Approximate amenability versus amenability for C*-algebras

Determine whether every approximately amenable C*-algebra is amenable.

Background

Approximate amenability is a Banach-algebra property defined by the pointwise norm-approximation of continuous derivations by inner derivations, without requiring the implementing net to be bounded. Amenability is the stronger property in which the corresponding approximating net can be chosen uniformly bounded.

The paper identifies the general implication from approximate amenability to amenability for C*-algebras as an unresolved question and then studies related implications for C*-algebras associated with locally compact quantum groups. Its main results establish amenability of the underlying quantum group under additional covariance, traceability, and approximate-amenability hypotheses, but do not resolve the general C*-algebra question.