An Approximate-Diagonal Characterization of Amenable Actions on C*Algebras

We introduce relative $h$-diagonals for crossed products $A{⋊}_{\alpha }\mathrm{\Gamma }$ by discrete groups, which are positive-form approximate diagonals in the Haagerup tensor product taken relative to $A$. We show that $\alpha$ is amenable if and only if $A{⋊}_{\alpha }\mathrm{\Gamma }$ admits a relative $h$-diagonal, an analogue of Johnson’s theorem for $L^1(G)$.