The property of amenability was originally introduced by John von Neumann for locally compact groups (see Amenable group). This property was later generalized to the case of locally compact quantum groups [BT03]. Sometimes amenabile C*-algebra is defined as a synonym for Nuclear C*-algebra.
Let be a compact quantum group and its discrete dual. The following are equivalent
If one of those equivalent conditions is satisfied, we call amenable and co-amenable.
Suppose that is unimodular. Then is amenable if and only if is injective.
Suppose that is a compact matrix quantum group with fundamental representation . Then is amenable if and only if is in the spectrum of , where . [Ban99a]
If is an amenable discrete quantum group then
A discrete quantum group is amenable if