Amenable fusion algebraic actions of discrete quantum groups on compact quantum spaces

Abstract

In this paper, we introduce actions of fusion algebras on unital C *algebras. Considering the co-representation ring of a compact quantum group as a fusion algebra, we define, as a special class of fusion algebraic actions, a newtype "point by point" action, called a canonical fusion algebraic (for short, CFA) action, of a discrete quantum group on a compact quantum space. Also we show that this CFA action and the commonly used action can be induced by each other. Furthermore, we define amenable CFA actions, and present some basic connections between amenable CFA actions and amenable discrete quantum groups. As an application, thinking of a state on a unital C * -algebra as a "probability measure" on a compact quantum space, we show that amenability for a discrete quantum group is equivalent to both of amenability for a CFA action and the existence of this kind of "probability measure" that is invariant under this action.