Amenable partial actions

We introduce and study various notions of amenability continuous (Borel) partial actions of locally compact (Borel) groups $G$ on topological (standard Borel) spaces. We also study amenability of partial representations of a locally compact group in a Banach space and show that a partial action on a measure space is amenable iff the corresponding Koopman partial representation on the corresponding $L^2$-space is amenable. We introduce the notion of induced partial representation from a closed subgroup and explore perseverance of amenability type properties under induction.