2008. AMS DOI record. Relevant: Chapter 4 on amenable actions and crossed products.